Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen RevisionVorhergehende ÜberarbeitungNächste Überarbeitung | Vorhergehende Überarbeitung | ||
| en:grundlagenhandbuch:schmelzefoerderung [2026/05/24 16:09] – [Extended Approximation - Model for the Calculation of Pressure Gradients] neelest | en:grundlagenhandbuch:schmelzefoerderung [2026/09/04 09:53] (aktuell) – [Flow in Simple Geometries] paal | ||
|---|---|---|---|
| Zeile 1: | Zeile 1: | ||
| - | ====== Melt conveying====== | + | ====== Melt Conveying====== |
| The basis of the general mathematical description of flow is the equilibrium of mass, momentum and energy. A flow is described, if at any place and at any time the velocity vector and the values pressure and temperature in the flow domain are known. For the calculation of these values, the conservative equations and the constitutive equations are combined | The basis of the general mathematical description of flow is the equilibrium of mass, momentum and energy. A flow is described, if at any place and at any time the velocity vector and the values pressure and temperature in the flow domain are known. For the calculation of these values, the conservative equations and the constitutive equations are combined | ||
| Zeile 360: | Zeile 360: | ||
| The length $Z_{ei}$ in the figure is calculated using this equation: | The length $Z_{ei}$ in the figure is calculated using this equation: | ||
| - | $$Z_{ei} = \frac{L_{ei}}{\sin(\varphi_s)} \tag{28}$$ | + | $$Z_{ei} = \frac{L_{ei}}{\sin(\varphi_s)}\tag{31}$$ |
| - | Using this model the flow resistance in the intermeshing region is taken into account | + | is calculated. The comparison, when considering the theoretical volumes $V_{fr1}$ |
| + | |||
| + | $$A_r = A_{zw}L_{Ele}\tag{32}$$ | ||
| + | |||
| + | with the theoretical volumes $V_{fr2}$ calculated according to the channel section flume model | ||
| + | |||
| + | $$V_{fr2} = k\bar{h}[b_{max}Z_{fr} + (b_{max}-e_{max})Z_{ei}]\tag{33}$$ | ||
| + | |||
| + | shows deviations of less than approx. +/- 3 vol% for realistic ratios of centre-to-centre distance to screw outer diameter and a lead of up to $i = 3$, depending on the geometric conditions, i.e.: | ||
| + | |||
| + | $$V_{fr1} \approx V_{fr2}\tag{34}$$ | ||
| + | |||
| + | By additionally incorporating | ||
| === Forced Conveying (Modified Model) === | === Forced Conveying (Modified Model) === | ||
| Zeile 372: | Zeile 384: | ||
| 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: | 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: | ||
| - | $$\dot{V}_{tot} = \dot{V}_{intermeshing area} + k \cdot \dot{V}_{channel} - \dot{V}_{gap} \tag{29}$$ | + | $$\dot{V}_{tot} = \dot{V}_{intermeshing area} + k \cdot \dot{V}_{channel} - \dot{V}_{gap} \tag{35}$$ |
| 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. | 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. | ||
| - | $$\dot{V}_{intermeshing area} = \bar{V}_{intermeshing area} \cdot n_0 \tag{30}$$ | + | $$\dot{V}_{intermeshing area} = \bar{V}_{intermeshing area} \cdot n_0 \tag{36}$$ |
| {{ : | {{ : | ||
| Zeile 397: | Zeile 409: | ||
| The geometry can be characterized using the following dimensionless numbers: | The geometry can be characterized using the following dimensionless numbers: | ||
| - | $$k1 = \frac{D_i}{D_z} \tag{1}$$ | + | $$k1 = \frac{D_i}{D_z} \tag{37}$$ |
| - | $$k2 = \frac{D_i}{D_a} \tag{2}$$ | + | $$k2 = \frac{D_i}{D_a} \tag{38}$$ |
| - | $$cl = \frac{a}{D_z/ | + | $$cl = \frac{a}{D_z/ |
| Analogous to the description of the pressure throughput relationship for conveying elements a dimensionless throughput, | Analogous to the description of the pressure throughput relationship for conveying elements a dimensionless throughput, | ||
| - | $$\pi_V = \frac{\dot{V}}{0.5 \cdot A_{frei} \cdot D_a \cdot n_0} \tag{4}$$ | + | $$\pi_V = \frac{\dot{V}}{0.5 \cdot A_{frei} \cdot D_a \cdot n_0} \tag{40}$$ |
| a dimensionless pressure gradient | a dimensionless pressure gradient | ||
| - | $$\pi_P = \frac{\Delta p \cdot R_z}{L \cdot n_0^n \cdot K} \tag{5}$$ | + | $$\pi_P = \frac{\Delta p \cdot R_z}{L \cdot n_0^n \cdot K} \tag{41}$$ |
| and the Power Law exponent $n$ was used for the calculation. | and the Power Law exponent $n$ was used for the calculation. | ||
| Zeile 423: | Zeile 435: | ||
| 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 $\pi_{\dot{\gamma}}$. | 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 $\pi_{\dot{\gamma}}$. | ||
| - | $$\pi_\gamma = \left(\frac{\dot{\gamma}}{n_0}\right)^n \tag{6}$$ | + | $$\pi_\gamma = \left(\frac{\dot{\gamma}}{n_0}\right)^n \tag{42}$$ |
| It depends on | It depends on | ||
| Zeile 432: | Zeile 444: | ||
| For the description we used a linear equation: | For the description we used a linear equation: | ||
| - | $$\pi_\gamma = A_0(k1, k2, cl, n) + A_1(k1, k2, cl, n) \cdot \pi_\gamma \tag{7}$$ | + | $$\pi_\gamma = A_0(k1, k2, cl, n) + A_1(k1, k2, cl, n) \cdot \pi_\gamma \tag{43}$$ |
| {{ : | {{ : | ||
| Zeile 440: | Zeile 452: | ||
| The pressure throughput behavior can be described using the following polynomial equation: | The pressure throughput behavior can be described using the following polynomial equation: | ||
| - | $$\pi_V = A_{B,1}(k1, k2, cl, n) \cdot \pi_P + A_{B,2}(k1, k2, cl, n) \cdot \pi_P^2 \tag{8}$$ | + | $$\pi_V = A_{B,1}(k1, k2, cl, n) \cdot \pi_P + A_{B,2}(k1, k2, cl, n) \cdot \pi_P^2 \tag{44}$$ |
| {{ : | {{ : | ||
| Zeile 446: | Zeile 458: | ||
| **Figure:** Comparison of approximated and numerically determined dimensionless flow rates | **Figure:** Comparison of approximated and numerically determined dimensionless flow rates | ||
| - | A comparison | + | The pressure-flow rate behaviour can be approximated by a second-order polynomial: |
| + | |||
| + | $$\pi_V = A_{B,1}(k1, k2, cl, n) \cdot \pi_P + A_{B,2}(k1, k2, cl, n) \cdot \pi_P^2\tag{45}$$ | ||
| + | |||
| + | A comparison | ||
| {{ : | {{ : | ||
| Zeile 474: | Zeile 490: | ||
| The pressure throughput behavior of the three different flow directions can be described by the following polynomial equation: | The pressure throughput behavior of the three different flow directions can be described by the following polynomial equation: | ||
| - | $$\pi_V = A_{R,0} \left(\frac{t}{D_a}, | + | $$\pi_V = A_{R,0} \left(\frac{t}{D_a}, |
| where the parameter $A_{R,0} = 0$ for neutral elements. In the figures the profiles of the dimensionless pressure gradients are shown. | where the parameter $A_{R,0} = 0$ for neutral elements. In the figures the profiles of the dimensionless pressure gradients are shown. | ||
| Zeile 494: | Zeile 510: | ||
| 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 | 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 | ||
| - | $$\dot{V}_{tooth\ mixing\ element} = \dot{V}_{disc} + i \cdot \dot{V}_{rectangle} \tag{2}$$ | + | $$\dot{V}_{tooth\ mixing\ element} = \dot{V}_{disc} + i \cdot \dot{V}_{rectangle} \tag{47}$$ |
| The following equation describes the pressure throughput relationship for turbine mixing elements results from the above equation: | The following equation describes the pressure throughput relationship for turbine mixing elements results from the above equation: | ||
| - | $$\pi_V = Y_0 + Y_1 \cdot \pi_P + Y_2 \cdot \pi_P^2 + Y_3 \cdot \pi_P^3 \tag{3}$$ | + | $$\pi_V = Y_0 + Y_1 \cdot \pi_P + Y_2 \cdot \pi_P^2 + Y_3 \cdot \pi_P^3 \tag{48}$$ |
| with: | with: | ||
| - | $$\pi_V = \frac{\dot{V}}{\frac{1}{2} \cdot n_0 \cdot A_{free} \cdot D_a} \tag{4}$$ | + | $$\pi_V = \frac{\dot{V}}{\frac{1}{2} \cdot n_0 \cdot A_{free} \cdot D_a} \tag{49}$$ |
| - | $$\pi_P = \frac{\Delta p}{L} \cdot \frac{\overline{s_R}}{K \cdot n_0^n} \tag{5}$$ | + | $$\pi_P = \frac{\Delta p}{L} \cdot \frac{\overline{s_R}}{K \cdot n_0^n} \tag{50}$$ |
