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--- en:grundlagenhandbuch:berechnung_der_dispergierguete [2026/05/12 21:00] – [Discussion and Results] neelest
+++ en:grundlagenhandbuch:berechnung_der_dispergierguete [2026/05/28 09:35] (aktuell) – [Agglomerate Disintegration] deppe2
@@ Zeile 15: / Zeile 15: @@
 For the work required to increase the surface area, the following applies to this fragmentation process, assuming an isothermal process and a brittle body [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[BC58]]] with surface energy $\sigma$:
-$$\sigma_M = \frac{\dot{V}_x}{\dot{V}_z}$$
+$$\sigma_M = \frac{\dot{V}_x}{\dot{V}_z} \tag{1}$$
 With the change in surface area over time:
-$$\overline{\sigma_M} = \frac{\sum_i(\sigma_M L_{Ele})_i}{\sum_i(L_{Ele})_i}$$
+$$\overline{\sigma_M} = \frac{\sum_i(\sigma_M L_{Ele})_i}{\sum_i(L_{Ele})_i} \tag{2}$$
 the following is obtained for the dispersion efficiency into two spherical particles:
-$$P = \frac{\partial W}{\partial t} = 2\sigma\pi d \frac{\partial \bar{d}}{\partial t}$$
+$$P = \frac{\partial W}{\partial t} = 2\sigma\pi d \frac{\partial \bar{d}}{\partial t} \tag{3}$$
 According to Rumpf [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Rum61]]], the surface energy is given by:
-$$\sigma = \sum_{n=1}^{n_p} \sum_{k=1}^{k} \frac{F_H}{d} = n_p k \frac{\overline{F_H}}{d}$$
+$$\sigma = \sum_{n=1}^{n_p} \sum_{k=1}^{k} \frac{F_H}{d} = n_p k \frac{\overline{F_H}}{d} \tag{4}$$
 Here, $F_H$ denotes the effective adhesive forces, $n_p$ the number of primary particles, and $k$ the number of contact points of the primary particles in the interface to be taken into account.
@@ Zeile 33: / Zeile 33: @@
 The minimum power required for a disintegration process is given by:
-$$P_{min} = \pi n_p k \overline{F_H} \frac{\partial \bar{d}}{\partial t}$$
+$$P_{min} = \pi n_p k \overline{F_H} \frac{\partial \bar{d}}{\partial t} \tag{5}$$
 In addition to the power, the stresses to which a sphere is subjected in a shear flow are considered in the next step, analogous to [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Raa61]]], [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[BT74]]] with regard to the energy input. The particle rotates in the shear flow and is thereby subjected to alternating tensile and compressive stresses. According to Raasch [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Raa61]]], the angular velocity $\omega$ is derived from the prevailing shear velocity of the surrounding flow:
-$$\omega = \frac{1}{2} \dot{\gamma} \tag{1}$$
+$$\omega = \frac{1}{2} \dot{\gamma}  \tag{6}$$
-The power at the sphere’s surface, as shown in Figure 10.2, is formulated as follows according to Raasch [[grundlagenhandbuch:berechnung_der_dispergierguete#literatur |[Raa61]]] when considering a pure shear flow:
+The power at the sphere’s surface, as shown in Figure 10.2, is formulated as follows according to Raasch [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Raa61]]] when considering a pure shear flow:
-$$P = \int_{A_{ober}} \left[\sigma_r v_r + \tau_{r\varphi} V_\varphi + \tau_{r\vartheta} V_\vartheta\right] dA$$
+$$P = \int_{A_{ober}} \left[\sigma_r v_r + \tau_{r\varphi} V_\varphi + \tau_{r\vartheta} V_\vartheta\right] dA \tag{7}$$
 The following applies to the surface differential:
-$$dA = r^2 \cos \vartheta \, d\vartheta d\varphi$$
+$$dA = r^2 \cos \vartheta \, d\vartheta d\varphi \tag{8}$$