| and | and | ||
| - | $$Y_0 = A_{R,0} \cdot i \cdot \pi_{geo, | + | $$Y_0 = A_{R,0} \cdot i \cdot \pi_{geo,V}\tag{51}$$ |
| - | $$Y_1 = A_{B,1} + A_{R,1} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo,p} \tag{6}$$ | + | $$Y_1 = A_{B,1} + A_{R,1} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo,p} \tag{52}$$ |
| - | $$Y_2 = A_{B,2} + A_{R,2} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo, | + | $$Y_2 = A_{B,2} + A_{R,2} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo, |
| - | $$Y_3 = A_{R,3} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo, | + | $$Y_3 = A_{R,3} \cdot i \cdot \pi_{geo,V} \cdot \pi_{geo, |
| The coefficients $p_{geo,V}$ and $p_{geo,p}$ are combined to couple the flow rates and the pressure gradients. They are defined as follows: | The coefficients $p_{geo,V}$ and $p_{geo,p}$ are combined to couple the flow rates and the pressure gradients. They are defined as follows: | ||
| - | $$\pi_{geo, | + | $$\pi_{geo, |
| and: | and: | ||
| - | $$\pi_{geo, | + | $$\pi_{geo, |
| Depending on the conveying direction different pressure throughput behaviors are resulting (see figures). | Depending on the conveying direction different pressure throughput behaviors are resulting (see figures). | ||
| Zeile 559: | Zeile 575: | ||
| The continuity law is the mathematical formulation of a mass balance in a fixed control room. It states that the mass saved in a volume corresponds to the difference of inflowing and outflowing mass flows. | The continuity law is the mathematical formulation of a mass balance in a fixed control room. It states that the mass saved in a volume corresponds to the difference of inflowing and outflowing mass flows. | ||
| - | $$\frac{\partial \rho}{\partial t} + \nabla(\rho \cdot \vec{v}) = 0$$ | + | $$\frac{\partial \rho}{\partial t} + \nabla(\rho \cdot \vec{v}) = 0\tag{57}$$ |
| Here $\rho$ is the density of the polymer melt, t the time and $\vec{v}$ the velocity vector. | Here $\rho$ is the density of the polymer melt, t the time and $\vec{v}$ the velocity vector. | ||
| Zeile 567: | Zeile 583: | ||
| The equation of motion is the difference between the momentum going in a volume element and that coming out of it plus the forces effecting the system (e.g. because of gravity). | The equation of motion is the difference between the momentum going in a volume element and that coming out of it plus the forces effecting the system (e.g. because of gravity). | ||
| - | $$\frac{\partial}{\partial t}(\rho \cdot \vec{v}) = -\nabla(\rho \cdot \vec{v} \cdot \vec{v}) - \nabla p - \nabla \tau + \rho \cdot \vec{a}$$ | + | $$\frac{\partial}{\partial t}(\rho \cdot \vec{v}) = -\nabla(\rho \cdot \vec{v} \cdot \vec{v}) - \nabla p - \nabla \tau + \rho \cdot \vec{a}\tag{58}$$ |
| **Law of conservation of energy** | **Law of conservation of energy** | ||
| Zeile 573: | Zeile 589: | ||
| A formulation of the balancing of the heat quantity saved in the control room compared to inflowing and outflowing heat flows is the principle of conservation of energy. | A formulation of the balancing of the heat quantity saved in the control room compared to inflowing and outflowing heat flows is the principle of conservation of energy. | ||
| - | $$\rho \cdot c \cdot \left(\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T\right) = -\nabla \vec{q} + \tau:\nabla \vec{v} + \phi$$ | + | $$\rho \cdot c \cdot \left(\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T\right) = -\nabla \vec{q} + \tau:\nabla \vec{v} + \phi\tag{59}$$ |
| === Materials law === | === Materials law === | ||
| Zeile 583: | Zeile 599: | ||
| **Power law** | **Power law** | ||
| - | $$\eta(\dot{\gamma}) = K \cdot a_T \cdot \dot{\gamma}^{n-1}$$ | + | $$\eta(\dot{\gamma}) = K \cdot a_T \cdot \dot{\gamma}^{n-1}\tag{60}$$ |
| **Carreau-approach** | **Carreau-approach** | ||
| - | $$\eta(\dot{\gamma}) = A \cdot a_T \cdot (1 + (B \cdot a_T \cdot \dot{\gamma})^c)^{-1}$$ | + | $$\eta(\dot{\gamma}) = A \cdot a_T \cdot (1 + (B \cdot a_T \cdot \dot{\gamma})^c)^{-1}\tag{61}$$ |
| **Approach according to Carreau-Michaeli** | **Approach according to Carreau-Michaeli** | ||
| - | $$\eta(\dot{\gamma}) = A \cdot a_T \cdot (1 + B \cdot a_T \cdot \dot{\gamma})^{-c}$$ | + | $$\eta(\dot{\gamma}) = A \cdot a_T \cdot (1 + B \cdot a_T \cdot \dot{\gamma})^{-c}\tag{62}$$ |
| **Yasuda-approach** | **Yasuda-approach** | ||
| - | $$\eta(\dot{\gamma}) = A \cdot a_T \cdot (1 + (B \cdot a_T \cdot \dot{\gamma})^c)^{-\frac{c}{a}}$$ | + | $$\eta(\dot{\gamma}) = A \cdot a_T \cdot (1 + (B \cdot a_T \cdot \dot{\gamma})^c)^{-\frac{c}{a}}\tag{63}$$ |
| Approaches to describe the temperature dependency of the viscosity: | Approaches to describe the temperature dependency of the viscosity: | ||
| **simplified Arrhenius-approach** | **simplified Arrhenius-approach** | ||
| - | $$\ln(a_T) = -\beta \cdot (T - T_{ref})$$ | + | $$\ln(a_T) = -\beta \cdot (T - T_{ref})\tag{64}$$ |
| **Arrhenius-approach** | **Arrhenius-approach** | ||
| - | $$\ln(a_T) = -\frac{E}{R} \left(\frac{1}{T} - \frac{1}{T_{ref}}\right)$$ | + | $$\ln(a_T) = -\frac{E}{R} \left(\frac{1}{T} - \frac{1}{T_{ref}}\right)\tag{65}$$ |
| **WLF-approach** | **WLF-approach** | ||
| - | $$\log(a_T) = \frac{8.86 \cdot (T_{ref} - T_S)}{101.6 + T_{ref} - T_S} - \frac{8.86 \cdot (T - T_S)}{101.6 + T - T_S}$$ | + | $$\log(a_T) = \frac{8.86 \cdot (T_{ref} - T_S)}{101.6 + T_{ref} - T_S} - \frac{8.86 \cdot (T - T_S)}{101.6 + T - T_S}\tag{66}$$ |
| Mathematically the power law is easy to manage. Yet, using this approach has the advantage that this model is able to describe the flow behavior only in certain sections of the flow curve sufficiently well. A description of the whole flow curve in general with the help of this law is not possible. The first three approaches offer better descriptions. Yet, these have the advantage that they are mathematically difficult to manage. In the following the power law is used because of these disadvantages. Its parameter is locally adapted to the Carreau-Michaeli-law. The basis is that the average shear rate is known in the viewed flow section. | Mathematically the power law is easy to manage. Yet, using this approach has the advantage that this model is able to describe the flow behavior only in certain sections of the flow curve sufficiently well. A description of the whole flow curve in general with the help of this law is not possible. The first three approaches offer better descriptions. Yet, these have the advantage that they are mathematically difficult to manage. In the following the power law is used because of these disadvantages. Its parameter is locally adapted to the Carreau-Michaeli-law. The basis is that the average shear rate is known in the viewed flow section. | ||
| Zeile 617: | Zeile 633: | ||
| The analysis of existing models to specify the pressure-throughput behavior of rectangular and twin screw flow channels gives evidence that these models are only suitable to a limited extent to describe mixing elements. This shall be explained on the basis of the results of a three-dimensional finite element simulation of a threaded-mixing element. | The analysis of existing models to specify the pressure-throughput behavior of rectangular and twin screw flow channels gives evidence that these models are only suitable to a limited extent to describe mixing elements. This shall be explained on the basis of the results of a three-dimensional finite element simulation of a threaded-mixing element. | ||
| - | {{ : | + | {{ : |
| **Figure:** Velocity profiles in a threaded mixing element. | **Figure:** Velocity profiles in a threaded mixing element. | ||
| Zeile 633: | Zeile 649: | ||
| To specify the throughput behavior of the screw elements, first of all, the basic geometry (twin screw channel, rectangular channel and disk elements) is analyzed and modelled. Subsequently, | To specify the throughput behavior of the screw elements, first of all, the basic geometry (twin screw channel, rectangular channel and disk elements) is analyzed and modelled. Subsequently, | ||
| - | ^ Geometry ^ Parameter ^ Formula ^ | + | {{ : |
| - | | Rectangular channels | Ratio of channel width to channel depth | $\frac{b}{h}$ | | + | |
| - | | | Ratio of pitch to screw diameter | $\frac{t}{D_s} = \pi \cdot \tan(\varphi_s)$ | | + | |
| - | | Twin screw channels | Ratio of axial distance to screw diameter | $CL = \frac{2 \cdot a}{D_s}$ | | + | |
| - | | | Ratio of pitch to screw diameter | $\frac{t}{D_s} = \pi \cdot \tan(\varphi_s)$ | | + | |
| - | | | Number of flights | $i$ | | + | |
| - | | Disk elements | Ratio of axial distance to cylinder diameter | $CL = \frac{2 \cdot a}{D_z}$ | | + | |
| - | | | Ratio of the diameter of the larger disk to the cylinder diameter | $k_1 = \frac{D_a}{D_z}$ | | + | |
| - | | | Ratio of the diameter of the smaller disk to the diameter of the larger disk | $k_2 = \frac{D_i}{D_a}$ | | + | |
| **Table:** Dimensionless parameters for the characterization of the geometry. | **Table:** Dimensionless parameters for the characterization of the geometry. | ||
| Zeile 658: | Zeile 666: | ||
| * In case of a Newtonian fluid (n=1), the influence of the helix angle on the pressure-throughput behavior decreases with an increasing ratio of b/h. | * In case of a Newtonian fluid (n=1), the influence of the helix angle on the pressure-throughput behavior decreases with an increasing ratio of b/h. | ||
| - | $$\pi_V = \left(\frac{b}{h} \rightarrow \infty, n = 1\right) = f(\varphi_s) \tag{1}$$ | + | $$\pi_V = \left(\frac{b}{h} \rightarrow \infty, n = 1\right) = f(\varphi_s) \tag{67}$$ |
| - | Due to the closely intermeshing geometry this requirement cannot be transferred to twin screw channels, because channel width and channel height can-not be varied independently. They are rather dependent on the pitch, the axial distance, the screw diameter, and the number of flights (see figure). Equation | + | Due to the closely intermeshing geometry this requirement cannot be transferred to twin screw channels, because channel width and channel height can-not be varied independently. They are rather dependent on the pitch, the axial distance, the screw diameter, and the number of flights (see figure). Equation |
| - | {{ : | + | {{ :en: |
| **Figure:** Influence of the helix angle on the geometry of twin screw channels. | **Figure:** Influence of the helix angle on the geometry of twin screw channels. | ||
| Zeile 669: | Zeile 677: | ||
| * the dimensionless volume throughput | * the dimensionless volume throughput | ||
| - | $$\pi_\dot{V} = \frac{\dot{V}}{\frac{1}{2} \cdot h \cdot b \cdot v_0 \cdot \cos(\varphi_s)} | + | $\pi_\dot{V} = \frac{\dot{V}}{\frac{1}{2} \cdot h \cdot b \cdot v_0 \cdot \cos(\varphi_s)}$ |
| * the dimensionless pressure gradient | * the dimensionless pressure gradient | ||
| - | $$\pi_p = \frac{\Delta p}{\Delta z} \cdot \frac{h^{1+n}}{6 \cdot K \cdot v_0^n \cdot \cos^n(\varphi_s)} | + | $\pi_p = \frac{\Delta p}{\Delta z} \cdot \frac{h^{1+n}}{6 \cdot K \cdot v_0^n \cdot \cos^n(\varphi_s)}$ |
| are incompatible with this requirement, | are incompatible with this requirement, | ||
| - | $$\pi_\dot{V} = \frac{\dot{V}}{h \cdot b \cdot v_0} \tag{4}$$ | + | $$\pi_\dot{V} = \frac{\dot{V}}{h \cdot b \cdot v_0} \tag{68}$$ |
| and | and | ||
| - | $$\pi_p = \frac{\Delta p}{\Delta z} \cdot \frac{h^{1+n}}{K \cdot v_0^n} \tag{5}$$ | + | $$\pi_p = \frac{\Delta p}{\Delta z} \cdot \frac{h^{1+n}}{K \cdot v_0^n} \tag{69}$$ |
| To describe the process behavior of disk elements the following parameters are applied: | To describe the process behavior of disk elements the following parameters are applied: | ||
| - | $$\pi_\dot{V} = \frac{\dot{V}}{A_{fr} \cdot D_a \cdot n_0} \tag{6}$$ | + | $$\pi_\dot{V} = \frac{\dot{V}}{A_{fr} \cdot D_a \cdot n_0} \tag{70}$$ |
| - | $$\pi_p = \frac{\Delta p}{L} \cdot \frac{D_a}{K \cdot n_0^n} \tag{7}$$ | + | $$\pi_p = \frac{\Delta p}{L} \cdot \frac{D_a}{K \cdot n_0^n} \tag{71}$$ |
| The subsequent modelling of the pressure-throughput behavior of rectangular and twin screw channels shall form the basis of the modelling of the most diverse screw elements. The models of both channel types are based on the same procedure. The characteristic curve family can be separated into 3 sections (see figure). | The subsequent modelling of the pressure-throughput behavior of rectangular and twin screw channels shall form the basis of the modelling of the most diverse screw elements. The models of both channel types are based on the same procedure. The characteristic curve family can be separated into 3 sections (see figure). | ||
| Zeile 694: | Zeile 702: | ||
| * **Section III:** In this area a negative pressure gradient and a positive volume throughput can be observed. | * **Section III:** In this area a negative pressure gradient and a positive volume throughput can be observed. | ||
| * | * | ||
| - | {{ : | + | {{ : |
| **Figure:** Boundary conditions for the modelling of the pressure-throughput behavior. | **Figure:** Boundary conditions for the modelling of the pressure-throughput behavior. | ||
| Zeile 737: | Zeile 745: | ||
| === Rectangular Channels === | === Rectangular Channels === | ||
| - | To describe the pressure-throughput behavior of rectangular channels the characteristic curve family is separated into three sections (see figure). For this, all those models were applied as specified in Eqs. (1) to (3). The parameters are dependent on the geometry and on the exponent of the power law. A detailed specification of the model equation can be found in the appendix C1.1 in [[en: | + | To describe the pressure-throughput behavior of rectangular channels the characteristic curve family is separated into three sections (see figure). For this, all those models were applied as specified in Eqs. 72 to 74. The parameters are dependent on the geometry and on the exponent of the power law. A detailed specification of the model equation can be found in the appendix C1.1 in [[en: |
| **Section I:** | **Section I:** | ||
| - | $$\pi_\dot{V} = \left(1 - \frac{\pi_p}{\pi_p|_{\pi_\dot{V}=0}}\right) \cdot \left(C_{RE, | + | $$\pi_\dot{V} = \left(1 - \frac{\pi_p}{\pi_p|_{\pi_\dot{V}=0}}\right) \cdot \left(C_{RE, |
| **Section II:** | **Section II:** | ||
| - | $$\pi_\dot{V} = \left(1 - \frac{\pi_p}{\pi_p|_{\pi_\dot{V}=0}}\right) \cdot \left(\pi_\dot{V}|_{\pi_p=0} + C_{RE,II,1} \cdot \pi_p + C_{RE,II,2} \cdot \pi_p^2\right) \tag{2}$$ | + | $$\pi_\dot{V} = \left(1 - \frac{\pi_p}{\pi_p|_{\pi_\dot{V}=0}}\right) \cdot \left(\pi_\dot{V}|_{\pi_p=0} + C_{RE,II,1} \cdot \pi_p + C_{RE,II,2} \cdot \pi_p^2\right) \tag{73}$$ |
| **Section III:** | **Section III:** | ||
| - | $$\pi_\dot{V} = \pi_\dot{V}|_{\pi_p=0} + C_{RE, | + | $$\pi_\dot{V} = \pi_\dot{V}|_{\pi_p=0} + C_{RE, |
| with | with | ||
| - | $$\pi_p|_{\pi_\dot{V}=0} = C_{Re,S,1} \cdot \cos(\varphi_s) + C_{Re,S,2} \cdot \cos^3(\varphi_s) + C_{Re,S,3} \cdot \cos^5(\varphi_s) \tag{4}$$ | + | $$\pi_p|_{\pi_\dot{V}=0} = C_{Re,S,1} \cdot \cos(\varphi_s) + C_{Re,S,2} \cdot \cos^3(\varphi_s) + C_{Re,S,3} \cdot \cos^5(\varphi_s) \tag{75}$$ |
| - | $$\pi_\dot{V}|_{\pi_p=0} = C_{Re,S,1} \cdot \cos(\varphi_s) + C_{Re,S,2} \cdot \cos^3(\varphi_s) + C_{Re,S,3} \cdot \cos^5(\varphi_s) \tag{5}$$ | + | |
| + | $$\pi_\dot{V}|_{\pi_p=0} = C_{Re,S,1} \cdot \cos(\varphi_s) + C_{Re,S,2} \cdot \cos^3(\varphi_s) + C_{Re,S,3} \cdot \cos^5(\varphi_s) \tag{76}$$ | ||