 {{ :grundlagenhandbuch:berechnung_der_dispergierguete:desagglomeration:de_sigma150_dlg_grundlagenhandbuch_berechnung_der_dispergierguete_002.png?nolink |}}
-**Figure:** Linear shear flow around a sphere [[basics_manual:calculation_of_dispersion_quality#literature |[Raa61]]]
+**Figure:** Linear shear flow around a sphere [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Raa61]]]
-The stresses arising on the sphere’s surface during the rotation shown about the z-axis are (cf. [[basics_manual:calculation_of_dispersion_quality#literature | [Raa61]]], [[basic_manual:calculation_of_dispersion_quality#literature |[HFM92]]]):
+The stresses arising on the sphere’s surface during the rotation shown about the z-axis are (cf. [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Raa61]]], [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[HFM92]]]):
-$$\sigma_r = \frac{5}{2} \eta \dot{\gamma} \sin 2\varphi \cos^2 \vartheta$$
+$$\sigma_r = \frac{5}{2} \eta \dot{\gamma} \sin 2\varphi \cos^2 \vartheta \tag{9}$$
-$$\tau_{r\varphi} = \frac{5}{2} \eta \dot{\gamma} \cos 2\varphi \cos \vartheta$$
+$$\tau_{r\varphi} = \frac{5}{2} \eta \dot{\gamma} \cos 2\varphi \cos \vartheta \tag{10}$$
-$$\tau_{r\vartheta} = -\frac{5}{2} \eta \dot{\gamma} \sin 2\varphi \sin \vartheta \cos \vartheta$$
+$$\tau_{r\vartheta} = -\frac{5}{2} \eta \dot{\gamma} \sin 2\varphi \sin \vartheta \cos \vartheta \tag{11}$$
 The following applies to the velocities:
-$$v_r = \frac{\dot{\gamma}}{4} \left(2r - 5\frac{r_0^3}{r^2} + 3\frac{r_0^5}{r^4}\right) \sin 2\varphi \cos^2 \vartheta$$
+$$v_r = \frac{\dot{\gamma}}{4} \left(2r - 5\frac{r_0^3}{r^2} + 3\frac{r_0^5}{r^4}\right) \sin 2\varphi \cos^2 \vartheta \tag{12}$$
-$$V_\varphi = \frac{\dot{\gamma}}{4}\left[\left(r - \frac{r_0^5}{r^4}\right) \cos 2\varphi - r\right] \cos \vartheta$$
+$$V_\varphi = \frac{\dot{\gamma}}{4}\left[\left(r - \frac{r_0^5}{r^4}\right) \cos 2\varphi - r\right] \cos \vartheta \tag{13}$$
-$$V_\vartheta = -\frac{\dot{\gamma}}{4} \left(r - \frac{r_0^5}{r^4}\right) \sin 2\varphi \cos 2\vartheta$$
+$$V_\vartheta = -\frac{\dot{\gamma}}{4} \left(r - \frac{r_0^5}{r^4}\right) \sin 2\varphi \cos 2\vartheta \tag{14}$$
 To calculate the power resulting from the fragmentation process, we now consider the limit at the sphere’s surface for $r = r_0$. The equation then becomes:
-$$P = \int_{A_{upper}} \left[\frac{5}{2} \eta \dot{\gamma} \cos 2\varphi \cos \vartheta\right] \left[-\frac{\dot{\gamma}}{2} r_0 \cos \vartheta\right] r_0^2 \cos \vartheta \, d\vartheta d\varphi$$
+$$P = \int_{A_{upper}} \left[\frac{5}{2} \eta \dot{\gamma} \cos 2\varphi \cos \vartheta\right] \left[-\frac{\dot{\gamma}}{2} r_0 \cos \vartheta\right] r_0^2 \cos \vartheta \, d\vartheta d\varphi \tag{15}$$
 The figure shows the power distribution on the spherical surface.
@@ Zeile 77: / Zeile 77: @@
 **Figure:** Power distribution on the spherical surface
-For the process of spontaneous disintegration, the shear stress and the maximum value of the power input, as shown in Figure 10.3, are of interest, subject to the boundary condition that $\tau_{r\varphi} = \tau_{crit}$ (critical shear stress, which leads to disintegration when the cohesive forces are exceeded [[grundlagenhandbuch:berechnung_der_dispergierguete# literature |[BT74]]]), are of interest. The agglomerate is treated here as an isotropic body, so that, regardless of the surface coordinate, exceeding the critical shear stress results in spontaneous disintegration.