| In the figure the calculated dimensionless pressure-throughput characteristics are contrasted with the Polyflow results of varying exponents of the power law. It can be seen, that the model offers a sufficiently good description for a wide area. | In the figure the calculated dimensionless pressure-throughput characteristics are contrasted with the Polyflow results of varying exponents of the power law. It can be seen, that the model offers a sufficiently good description for a wide area. | ||
| - | {{ : | + | {{ : |
| **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the power law. | **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the power law. | ||
| Zeile 760: | Zeile 769: | ||
| The characteristic curves of conveying and reconveying channels are related to each other in a centrosymmetrical way | The characteristic curves of conveying and reconveying channels are related to each other in a centrosymmetrical way | ||
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| i.e. if the characteristic curves of the conveying channels are known, those of the reconveying channels can be deduced from this basis. | i.e. if the characteristic curves of the conveying channels are known, those of the reconveying channels can be deduced from this basis. | ||
| - | The characteristic curve to specify the transportation | + | The characteristic curve to specify the neutral elements correspond with the specification in zone III, as the drag conveying capacity equals zero and as, therefore, the element is always overrun. The figure displays the pressure-throughput characteristics for the three types of transport and varying exponents of the power law. It can be recognized that the above outlined transformation of the characteristics allow a sufficiently good description of the Poly-flow results. |
| - | {{ : | + | {{ : |
| **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics of conveying, reconveying and neutral rectangular channels. | **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics of conveying, reconveying and neutral rectangular channels. | ||
| Zeile 772: | Zeile 781: | ||
| === Twin Screw Channels === | === Twin Screw Channels === | ||
| - | In an analogous way to the modelling procedure of rectangular channels in the modelling of twin screw channels also three areas can be separated (see figure). For the description of the particular zones all those models are used as specified in Eqs. (7) to (9). The parameters are dependent on the geometry and on the exponent of the power law. A detailed specification of the model equation can be found in the appendix C2.1 in [[en: | + | In an analogous way to the modelling procedure of rectangular channels in the modelling of twin screw channels also three areas can be separated (see figure). For the description of the particular zones all those models are used as specified in Eqs. 78 to 80. The parameters are dependent on the geometry and on the exponent of the power law. A detailed specification of the model equation can be found in the appendix C2.1 in [[en: |
| **Section I:** | **Section I:** | ||
| - | $$\pi_\dot{V} = \left(1 - \frac{\pi_p}{\pi_p|_{\pi_\dot{V}=0}}\right) \cdot \left(C_{DSE, | + | $$\pi_\dot{V} = \left(1 - \frac{\pi_p}{\pi_p|_{\pi_\dot{V}=0}}\right) \cdot \left(C_{DSE, |
| **Section II:** | **Section II:** | ||
| - | $$\pi_\dot{V} = \left(1 - \frac{\pi_p}{\pi_p|_{\pi_\dot{V}=0}}\right) \cdot \left(C_{DSE, | + | $$\pi_\dot{V} = \left(1 - \frac{\pi_p}{\pi_p|_{\pi_\dot{V}=0}}\right) \cdot \left(C_{DSE, |
| **Section III:** | **Section III:** | ||
| - | $$\pi_\dot{V} = \pi_\dot{V}|_{\pi_p=0} + C_{DSE, | + | $$\pi_\dot{V} = \pi_\dot{V}|_{\pi_p=0} + C_{DSE, |
| with | with | ||
| - | $$\pi_\dot{V}|_{\pi_p=0} = \frac{1}{C_{DSE, | + | $$\pi_\dot{V}|_{\pi_p=0} = \frac{1}{C_{DSE, |
| - | $$\pi_p|_{\pi_\dot{V}=0} = \left(C_{DSE, | + | $$\pi_p|_{\pi_\dot{V}=0} = \left(C_{DSE, |
| In the figure the dimensionless pressure-throughput characteristics are illustrated for varying exponents of the power law. It can be recognized that the model yields a sufficiently good description for a wide area. | In the figure the dimensionless pressure-throughput characteristics are illustrated for varying exponents of the power law. It can be recognized that the model yields a sufficiently good description for a wide area. | ||
| - | {{ : | + | {{ : |
| **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the Power Law. | **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the Power Law. | ||
| Zeile 797: | Zeile 806: | ||
| The description model of the throughput behavior of twin screw channels can be applied to conveying and reconveying channels. A comparison of the model predictions and the Polyflow results for both conveying and reconveying channels is illustrated in the figure. For all three transportation types a sufficiently good description can be identified. | The description model of the throughput behavior of twin screw channels can be applied to conveying and reconveying channels. A comparison of the model predictions and the Polyflow results for both conveying and reconveying channels is illustrated in the figure. For all three transportation types a sufficiently good description can be identified. | ||
| - | {{ : | + | {{ : |
| **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for conveying, reconveying and transportation neutral twin screw channels. | **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for conveying, reconveying and transportation neutral twin screw channels. | ||
| Zeile 805: | Zeile 814: | ||
| Contrary to the previous geometries for the description of the throughput behavior of disk elements only one zone is examined. The motive for doing this is the fact that disk elements are neutral elements and, therefore, are always pressure consumers. Accordingly, | Contrary to the previous geometries for the description of the throughput behavior of disk elements only one zone is examined. The motive for doing this is the fact that disk elements are neutral elements and, therefore, are always pressure consumers. Accordingly, | ||
| - | $$\pi_\dot{V} = C_{SE,1} \cdot \pi_p + C_{SE,2} \cdot \pi_p^2 + C_{SE,3} \cdot \pi_p^3 \tag{12}$$ | + | $$\pi_\dot{V} = C_{SE,1} \cdot \pi_p + C_{SE,2} \cdot \pi_p^2 + C_{SE,3} \cdot \pi_p^3 \tag{83}$$ |
| - | {{ : | + | {{ : |
| **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the Power Law. | **Figure:** Comparison of the calculated and the predicted pressure-throughput characteristics for varying exponents of the Power Law. | ||
| - | In the figure the dimensionless pressure-throughput characteristics are delineated for varying exponents of the power law. It can be recognized that the model gives a sufficiently good description for a wide area. The parameters in Equation | + | In the figure the dimensionless pressure-throughput characteristics are delineated for varying exponents of the power law. It can be recognized that the model gives a sufficiently good description for a wide area. The parameters in Equation |
| ==== Pressure-Throughput Model for Threaded Elements ==== | ==== Pressure-Throughput Model for Threaded Elements ==== | ||
| Zeile 825: | Zeile 834: | ||
| To be able to represent this effect, first of all, the flow in the intermeshing zone and the flow in the screw channels are modelled separately before they are ultimately linked with each other. | To be able to represent this effect, first of all, the flow in the intermeshing zone and the flow in the screw channels are modelled separately before they are ultimately linked with each other. | ||
| - | $$\dot{V}_{GE} = \dot{V}_{intermeshing\ area,GE} + \dot{V}_{channel, | + | $$\dot{V}_{GE} = \dot{V}_{intermeshing\ area,GE} + \dot{V}_{channel, |