+For the process of spontaneous disintegration, the shear stress and the maximum value of the power input, as shown in Figure 10.3, are of interest, subject to the boundary condition that $\tau_{r\varphi} = \tau_{crit}$ (critical shear stress, which leads to disintegration when the cohesive forces are exceeded [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[BT74]]]), are of interest. The agglomerate is treated here as an isotropic body, so that, regardless of the surface coordinate, exceeding the critical shear stress results in spontaneous disintegration.
-The maximum possible shear stress that can occur on the sphere’s surface is given by Eq. (10.16):
+The maximum possible shear stress that can occur on the sphere’s surface is given by Eq. (16):
-$$\tau_{max} = \frac{5}{2} \eta_{Fluid} \dot{\gamma}$$
+$$\tau_{max} = \frac{5}{2} \eta_{Fluid} \dot{\gamma} \tag{16}$$
 The circumferential velocity is given by the equation:
-$$V_\varphi = r_0 \omega = r_0 \frac{\dot{\gamma}}{2}$$
+$$V_\varphi = r_0 \omega = r_0 \frac{\dot{\gamma}}{2} \tag{17}$$
 As a result of these considerations, for a sphere (agglomerate) with diameter $\bar{d}$, the maximum power that can occur is:
-$$P_{max} = \frac{5}{4} \eta_{Fluid} \dot{\gamma}^2 \pi r_0^3 = \frac{5}{32} \tau_{Fluid} \dot{\gamma} \pi \bar{d}^3$$
+$$P_{max} = \frac{5}{4} \eta_{Fluid} \dot{\gamma}^2 \pi r_0^3 = \frac{5}{32} \tau_{Fluid} \dot{\gamma} \pi \bar{d}^3 \tag{18}$$
 Equating the two power expressions yields the following differential equation:
-$$\frac{5}{32} \tau_{Fluid} \dot{\gamma} \pi \bar{d}^3 = 2\pi \cdot n_p k \overline{F_H} \frac{\partial \bar{d}}{\partial t}$$
+$$\frac{5}{32} \tau_{Fluid} \dot{\gamma} \pi \bar{d}^3 = 2\pi \cdot n_p k \overline{F_H} \frac{\partial \bar{d}}{\partial t} \tag{19}$$
-If we consider the interface of the agglomerate, the number of particles at this interface depends on the surface porosity. The surface porosity $\psi_F$ is calculated according to [[grundlagenhandbuch:berechnung_der_dispergierguete#literatur |[PEW93]]] as the ratio of the area $A_H$ not occupied by the solid to the total area $A$:
+If we consider the interface of the agglomerate, the number of particles at this interface depends on the surface porosity. The surface porosity $\psi_F$ is calculated according to [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[PEW93]]] as the ratio of the area $A_H$ not occupied by the solid to the total area $A$:
-$$\psi_F = \frac{A_H}{A} = 1 - \frac{n_p A_p}{A}$$
+$$\psi_F = \frac{A_H}{A} = 1 - \frac{n_p A_p}{A} \tag{20}$$
 From this equation, it is possible to derive a relationship for calculating the number of particles in the interface as a function of the surface porosity and the mean agglomerate diameter:
-$$n_p = (1 - \psi_F) \frac{A}{A_p} = (1 - \psi_F) \frac{\bar{d}^2}{d_p^2}$$
+$$n_p = (1 - \psi_F) \frac{A}{A_p} = (1 - \psi_F) \frac{\bar{d}^2}{d_p^2} \tag{21}$$
 Substituting the values gives the equation:
-$$\frac{5}{32} \tau_{Fluid} \dot{\gamma} \pi \bar{d} = 2(1 - \psi_F) \frac{1}{d_p^2} k \pi \overline{F_H} \frac{\partial \bar{d}}{\partial t}$$
+$$\frac{5}{32} \tau_{Fluid} \dot{\gamma} \pi \bar{d} = 2(1 - \psi_F) \frac{1}{d_p^2} k \pi \overline{F_H} \frac{\partial \bar{d}}{\partial t} \tag{22}$$
 The considerations outlined so far apply to shear fractures in a pure shear flow. However, this does not occur in extruders; rather, in fully filled screw sections, there is always a superposition of compressional and drag flow. Here, the superposition of a pressure gradient in a screw channel results in a shift in the stress state, as shown in Figure 10.4 (Mohr’s stress circle). Accordingly, under sufficiently high pressures, fragmentation can no longer occur under tensile stress. Since experiments have shown agglomerate disintegration particularly in pressurised zones, shear stresses are therefore responsible for the disintegration processes. In partially filled screw sections, too, one is dealing with a multidimensional flow.