| === Channel Area === | === Channel Area === | ||
| Zeile 831: | Zeile 840: | ||
| For the description of the flow in the channel area a channel model is used, in which the screw and the barrel are uncoiled along the barrel wall. Additionally, | For the description of the flow in the channel area a channel model is used, in which the screw and the barrel are uncoiled along the barrel wall. Additionally, | ||
| - | {{ : | + | {{ : |
| **Figure:** Conveyor channel model for double-flighted conveying threaded elements | **Figure:** Conveyor channel model for double-flighted conveying threaded elements | ||
| Zeile 837: | Zeile 846: | ||
| The velocity $v_0$ of the uncoiled barrel (see figure) equals the peripheral velocity at the outside diameter of the screw, which rotates with a speed of $n_0$. | The velocity $v_0$ of the uncoiled barrel (see figure) equals the peripheral velocity at the outside diameter of the screw, which rotates with a speed of $n_0$. | ||
| - | $$v_0 = \pi \cdot D_s \cdot n_0 \tag{2}$$ | + | $$v_0 = \pi \cdot D_s \cdot n_0 \tag{85}$$ |
| The modeling of the throughput behavior is now based on a volume flow balance at the control rooms ABC in Figure. | The modeling of the throughput behavior is now based on a volume flow balance at the control rooms ABC in Figure. | ||
| - | $$\dot{V}_{channel, | + | $$\dot{V}_{channel, |
| === Flow in the Channel === | === Flow in the Channel === | ||
| Zeile 847: | Zeile 856: | ||
| For the description of the flow in the conveyor channel $\dot{V}_z$ the approach specified in chapter is applied. | For the description of the flow in the conveyor channel $\dot{V}_z$ the approach specified in chapter is applied. | ||
| - | $$\dot{V}_{z, | + | $$\dot{V}_{z, |
| It has to be taken into consideration that the calculation of $\pi_{V,z}$ is performed in sections: | It has to be taken into consideration that the calculation of $\pi_{V,z}$ is performed in sections: | ||
| Zeile 855: | Zeile 864: | ||
| \left(1 - \frac{\pi_{p, | \left(1 - \frac{\pi_{p, | ||
| \pi_V|_{\pi_{p, | \pi_V|_{\pi_{p, | ||
| - | \end{cases} \tag{5}$$ | + | \end{cases} \tag{88}$$ |
| with | with | ||
| - | $$\pi_{p,z} = \frac{\Delta p}{Z_{Tv}} \cdot \frac{\overline{h}^{1+n}}{K \cdot v_0^n} \tag{6}$$ | + | $$\pi_{p,z} = \frac{\Delta p}{Z_{Tv}} \cdot \frac{\overline{h}^{1+n}}{K \cdot v_0^n} \tag{89}$$ |
| - | Accordingly, | + | Accordingly, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| === Flow in the Radial Clearance === | === Flow in the Radial Clearance === | ||
| Zeile 869: | Zeile 878: | ||
| For the description of the leakage flow $\dot{V}_x$ the approach for rectangular channels is used as specified in Chapter Flow in the Channel. | For the description of the leakage flow $\dot{V}_x$ the approach for rectangular channels is used as specified in Chapter Flow in the Channel. | ||
| - | $$\dot{V}_{x, | + | $$\dot{V}_{x, |
| It has to be taken into consideration that the calculation of $\pi_{V,x}$ is performed in sections: | It has to be taken into consideration that the calculation of $\pi_{V,x}$ is performed in sections: | ||
| Zeile 879: | Zeile 888: | ||
| \\ | \\ | ||
| \left(\pi_V|_{\pi_{p, | \left(\pi_V|_{\pi_{p, | ||
| - | \end{cases}$$ | + | \end{cases}\tag{92}$$ |
| with | with | ||
| - | $$\pi_{p,x} = \frac{\Delta p}{e_{max}} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} \tag{10}$$ | + | $$\pi_{p,x} = \frac{\Delta p}{e_{max}} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} \tag{93}$$ |
| - | By virtue of clarity a general notation is also chosen | + | By virtue of clarity a general notation is also chosen: |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| === Linkage of the Volume Flow Rates === | === Linkage of the Volume Flow Rates === | ||
| - | Are the equations | + | If we solve the equations |
| $$\dot{V}_{channel, | $$\dot{V}_{channel, | ||
| $$+ A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p, | $$+ A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p, | ||
| - | $$+ A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p, | + | $$+ A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p, |
| Is this equation standardized to $k \cdot (\overline{h} \cdot b_{max} \cdot v_0)$, then follows: | Is this equation standardized to $k \cdot (\overline{h} \cdot b_{max} \cdot v_0)$, then follows: | ||
| $$\pi_{V, | $$\pi_{V, | ||
| - | $$+ A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p, | + | $$+ A_{DSE,3} \cdot \pi_{p,z}^3 + A_{DSE,4} \cdot \pi_{p, |
| - | $$+ A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p, | + | $$+ A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p, |
| As a next measure it proves necessary to build a linkage of the pressure gradients in channel direction and above the screw flight. For this purpose the approach proposed by Ansahl | As a next measure it proves necessary to build a linkage of the pressure gradients in channel direction and above the screw flight. For this purpose the approach proposed by Ansahl | ||
| - | $$\Delta p_x = \frac{\Delta p \cdot b_{max} + e_{max}}{\tan(\varphi_s)} \tag{14}$$ | + | $$\Delta p_x = \frac{\Delta p \cdot b_{max} + e_{max}}{\tan(\varphi_s)} \tag{97}$$ |
| - | Is Equation | + | $$\pi_{p,x} = \frac{\Delta p \cdot b_{max} + e_{max} \cdot s_R^{1+n}}{\Delta z \cdot e_{max} \cdot \tan(\varphi_s) \cdot K \cdot v_0^n} = \pi_{geo, |
| - | $$\pi_{p,x} = \frac{\Delta p \cdot b_{max} + e_{max} \cdot s_R^{1+n}}{\Delta z \cdot e_{max} \cdot \tan(\varphi_s) \cdot K \cdot v_0^n} = \pi_{geo, | ||
| - | with | + | $$\pi_{geo, |
| - | + | ||
| - | $$\pi_{geo, | + | |
| - | From Equation (15) applied to (13) derives | + | This gives us the equation |
| $$\pi_{V, | $$\pi_{V, | ||
| $$+ \left(A_{DSE, | $$+ \left(A_{DSE, | ||
| $$+ \left(A_{DSE, | $$+ \left(A_{DSE, | ||
| $$+ \left(A_{DSE, | $$+ \left(A_{DSE, | ||
| - | $$+ \left(A_{DSE, | + | $$+ \left(A_{DSE, |
| with | with | ||
| - | $$\pi_1 = \frac{(s_R \cdot b_{steg} \cdot v_0)}{k \cdot (\overline{h} \cdot b_{max} \cdot v_0)} \tag{18}$$ | + | $$\pi_1 = \frac{(s_R \cdot b_{steg} \cdot v_0)}{k \cdot (\overline{h} \cdot b_{max} \cdot v_0)} \tag{101}$$ |
| === Intermeshing Zone === | === Intermeshing Zone === | ||
| Zeile 930: | Zeile 936: | ||
| In the literature diverse approaches to describe the flow in the intermeshing zone have been developed. In the following the approach of Booy [[en: | In the literature diverse approaches to describe the flow in the intermeshing zone have been developed. In the following the approach of Booy [[en: | ||
| - | $$\dot{V}_{intermeshing\ area,GE} = A_{intermeshing\ area} \cdot t \cdot n_0 \tag{19}$$ | + | $$\dot{V}_{intermeshing\ area,GE} = A_{intermeshing\ area} \cdot t \cdot n_0 \tag{102}$$ |
| - | On the basis of measurements and flow simulations Bakalis | + | On the basis of measurements and flow simulations Bakalis |
| **Pressure - Throughput Model** | **Pressure - Throughput Model** | ||
| - | As a final step the volume throughput in the conveyor channel model (Eq. 20) is to be superposed with the volume throughput in the intermeshing zone (Eq. 19): | + | As a final step the volume throughput in the conveyor channel model is to be superposed with the volume throughput in the intermeshing zone: |
| - | $$\dot{V}_{GE} = \dot{V}_{intermeshing, | + | $$\dot{V}_{GE} = \dot{V}_{intermeshing, |
| In the figure the theoretical pressure-throughput characteristics are illustrated for conveying and reconveying threaded elements. | In the figure the theoretical pressure-throughput characteristics are illustrated for conveying and reconveying threaded elements. | ||
| - | {{ : | + | {{ : |