-Tensile strength, as a material property, may only be used in the case of pure cleavage fractures. For a cleavage fracture perpendicular to the principal tensile stress, the normal stress hypothesis applies. However, this cannot be applied, as pressures significantly higher than the agglomerate strength typically occur in the melt. The extended shear stress hypothesis is appropriate here. It assumes different yield shear stresses. The envelope of the corresponding Mohr’s stress circles is then the yield strength $\tau = f(\sigma)$ [[basics_handbook:calculation_of_dispersion_quality#literature |[Dub91]]]. This represents a comprehensive material characteristic. Here, due to incomplete material properties, the envelope is replaced by three straight lines (Figure 10.4).
+Tensile strength, as a material property, may only be used in the case of pure cleavage fractures. For a cleavage fracture perpendicular to the principal tensile stress, the normal stress hypothesis applies. However, this cannot be applied, as pressures significantly higher than the agglomerate strength typically occur in the melt. The extended shear stress hypothesis is appropriate here. It assumes different yield shear stresses. The envelope of the corresponding Mohr’s stress circles is then the yield strength $\tau = f(\sigma)$ [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Dub91]]]. This represents a comprehensive material characteristic. Here, due to incomplete material properties, the envelope is replaced by three straight lines (Figure 10.4).
 {{ :en:grundlagenhandbuch:en_sigma150_dlg_grundlagenhandbuch_berechnung_der_dispergierguete_005.png?nolink |}}
@@ Zeile 117: / Zeile 117: @@
 Due to the high pressures occurring in continuous mixers, shear failures are more than likely to occur. From this follows:
-$$\tau_{Sch} = \frac{1}{2} \sigma_v = \sigma_3 - \sigma_1 \tag{10}$$
+$$\tau_{Sch} = \frac{1}{2} \sigma_v = \sigma_3 - \sigma_1 \tag{23}$$
 With σ1 and σ3 being the maximum principle stresses. We assume the hoop stress σv to be almost equal to the tensile strength σz, which we predict using Rumpf's [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Rum61]]] equation:
-$$\tau_{Sch} = \frac{1}{2} \sigma_z = \frac{(1 - \varepsilon_A) \overline{F_{adh}}}{2\pi \cdot d_p^2} \tag{11}$$
+$$\tau_{Sch} = \frac{1}{2} \sigma_z = \frac{(1 - \varepsilon_A) \overline{F_{adh}}}{2\pi \cdot d_p^2} \tag{24}$$
 Inserting the agglomerate strength according to Rumpf (eqn. (11) results in the following differential equation for the agglomerate rupture:
-$$\frac{\partial \bar{d}}{\partial t} = \frac{5}{64} \cdot \frac{\tau_{Fluid} \cdot \dot{\gamma}}{\tau_{Sch} \cdot \pi} \cdot \bar{d} \tag{12}$$
+$$\frac{\partial \bar{d}}{\partial t} = \frac{5}{64} \cdot \frac{\tau_{Fluid} \cdot \dot{\gamma}}{\tau_{Sch} \cdot \pi} \cdot \bar{d} \tag{25}$$
+==== Erosion ====
+In order for primary particles to break away (erode) from the agglomerate, the shear stresses acting on them must overcome the adhesive forces of the primary particles bound to the surface of the agglomerate.