| **Figure:** Pressure-throughput behavior of conveying and reconveying threaded elements. | **Figure:** Pressure-throughput behavior of conveying and reconveying threaded elements. | ||
| Zeile 948: | Zeile 954: | ||
| In the figure the model predictions are compared with the experimental results. The experiments were performed with a model extruder using silicon oil as melt. | In the figure the model predictions are compared with the experimental results. The experiments were performed with a model extruder using silicon oil as melt. | ||
| - | {{ : | + | {{ : |
| **Figure:** Comparison of calculated and measured pressure gradients in threaded elements. | **Figure:** Comparison of calculated and measured pressure gradients in threaded elements. | ||
| Zeile 957: | Zeile 963: | ||
| In the figure the variation of the axial velocities in a conveying, closely intermeshing thread mixing element are illustrated on two different levels. Out of the variation of the velocities it can be noticed that the flow in the grooves and in the intermeshing zone of both screws significantly differs from the flow in the channel area. In the intermeshing zone axial velocities can be detected, which are considerably higher than those in the channel area. In the grooves, however, another effect can be observed. As the grooves are of a back-flow design (in this case), in this area a back-flow is to be recognized accordingly. In the modelling, at first, a distinction is made between the intermeshing zone and the channel zone. | In the figure the variation of the axial velocities in a conveying, closely intermeshing thread mixing element are illustrated on two different levels. Out of the variation of the velocities it can be noticed that the flow in the grooves and in the intermeshing zone of both screws significantly differs from the flow in the channel area. In the intermeshing zone axial velocities can be detected, which are considerably higher than those in the channel area. In the grooves, however, another effect can be observed. As the grooves are of a back-flow design (in this case), in this area a back-flow is to be recognized accordingly. In the modelling, at first, a distinction is made between the intermeshing zone and the channel zone. | ||
| - | $$\dot{V}_{dGME} = \dot{V}_{channel, | + | $$\dot{V}_{dGME} = \dot{V}_{channel, |
| The influence of the grooves on the behavior of the elements is then taken account of in the description of the separate areas. | The influence of the grooves on the behavior of the elements is then taken account of in the description of the separate areas. | ||
| - | {{ : | + | {{ : |
| **Figure:** Calculated velocity fields in a tightly intermeshing threaded mixing element. | **Figure:** Calculated velocity fields in a tightly intermeshing threaded mixing element. | ||
| Zeile 967: | Zeile 973: | ||
| === Channel Model for the Channel Area === | === Channel Model for the Channel Area === | ||
| - | The channel model for tightly intermeshing threaded mixing elements can be compared to that of threaded elements. The flow in the grooves was accounted for by an additional volume throughput in the balance | + | The channel model for tightly intermeshing threaded mixing elements can be compared to that of threaded elements. The flow in the grooves was accounted for by an additional volume throughput in the balance |
| - | {{ : | + | {{ :en: |
| **Figure:** Channel model for a reconveying, | **Figure:** Channel model for a reconveying, | ||
| Zeile 975: | Zeile 981: | ||
| A volume flow balance in the control room as depicted in the figure results in the following equation: | A volume flow balance in the control room as depicted in the figure results in the following equation: | ||
| - | $$\dot{V}_{channel, | + | $$\dot{V}_{channel, |
| For the description of the flow in channel direction and in the grooves all models developed in chapter Rectangular channels and chapter Twin Screw Channels respectively are used in a general form: | For the description of the flow in channel direction and in the grooves all models developed in chapter Rectangular channels and chapter Twin Screw Channels respectively are used in a general form: | ||
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| with | with | ||
| - | $$\dot{V}_{z, | + | $$\dot{V}_{z, |
| - | $$\dot{V}_{x, | + | $$\dot{V}_{x, |
| - | $$\dot{V}_{groove, | + | $$\dot{V}_{groove, |
| and | and | ||
| - | $$\pi_{p,z} = \frac{\Delta p}{\Delta z} \cdot \frac{\bar{h}^{1+n}}{K \cdot v_0^n} \tag{9}$$ | + | $$\pi_{p,z} = \frac{\Delta p}{\Delta z} \cdot \frac{\bar{h}^{1+n}}{K \cdot v_0^n} \tag{112}$$ |
| - | $$\pi_{p,x} = \frac{\Delta p}{x} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} \tag{10}$$ | + | $$\pi_{p,x} = \frac{\Delta p}{x} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} \tag{113}$$ |
| - | $$\pi_{p,N} = \frac{\Delta p}{x} \cdot \frac{h_N}{K \cdot v_0^n} \tag{11}$$ | + | $$\pi_{p,N} = \frac{\Delta p}{x} \cdot \frac{h_N}{K \cdot v_0^n} \tag{114}$$ |
| and | and | ||
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | If now the Equations (3) to (8) are inserted into Equation (2), one receives: | + | Substituting gives: |
| $$\dot{V}_{channel, | $$\dot{V}_{channel, | ||
| Zeile 1015: | Zeile 1021: | ||
| $$+A_{DSE, | $$+A_{DSE, | ||
| - | $$+A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p, | + | $$+A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p, |
| $$+l \cdot (h_N \cdot b_N \cdot v_0) \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p, | $$+l \cdot (h_N \cdot b_N \cdot v_0) \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p, | ||
| - | $$+A_{RE, | + | $$+A_{RE, |
| Is this equation standardized to $k \cdot (\bar{h} \cdot b_{max} \cdot v_0)$, then follows: | Is this equation standardized to $k \cdot (\bar{h} \cdot b_{max} \cdot v_0)$, then follows: | ||
| Zeile 1027: | Zeile 1033: | ||
| $$+A_{DSE, | $$+A_{DSE, | ||
| - | $$+A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p, | + | $$+A_{RE,1} \cdot \pi_{p,x} + A_{RE,2} \cdot \pi_{p,x}^2 + A_{RE,3} \cdot \pi_{p,x}^3 + A_{RE,4} \cdot \pi_{p, |
| $$+ \frac{l \cdot (h_N \cdot b_N \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p, | $$+ \frac{l \cdot (h_N \cdot b_N \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot \pi_{p,N} + A_{RE,N,2} \cdot \pi_{p, | ||
| - | $$+A_{RE, | + | $$+A_{RE, |
| To obtain a comprehensive description of the process behavior the pressure gradient in conveyor channel direction is to be linked with the pressure gradients above the radial clearance and with the pressure gradient in the groove. For this purpose the approach proposed by Ansahl | To obtain a comprehensive description of the process behavior the pressure gradient in conveyor channel direction is to be linked with the pressure gradients above the radial clearance and with the pressure gradient in the groove. For this purpose the approach proposed by Ansahl | ||
| - | $$\pi_{p,x} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo, | + | $$\pi_{p,x} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo, |
| - | $$\pi_{p,x} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,N} \cdot \pi_{p,z} \tag{18}$$ | + | $$\pi_{p,N} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \frac{h_N^{1+n}}{K \cdot v_0^n} = \pi_{geo,N} \cdot \pi_{p, |
| - | with | + | $$\pi_{p,x} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \frac{s_R^{1+n}}{K \cdot v_0^n} = \pi_{geo,x} \cdot \pi_{p, |
| - | $$\pi_{geo,GE} = \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \left(\frac{s_R}{\bar{h}}\right)^{1+n} \tag{19}$$ | + | $$\pi_{p,N} = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) \cdot \left(\frac{h_N}{\bar{h}}\right)^{1+n} = \pi_{geo,N} \cdot \pi_{p,z}\tag{123}$$ |
| - | $$\pi_{geo, | + | with: |
| - | Out of Equation (19), (20) and (16) one receives: | + | $$\pi_{geo,GE} = \frac{b_{max} + e_{max}}{e_{max} \cdot \tan(\varphi_s)} \cdot \left(\frac{s_R}{\bar{h}}\right)^{1+n}\tag{124}$$ |
| - | $$\pi_{\dot{V}, | + | $$\pi_{geo,N} = \frac{\Delta p \cdot b_{max} + e_{max}}{\Delta z \cdot e_{max} \cdot \tan(\varphi_s)} \cdot \cos\left(\frac{\pi}{2} - \varphi_s - \varphi_N\right) |
| - | $$+A_{DSE,4} \cdot \pi_{p, | + | From the equations, we obtain: |
| - | $$+A_{RE,2} \cdot (\pi_{p,z} \cdot \pi_{geo})^2 + A_{RE,3} \cdot (\pi_{p,z} \cdot \pi_{geo})^3 + A_{RE,4} \cdot (\pi_{p,z} \cdot \pi_{geo})^5)(21)$$ | + | $$\pi_{ \dot V, |
| - | + | $$\left.+A_{DSE, | |
| - | $$+\pi_2 \cdot (A_{RE,N,0} + A_{RE,N,1} \cdot (\pi_{p,z} \cdot \pi_{geo, | + | $$\left.+A_{RE,2} \cdot \left(\pi_{p,z} \cdot \pi_{geo}\right)^2 + A_{RE,3} \cdot \left(\pi_{p,z} \cdot \pi_{geo}\right)^3 + A_{RE,4} \cdot \left(\pi_{p,z} \cdot \pi_{geo}\right)^5\right)$$ |
| - | + | $$+\pi_2 \cdot \left(A_{RE,N,0} + A_{RE,N,1} \cdot \left(\pi_{p,z} \cdot \pi_{geo,N}\right) + A_{RE,N,2} \cdot \left(\pi_{p,z} \cdot \pi_{geo,N}\right)^2\right.$$ | |
| - | $$+A_{RE, | + | $$\left.+A_{RE,N,3} \cdot \left(\pi_{p,z} \cdot \pi_{geo,N}\right)^3 + A_{RE,N,4} \cdot \left(\pi_{p,z} \cdot \pi_{geo,N}\right)^5\right)\tag{126}$$ |
| with | with | ||
| - | $$\pi_1 = \frac{(s_R \cdot b_{thread} \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \tag{22}$$ | + | $$\pi_1 = \frac{\left(s_R \cdot b_{Steg} \cdot v_0\right)}{k \cdot \left(\bar{h} \cdot b_{max} \cdot v_0\right)}\tag{127}$$ |
| - | + | ||
| - | $$\pi_2 = \frac{l \cdot (h_N \cdot b_N \cdot v_0)}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \tag{23}$$ | + | |
| - | + | ||
| - | the equation to calculate the pressure-throughput behavior of the channel area of threaded elements: | + | |
| - | + | ||
| - | $$\pi_{\dot{V}, | + | |
| - | + | ||
| - | $$+(A_{DSE, | + | |
| - | $$+(A_{DSE,2} + \pi_1 \cdot \pi_{geo}^2 | + | $$\pi_2 = \frac{l |
| - | $$+(A_{DSE, | + | the equation for calculating the pressure-flow characteristics of the channel section of threaded elements: |
| - | $$+(A_{DSE, | + | $$\pi_{\dot V,GE} = \left(A_{DSE, |
| + | $$+\left(A_{DSE, | ||
| + | $$+\left(A_{DSE, | ||
| + | $$+\left(A_{DSE, | ||
| + | $$+\left(A_{DSE,4} + \pi_1 \cdot \pi_{geo}^5 \cdot A_{RE,4} + \pi_2 \cdot \pi_{geo, | ||
| === Intermeshing Zone === | === Intermeshing Zone === | ||
| Zeile 1081: | Zeile 1083: | ||
| For the description of the throughput behavior in the intermeshing zone it is assumed that the throughput behavior is substantially dominated by a chamber flow. The grooves in the intermeshing zone alone result in an influence of the pressure gradient on the flow. | For the description of the throughput behavior in the intermeshing zone it is assumed that the throughput behavior is substantially dominated by a chamber flow. The grooves in the intermeshing zone alone result in an influence of the pressure gradient on the flow. | ||
| - | $$\dot{V}_{intermeshing\, | + | $$\dot{V}_{intermeshing\, |
| To specify the groove flow the description of the rectangular channels is applied: | To specify the groove flow the description of the rectangular channels is applied: | ||
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | {{ : | + | {{ : |
| **Figure:** Axial velocities in the intermeshing zone of a closely intermeshing threaded mixing element. | **Figure:** Axial velocities in the intermeshing zone of a closely intermeshing threaded mixing element. | ||
| Zeile 1093: | Zeile 1095: | ||
| For the linkage of the pressure gradients in the intermeshing zone and in the conveyor channel area it is assumed that the back pressure in the intermeshing zone is negligible and that the same approach can thus be used as above. Consequently, | For the linkage of the pressure gradients in the intermeshing zone and in the conveyor channel area it is assumed that the back pressure in the intermeshing zone is negligible and that the same approach can thus be used as above. Consequently, | ||
| - | $$\pi_{p, | + | $$\pi_{p, |
| with | with | ||
| - | $$\pi_{geo, | + | $$\pi_{geo, |
| === Linkage of the Models === | === Linkage of the Models === | ||
| Zeile 1107: | Zeile 1109: | ||
| $$+(A_{DSE, | $$+(A_{DSE, | ||
| - | $$+(A_{DSE, | + | $$+(A_{DSE, |
| $$+(A_{DSE, | $$+(A_{DSE, | ||
| - | $$+(A_{DSE, | + | $$+(A_{DSE, |
| with | with | ||
| - | $$\pi_3 = \frac{l_{zw} \cdot (h_N \cdot b_N \cdot v_{zw})}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \tag{30}$$ | + | $$\pi_3 = \frac{l_{zw} \cdot (h_N \cdot b_N \cdot v_{zw})}{k \cdot (\bar{h} \cdot b_{max} \cdot v_0)} \tag{135}$$ |
| The figure displays a comparison of the measured dimensionless pressure gradient and those calculated by means of the model for a conveying and for a reconveying threaded mixing element. Despite the multiplicity of simplifications it can be observed that the model is in a position to describe the experimental results with sufficient accurateness. | The figure displays a comparison of the measured dimensionless pressure gradient and those calculated by means of the model for a conveying and for a reconveying threaded mixing element. Despite the multiplicity of simplifications it can be observed that the model is in a position to describe the experimental results with sufficient accurateness. | ||
| - | {{ : | + | {{ : |
| **Figure:** Comparison of the experimentally investigated and the calculated dimensionless pressure gradients. | **Figure:** Comparison of the experimentally investigated and the calculated dimensionless pressure gradients. | ||
| Zeile 1129: | Zeile 1131: | ||
| When examining the flow in the intermeshing zone of these elements it directly attracts the attention that the axial velocities strongly deviate along the element pair. In those areas in the figure marked with an " | When examining the flow in the intermeshing zone of these elements it directly attracts the attention that the axial velocities strongly deviate along the element pair. In those areas in the figure marked with an " | ||
| - | {{ : | + | {{ : |
| **Figure:** Distribution of the axial velocities in non-intermeshing threaded mixing elements. | **Figure:** Distribution of the axial velocities in non-intermeshing threaded mixing elements. | ||
| Zeile 1137: | Zeile 1139: | ||
| The unwinding of the non-intermeshing threaded mixing element into the section results in the conveyor channel model as depicted in the figure. | The unwinding of the non-intermeshing threaded mixing element into the section results in the conveyor channel model as depicted in the figure. | ||
| - | {{ : | + | {{ : |
| **Figure:** Conveyor channel model for non-intermeshing threaded mixing elements. | **Figure:** Conveyor channel model for non-intermeshing threaded mixing elements. | ||
| Zeile 1143: | Zeile 1145: | ||
| The balance of the volume flow rates in the control room ABCD yields: | The balance of the volume flow rates in the control room ABCD yields: | ||
| - | $$\dot{V}_{tGME} = (k \cdot \dot{V}_{z, | + | $$\dot{V}_{tGME} = (k \cdot \dot{V}_{z, |
| with | with | ||
| - | $$\dot{V}_{z, | + | $$\dot{V}_{z, |
| - | $$\dot{V}_{x, | + | $$\dot{V}_{x, |
| - | $$\dot{V}_{groove, | + | $$\dot{V}_{groove, |
| - | $$\dot{V}_{z, | + | $$\dot{V}_{z, |
| - | $$\dot{V}_{x, | + | $$\dot{V}_{x, |
| - | $$\dot{V}_{groove, | + | $$\dot{V}_{groove, |
| The dimensionless volume flow rates are defined as follows: | The dimensionless volume flow rates are defined as follows: | ||
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$+A_{DSE, | + | $$+A_{DSE, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$+A_{RE, | + | $$+A_{RE, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$+A_{RE, | + | $$+A_{RE, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$+A_{DSE, | + | $$+A_{DSE, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$+A_{RE, | + | $$+A_{RE, |