+{{ :grundlagenhandbuch:berechnung_der_dispergierguete:desagglomeration:de_sigma150_dlg_grundlagenhandbuch_berechnung_der_dispergierguete_005.png?nolink |}}
+**Figure:** Schematic representation of erosion
+The power required to detach individual particles from an agglomerate results, as with fragmentation, from the creation of new surfaces. The power for the erosion process under consideration can be expressed as follows, in accordance with the figure:
+$$P = \sigma \frac{\partial A}{\partial t} \tag{26}$$
+The surface area of an agglomerate consisting of primary particles with diameter $d_P$ is approximately:
+$$A = n_0 \pi \frac{d_p^2}{2} \tag{27}$$
+To calculate the number of primary particles $n_0$ on the surface of the agglomerates, it is assumed that the primary particles are uniformly distributed throughout the agglomerate and that the porosity at the surface is therefore the same as the porosity at any cross-section:
+$$n_0 \pi \frac{d_p^2}{2} \cong (1 - \psi_F) \pi \bar{d}^2 \tag{28}$$
+$$n_0 = \frac{2(1 - \psi_F) \bar{d}^2}{d_p^2} \tag{29}$$
+It follows that for the change over time in the surface area of an agglomerate from which particles are removed from the outer shell:
+$$\frac{\partial A}{\partial t} = \pi (1 - \psi_F) \bar{d} \frac{\partial \bar{d}}{\partial t} \tag{30}$$
+The minimum power required for erosion is therefore:
+$$P_{min} = \sigma \pi (1 - \psi_F) \bar{d} \frac{\partial \bar{d}}{\partial t} \tag{31}$$
+Here, too, the power is equated with the maximum instantaneous power delivered by the fluid:
+$$\frac{5}{32} \tau_{Fluid} \dot{\gamma} \pi \bar{d}^3 = \sigma \pi (1 - \psi_F) \bar{d} \frac{\partial \bar{d}}{\partial t} \tag{32}$$
+If we now substitute the equation for the surface energy into the equation, analogous to the fragmentation, we obtain:
+$$\frac{5}{32} \tau_{Fluid} \dot{\gamma} \bar{d} = 2n_0 k \overline{F_H} (1 - \psi_F) \frac{\partial \bar{d}}{\partial t} \tag{33}$$
+Using the number of primary particles on the surface of an agglomerate (spherical in shape), we obtain:
+$$\frac{5}{64} \tau_{Fluid} \dot{\gamma} \bar{d} = 2 \frac{(1 - \psi_F)}{\bar d_p^2} k \overline{F_H} (1 - \psi_F) \frac{\partial \bar{d}}{\partial t} \tag{34}$$
+If we substitute the shear strength defined above, the following results for the erosion process:
+$$\frac{\partial \bar{d}}{\partial t} = \frac{5 \tau_{Fluid} \dot{\gamma}}{256 \pi (1 - \psi_F) \tau_{Sch}} \bar{d} \tag{35}$$
 ===== Clustering =====
@@ Zeile 142: / Zeile 186: @@
 Assuming that the agglomerates have similar mean spherical diameters at the same times, a monitoring cross-section can be defined for the agglomeration process, within which agglomerates can come into contact:
-$$S_{üb} = \frac{\pi}{4} (d_M + d)^2$$
+$$S_{üb} = \frac{\pi}{4} (d_M + d)^2 \tag{36}$$
 {{ :grundlagenhandbuch:berechnung_der_dispergierguete:de_sigma150_dlg_grundlagenhandbuch_berechnung_der_dispergierguete_007.png?nolink |}}
@@ Zeile 150: / Zeile 194: @@
 If we now apply this analysis to a slice of the volume element with length $\partial z$ and cross-sectional area $S_D$, there are $\partial n_A$ agglomerates within this slice:
-$$\partial n_A = n_A \frac{S_D \partial z}{V}$$
+$$\partial n_A = n_A \frac{S_D \partial z}{V} \tag{37}$$
 In relation to the monitoring cross-section, this yields a differential number of agglomerates $n_M$ that may experience a collision:
-$$\partial n_M = \frac{\partial n_A S_{mon}}{S_D}$$
+$$\partial n_M = \frac{\partial n_A S_{mon}}{S_D} \tag{38}$$
 Substituting the equations, we obtain:
-$$\partial n_M = n_A \frac{\pi (d_M + d)^2}{4 V} \partial z$$
+$$\partial n_M = n_A \frac{\pi (d_M + d)^2}{4 V} \partial z \tag{39}$$
 By substituting the solid concentration
-$$c_F = \frac{V_F}{V}$$
+$$c_F = \frac{V_F}{V} \tag{40}$$
 and the solid volume
-$$V_F = n_{A0} V_{A0}$$
+$$V_F = n_{A0} V_{A0} \tag{41}$$
 we obtain an expression for the rate of change of the number of particles in the volume element:
-$$\frac{\partial n_M}{\partial t} = \frac{\partial}{\partial t} \left(\frac{n_A \pi \left(\bar{d}_0 + \bar{d}\right)^2}{n_{A0} 4 V_{A0}} c_F \partial z\right)$$
+$$\frac{\partial n_M}{\partial t} = \frac{\partial}{\partial t} \left(\frac{n_A \pi \left(\bar{d}_0 + \bar{d}\right)^2}{n_{A0} 4 V_{A0}} c_F \partial z\right) \tag{42}$$
 The number $n_a$ of particles participating in the agglomeration is given by the ratio of the particle volume at time $t > t_0$ to the particle volume at time $t = t_0$:
-$$n_a = \frac{\bar{V}_A}{\bar{V}_{A0}} = \frac{\pi \bar{d}^3}{6 \bar{V}_{A0}}$$
+$$n_a = \frac{\bar{V}_A}{\bar{V}_{A0}} = \frac{\pi \bar{d}^3}{6 \bar{V}_{A0}} \tag{43}$$
 Since the essential process in the agglomeration of particles is collision transfer, the change in the number of particles undergoing a collision must be equal to the change in the number of particles participating in the agglomeration. This yields:
-$$\frac{\partial n_M}{\partial t} = \frac{\partial n_a}{\partial t}$$
+$$\frac{\partial n_M}{\partial t} = \frac{\partial n_a}{\partial t} \tag{44}$$
 Substituting
-$$\frac{\partial n_a}{\partial t} = \frac{\pi 1 \partial \bar{d}^3}{6 \bar{V}_{A0} \partial t}$$
+$$\frac{\partial n_a}{\partial t} = \frac{\pi 1 \partial \bar{d}^3}{6 \bar{V}_{A0} \partial t} \tag{45}$$
 we obtain:
-$$\frac{\partial \bar{d}^3}{\partial t} = \frac{3 n_A}{2 n_{A0}} c_F \left(\bar{d}_0 + \bar{d}\right)^2 \frac{\partial z}{\partial t}$$
+$$\frac{\partial \bar{d}^3}{\partial t} = \frac{3 n_A}{2 n_{A0}} c_F \left(\bar{d}_0 + \bar{d}\right)^2 \frac{\partial z}{\partial t} \tag{46}$$
 Since the solid mass in the volume element does not change over time as the agglomerates coalesce, the ratio of the number of particles can be calculated from a mass balance:
-$$\frac{n_A}{n_{A0}} = \frac{\bar{d}^3}{\bar{d}_0^3}$$
+$$\frac{n_A}{n_{A0}} = \frac{\bar{d}^3}{\bar{d}_0^3} \tag{47}$$
 For two agglomerates to collide, it must also be ensured that there is a relative velocity between the two particles:
-$$\bar{v}_{z,rel} = \frac{\partial z}{\partial t}|_{avg} = \frac{1}{2} (v_2 - v_1)$$
+$$\bar{v}_{z,rel} = \frac{\partial z}{\partial t}|_{avg} = \frac{1}{2} (v_2 - v_1) \tag{48}$$
 If we now derive an expression for the shear velocity of the surrounding fluid
-$$\dot{\gamma} = \frac{\partial v_z}{\partial y} = \frac{2(v_2 - v_1)}{(\bar{d}_0 + \bar{d})}$$
+$$\dot{\gamma} = \frac{\partial v_z}{\partial y} = \frac{2(v_2 - v_1)}{(\bar{d}_0 + \bar{d})} \tag{49}$$
 and substitute this, we obtain the average relative velocity:
-$$\frac{\partial z}{\partial t}|_{avg} = \bar{v}_{z,rel} = \frac{1}{4} \dot{\gamma} (\bar{d} + \bar{d}_0)$$
+$$\frac{\partial z}{\partial t}|_{avg} = \bar{v}_{z,rel} = \frac{1}{4} \dot{\gamma} (\bar{d} + \bar{d}_0) \tag{50}$$
 Substituting gives:
-$$\frac{\partial \bar{d}^3}{\partial t} = \frac{1 \bar{d}^3}{8 \bar{d}_0^3} c_F \dot{\gamma} \left(\bar{d}_0 + \bar{d}\right)^2$$
+$$\frac{\partial \bar{d}^3}{\partial t} = \frac{1 \bar{d}^3}{8 \bar{d}_0^3} c_F \dot{\gamma} \left(\bar{d}_0 + \bar{d}\right)^2 \tag{51}$$
-If, analogous to the dispersion analyses by Bolen and Colwell [[grundlagenhandbuch:berechnung_der_dispergierguete#literatur |[BC58]]], it is assumed that only two agglomerates adhere to each other at any one time, the diameter of the agglomerate after cluster formation is given by:
+If, analogous to the dispersion analyses by Bolen and Colwell [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[BC58]]], it is assumed that only two agglomerates adhere to each other at any one time, the diameter of the agglomerate after cluster formation is given by:
-$$\bar{d} = \sqrt[3]{2} \bar{d}_0$$
+$$\bar{d} = \sqrt[3]{2} \bar{d}_0 \tag{52}$$
 Finally, introducing the simplification yields the differential equation describing the cluster formation process:
-$$\frac{\partial \bar{d}}{\partial t} = \frac{1}{8} \left(1 + \sqrt[3]{2}\right) c_F \dot{\gamma} \bar{d}$$
+$$\frac{\partial \bar{d}}{\partial t} = \frac{1}{8} \left(1 + \sqrt[3]{2}\right) c_F \dot{\gamma} \bar{d} \tag{53}$$
 ===== Superimposition of Clustering and Breakdown Models =====
@@ Zeile 228: / Zeile 272: @@
 A closed-form solution incorporating all three cases can now be obtained from the superposition of the individual processes. If the differential equations describing cluster formation and agglomerate disintegration are superimposed, a differential change in diameter with time results from the sum of the changes in the individual processes:
-$$\frac{\partial \bar{d}}{\partial t} = \left(\frac{\partial \bar{d}}{\partial t}\right)_{cluster formation} - \left(\frac{\partial \bar{d}}{\partial t}\right)_{disintegration} - \left(\frac{\partial \bar{d}}{\partial t}\right)_{erosion}$$
+$$\frac{\partial \bar{d}}{\partial t} = \left(\frac{\partial \bar{d}}{\partial t}\right)_{cluster formation} - \left(\frac{\partial \bar{d}}{\partial t}\right)_{disintegration} - \left(\frac{\partial \bar{d}}{\partial t}\right)_{erosion} \tag{54}$$
 To solve the equation, the following dimensionless parameters are defined, which result from the individual processes:
@@ Zeile 234: / Zeile 278: @@
 **Dimensionless mean agglomerate diameter:**
-$$d^x = \frac{\bar{d}}{\bar{d}_0}$$
+$$d^x = \frac{\bar{d}}{\bar{d}_0} \tag{55}$$
 **Dimensionless exposure time:**
-$$t^x = \dot{\gamma} t$$
+$$t^x = \dot{\gamma} t \tag{56}$$
 **Disintegration constant:**
-$$C_{Ze} = \frac{5}{128 \pi}$$
+$$C_{Ze} = \frac{5}{128 \pi} \tag{57}$$
 **Cluster formation constant:**
-$$C_{Cl} = \frac{1}{8} \left(1 + \sqrt[3]{2}\right)$$
+$$C_{Cl} = \frac{1}{8} \left(1 + \sqrt[3]{2}\right) \tag{58}$$
 **Erosion constant:**
-$$C_{Er} = \frac{5}{256 (1 - \psi_F) \pi}$$
+$$C_{Er} = \frac{5}{256 (1 - \psi_F) \pi} \tag{59}$$
 **Dimensionless stress:**
-$$\pi_B = \frac{\tau_{Fluid}}{\tau_{Sch}}$$
+$$\pi_B = \frac{\tau_{Fluid}}{\tau_{Sch}} \tag{60}$$
 Applying the dimensionless parameters yields the normalised differential equation:
-$$\frac{\partial d^x}{\partial t^x} = C_{Cl} c_F d^x - (C_{Er} + C_{Ze}) \pi_B d^x$$
+$$\frac{\partial d^x}{\partial t^x} = C_{Cl} c_F d^x - (C_{Er} + C_{Ze}) \pi_B d^x \tag{61}$$
 With the boundary condition that the initial agglomerate diameter
-$$d^x (t = t_0 = 0) = 1$$
+$$d^x (t = t_0 = 0) = 1 \tag{62}$$
 is, the solution is
-$$d^x(t^x) = \exp\{(C_{Cl} c_F - (C_{Er} + C_{Ze}) \pi_B) t^x\}$$
+$$d^x(t^x) = \exp\{(C_{Cl} c_F - (C_{Er} + C_{Ze}) \pi_B) t^x\} \tag{63}$$