| - | $$\pi_{\dot{V}, | + | $$\pi_{\dot{V}, |
| - | $$+A_{RE, | + | $$+A_{RE, |
| with the parameters defined in the table. | with the parameters defined in the table. | ||
| Zeile 1189: | Zeile 1191: | ||
| Simplifying assumed that the isobars operate at right angle to the screw axis the following equation applies: | Simplifying assumed that the isobars operate at right angle to the screw axis the following equation applies: | ||
| - | $$\frac{\Delta p_{fe}}{\Delta z} = \frac{\Delta p_{rfe}}{\Delta z} \Leftrightarrow \pi_{p, | + | $$\frac{\Delta p_{fe}}{\Delta z} = \frac{\Delta p_{rfe}}{\Delta z} \Leftrightarrow \pi_{p, |
| The pressure gradients in channel direction and at right angle to it are again linked in an analogous manner to the approaches introduced so far. | The pressure gradients in channel direction and at right angle to it are again linked in an analogous manner to the approaches introduced so far. | ||
| - | $$\Delta p_x = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{\tan(\varphi_s)} \tag{15}$$ | + | $$\Delta p_x = \frac{\Delta p}{\Delta z} \cdot \frac{b_{max} + e_{max}}{\tan(\varphi_s)} \tag{150}$$ |
| ** Dimensionless parameters for the description of the pressure-throughput behavior of non-intermeshing threaded mixing elements: ** | ** Dimensionless parameters for the description of the pressure-throughput behavior of non-intermeshing threaded mixing elements: ** | ||
| Zeile 1207: | Zeile 1209: | ||
| Consequently, | Consequently, | ||
| - | $$\pi_{p, | + | $$\pi_{p, |
| - | $$\pi_{p, | + | $$\pi_{p, |
| - | $$\pi_{p, | + | $$\pi_{p, |
| - | $$\pi_{p, | + | $$\pi_{p, |
| with: | with: | ||
| - | $$\pi_{geo, | + | $$\pi_{geo, |
| - | $$\pi_{geo, | + | $$\pi_{geo, |
| - | Are these equations inserted into the balance | + | Are these equations inserted into the balance, one receives the model to specify the pressure-throughput behavior in the conveyor channel area: |
| $$\pi_{\dot{V}, | $$\pi_{\dot{V}, | ||
| Zeile 1237: | Zeile 1239: | ||
| $$+(A_{DSE, | $$+(A_{DSE, | ||
| - | $$+\pi_2 \cdot \pi_{geo, | + | $$+\pi_2 \cdot \pi_{geo, |
| The comparison of the model predictions with the experimental results for distinct polymers confirms, that the model yields sufficient precision despite its multitude of simplifications. | The comparison of the model predictions with the experimental results for distinct polymers confirms, that the model yields sufficient precision despite its multitude of simplifications. | ||
| - | {{ : | + | {{ : |
| **Figure:** Comparison of the calculated and the measured dimensionless pressure gradients for a non-intermeshing threaded mixing element. | **Figure:** Comparison of the calculated and the measured dimensionless pressure gradients for a non-intermeshing threaded mixing element. | ||
| Zeile 1252: | Zeile 1254: | ||
| * The flow in the grooves can be described sufficiently accurate with a flow in the rectangular channels. | * The flow in the grooves can be described sufficiently accurate with a flow in the rectangular channels. | ||
| - | {{ : | + | {{ :en: |
| **Figure:** Real turbine mixing elements geometry and the geometrical substitution model. | **Figure:** Real turbine mixing elements geometry and the geometrical substitution model. | ||
| Zeile 1258: | Zeile 1260: | ||
| The description of the process behavior is based upon the superimposition of the volume flow rates in the grooves and in the disk area. | The description of the process behavior is based upon the superimposition of the volume flow rates in the grooves and in the disk area. | ||
| - | $$\dot{V}_{ZE} = \dot{V}_{disc} + l_N \cdot \dot{V}_{groove} \tag{1}$$ | + | $$\dot{V}_{ZE} = \dot{V}_{disc} + l_N \cdot \dot{V}_{groove} \tag{158}$$ |
| To specify the volume flow rate the approach is applied as introduced in chapter Pressure-Throughput Model for Threaded Elements: | To specify the volume flow rate the approach is applied as introduced in chapter Pressure-Throughput Model for Threaded Elements: | ||
| - | $$\dot{V}_{disc} = A_{free,SE} \cdot D_a \cdot n_0 \cdot (A_{SE,1} \cdot \pi_{p,SE} + A_{SE,2} \cdot \pi_{p, | + | $$\dot{V}_{disc} = A_{free,SE} \cdot D_a \cdot n_0 \cdot (A_{SE,1} \cdot \pi_{p,SE} + A_{SE,2} \cdot \pi_{p, |
| The modeling of the throughput behavior in the grooves is performed by virtue of the approaches presented in chapter Pressure-Throughput Model for Threaded Elements: | The modeling of the throughput behavior in the grooves is performed by virtue of the approaches presented in chapter Pressure-Throughput Model for Threaded Elements: | ||
| - | $$\dot{V}_{groove} = h_N \cdot b_N \cdot v_0 \cdot (A_{RE,0} + A_{RE,1} \cdot \pi_{p,N} + A_{RE,2} \cdot \pi_{p,N}^2 + A_{RE,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p, | + | $$\dot{V}_{groove} = h_N \cdot b_N \cdot v_0 \cdot (A_{RE,0} + A_{RE,1} \cdot \pi_{p,N} + A_{RE,2} \cdot \pi_{p,N}^2 + A_{RE,3} \cdot \pi_{p,N}^3 + A_{RE,4} \cdot \pi_{p, |
| Out of this results the following equation to describe the pressure-throughput behavior of toothed mixing elements: | Out of this results the following equation to describe the pressure-throughput behavior of toothed mixing elements: | ||
| - | $$\pi_{\dot V,ZE} = A_{ZE,0} + A_{ZE,1} \cdot \pi_p + A_{ZE,2} \cdot \pi_p^2 + A_{ZE,3} \cdot \pi_p^3 + A_{ZE,4} \cdot \pi_p^5 \tag{4}$$ | + | $$\pi_{\dot V,ZE} = A_{ZE,0} + A_{ZE,1} \cdot \pi_p + A_{ZE,2} \cdot \pi_p^2 + A_{ZE,3} \cdot \pi_p^3 + A_{ZE,4} \cdot \pi_p^5 \tag{161}$$ |
| with: | with: | ||
| - | $$\pi_{\dot V,ZE} = \frac{\dot{V}_{ZE}}{A_{fr} \cdot D_a \cdot n_0} \tag{5}$$ | + | $$\pi_{\dot V,ZE} = \frac{\dot{V}_{ZE}}{A_{fr} \cdot D_a \cdot n_0} \tag{162}$$ |
| - | $$\pi_p = \frac{\Delta p}{L} \cdot \frac{D_a}{K \cdot n_0^n} \tag{6}$$ | + | $$\pi_p = \frac{\Delta p}{L} \cdot \frac{D_a}{K \cdot n_0^n} \tag{163}$$ |
| and | and | ||
| - | $$A_{ZE,0} = A_{RE,0} \cdot i_N \cdot \pi_{geo,V} \tag{7}$$ | + | $$A_{ZE,0} = A_{RE,0} \cdot i_N \cdot \pi_{geo,V} \tag{164}$$ |
| - | $$A_{ZE,1} = A_{SE,1} + A_{RE,1} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p} \tag{8}$$ | + | $$A_{ZE,1} = A_{SE,1} + A_{RE,1} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo,p} \tag{165}$$ |
| - | $$A_{ZE,2} = A_{RE,2} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo, | + | $$A_{ZE,2} = A_{RE,2} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo, |
| - | $$A_{ZE,3} = A_{SE,2} + A_{RE,3} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo, | + | $$A_{ZE,3} = A_{SE,2} + A_{RE,3} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo, |
| - | $$A_{ZE,4} = A_{SE,3} + A_{RE,4} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo, | + | $$A_{ZE,4} = A_{SE,3} + A_{RE,4} \cdot i_N \cdot \pi_{geo,V} \cdot \pi_{geo, |
| The two factors $\pi_{geo, | The two factors $\pi_{geo, | ||
| - | $$\pi_{geo, | + | $$\pi_{geo, |
| and: | and: | ||
| - | $$\pi_{geo, | + | $$\pi_{geo, |
| The figure shows a comparison of the model predictions with the experimental results. It can be recognized that the model is in a position to describe the experimental results with satisfactory precision. | The figure shows a comparison of the model predictions with the experimental results. It can be recognized that the model is in a position to describe the experimental results with satisfactory precision. | ||
| - | {{ : | + | {{ : |
| **Figure:** Comparison of experimental and approximated dimensionless pressure gradients of conveying turbine mixing elements. | **Figure:** Comparison of experimental and approximated dimensionless pressure gradients of conveying turbine mixing elements. | ||