-When applying this relationship, the distinction between cases defined above must be taken into account. If the critical shear stresses for erosion or agglomerate fragmentation are not reached during application, the respective dimensionless constants $C_{Ze}$ and $C_{Er}$ in the equations must be set to zero. The limiting shear stresses depend on the specific application (polymer-filler combination) and cannot be specified in general terms [[grundlagenhandbuch:berechnung_der_dispergierguete#literatur |[Rum75]]], [[grundlagenhandbuch:berechnung_der_dispergierguete#literatur |[Kre64]]]. In addition to the yield shear stress, according to Martin [[grundlagenhandbuch:berechnung_der_dispergierguete#literatur |[Mar72]]], a minimum stress duration must be taken into account depending on the specific application, below which no fragmentation can occur even if the yield shear stress is exceeded.
+When applying this relationship, the distinction between cases defined above must be taken into account. If the critical shear stresses for erosion or agglomerate fragmentation are not reached during application, the respective dimensionless constants $C_{Ze}$ and $C_{Er}$ in the equations must be set to zero. The limiting shear stresses depend on the specific application (polymer-filler combination) and cannot be specified in general terms [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Rum75]]], [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Kre64]]]. In addition to the yield shear stress, according to Martin [[en:grundlagenhandbuch:berechnung_der_dispergierguete#references |[Mar72]]], a minimum stress duration must be taken into account depending on the specific application, below which no fragmentation can occur even if the yield shear stress is exceeded.
@@ Zeile 289: / Zeile 333: @@
 In order to discuss the results further, it is necessary at this point to define the critical solid volume concentration. Setting $d^*(t^*)=const.=1$ describes the critical solids concentration, at which there is neither an increase nor a decrease in the diameter of the particle.
-$$C_{F,krit} = \frac{C_{Er} + C_{Ze}}{C_{Cl}} \pi_B \tag{1}$$
+$$C_{F,krit} = \frac{C_{Er} + C_{Ze}}{C_{Cl}} \pi_B \tag{64}$$
 The diagram in the figure shows the critical solids concentration against the dimensionless stress and the agglomerate porosity. The area in which the agglomeration dominates the deagglomeration is situated on the upper side of the plot.
@@ Zeile 297: / Zeile 341: @@
 **Figure:** Critical solid volume concentration vs. dimensionless stress.
-Taking into account the different densities of polymers and fillers, the solids weight content is usually considerably higher than the volume content. It can be assumed the agglomerate break down will dominate the agglomerate clustering in most applications. This was taking into account the influence of the area void fraction.
+Here, too, it is theoretically possible that cluster formation outweighs agglomerate fragmentation. It can be seen that the region in which agglomerate aggregation dominates reaches the physical maximum value of the critical volume concentration even at a very low dimensionless stress. If one takes into account that, due to the different densities of the filler and the plastic melt, the mass concentration is significantly greater than the volume concentration, it can be assumed that, in this case, agglomerate fragmentation dominates over cluster formation and agglomerate breakdown effectively occurs. The porosity of the agglomerates, which is incorporated into the erosion constant, also plays a role in this context.
-The model describes the effects described in the literature (erosion, agglomerate rupture, agglomerate clustering).
+The physical model is capable of describing the effects of dispersion (erosion, agglomerate fragmentation, agglomeration) described in the literature by superimposing the physical models of the individual effects, which were also developed in this work.
 ===== References =====