Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen RevisionVorhergehende ÜberarbeitungNächste Überarbeitung | Vorhergehende Überarbeitung | ||
| en:grundlagenhandbuch:berechnung_der_dispergierguete [2026/05/12 20:38] – [Erosion] 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: | 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: | ||
| - | $$\sigma_M = \frac{\dot{V}_x}{\dot{V}_z}$$ | + | $$\sigma_M = \frac{\dot{V}_x}{\dot{V}_z} \tag{1}$$ |
| With the change in surface area over time: | 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: | 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: | According to Rumpf [[en: | ||
| - | $$\sigma = \sum_{n=1}^{n_p} \sum_{k=1}^{k} \frac{F_H}{d} = n_p k \frac{\overline{F_H}}{d}$$ | + | $$\sigma = \sum_{n=1}^{n_p} \sum_{k=1}^{k} \frac{F_H}{d} = n_p k \frac{\overline{F_H}}{d} \tag{4}$$ |
| 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. | 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: | 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: | 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: | ||
| - | $$\omega = \frac{1}{2} \dot{\gamma} \tag{1}$$ | + | $$\omega = \frac{1}{2} \dot{\gamma} |
| - | The power at the sphere’s surface, as shown in Figure 10.2, is formulated as follows according to Raasch [[grundlagenhandbuch: | + | The power at the sphere’s surface, as shown in Figure 10.2, is formulated as follows according to Raasch [[en:grundlagenhandbuch: |
| - | $$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}$$ |
| The following applies to the surface differential: | The following applies to the surface differential: | ||
| - | $$dA = r^2 \cos \vartheta \, d\vartheta d\varphi$$ | + | $$dA = r^2 \cos \vartheta \, d\vartheta d\varphi |
| {{ : | {{ : | ||
| - | **Figure:** Linear shear flow around a sphere [[basics_manual:calculation_of_dispersion_quality#literature | + | **Figure:** Linear shear flow around a sphere [[en:grundlagenhandbuch: |
| - | 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 | + | The stresses arising on the sphere’s surface during the rotation shown about the z-axis are (cf. [[en:grundlagenhandbuch: |
| - | $$\sigma_r = \frac{5}{2} \eta \dot{\gamma} \sin 2\varphi \cos^2 \vartheta$$ | + | $$\sigma_r = \frac{5}{2} \eta \dot{\gamma} \sin 2\varphi \cos^2 \vartheta |
| - | $$\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 |
| - | $$\tau_{r\vartheta} = -\frac{5}{2} \eta \dot{\gamma} \sin 2\varphi \sin \vartheta \cos \vartheta$$ | + | $$\tau_{r\vartheta} = -\frac{5}{2} \eta \dot{\gamma} \sin 2\varphi \sin \vartheta \cos \vartheta |
| The following applies to the velocities: | The following applies to the velocities: | ||
| - | $$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 |
| - | $$V_\varphi = \frac{\dot{\gamma}}{4}\left[\left(r - \frac{r_0^5}{r^4}\right) \cos 2\varphi - r\right] \cos \vartheta$$ | + | $$V_\varphi = \frac{\dot{\gamma}}{4}\left[\left(r - \frac{r_0^5}{r^4}\right) \cos 2\varphi - r\right] \cos \vartheta |
| - | $$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 |
| 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: | 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: | ||
| - | $$P = \int_{A_{upper}} \left[\frac{5}{2} \eta \dot{\gamma} \cos 2\varphi \cos \vartheta\right] \left[-\frac{\dot{\gamma}}{2} r_0 \cos \vartheta\right] r_0^2 \cos \vartheta \, d\vartheta d\varphi$$ | + | $$P = \int_{A_{upper}} \left[\frac{5}{2} \eta \dot{\gamma} \cos 2\varphi \cos \vartheta\right] \left[-\frac{\dot{\gamma}}{2} r_0 \cos \vartheta\right] r_0^2 \cos \vartheta \, d\vartheta d\varphi |
| The figure shows the power distribution on the spherical surface. | The figure shows the power distribution on the spherical surface. | ||
| Zeile 77: | Zeile 77: | ||
| **Figure:** Power distribution on the spherical surface | **Figure:** Power distribution on the spherical surface | ||
| - | For the process of spontaneous disintegration, | + | For the process of spontaneous disintegration, |
| - | The maximum possible shear stress that can occur on the sphere’s surface is given by Eq. (10.16): | + | The maximum possible shear stress that can occur on the sphere’s surface is given by Eq. (16): |
| - | $$\tau_{max} = \frac{5}{2} \eta_{Fluid} \dot{\gamma}$$ | + | $$\tau_{max} = \frac{5}{2} \eta_{Fluid} \dot{\gamma} \tag{16}$$ |
| The circumferential velocity is given by the equation: | The circumferential velocity is given by the equation: | ||
| - | $$V_\varphi = r_0 \omega = r_0 \frac{\dot{\gamma}}{2}$$ | + | $$V_\varphi = r_0 \omega = r_0 \frac{\dot{\gamma}}{2} \tag{17}$$ |
| As a result of these considerations, | As a result of these considerations, | ||
| - | $$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 |
| Equating the two power expressions yields the following differential equation: | 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, | + | If we consider the interface of the agglomerate, |
| - | $$\psi_F = \frac{A_H}{A} = 1 - \frac{n_p A_p}{A}$$ | + | $$\psi_F = \frac{A_H}{A} = 1 - \frac{n_p A_p}{A} \tag{20}$$ |
| 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: | 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: | 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, | 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, | ||
| - | 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 | + | 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: |
| {{ : | {{ : | ||
| Zeile 117: | Zeile 117: | ||
| Due to the high pressures occurring in continuous mixers, shear failures are more than likely to occur. From this follows: | 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' | 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' | ||
| - | $$\tau_{Sch} = \frac{1}{2} \sigma_z = \frac{(1 - \varepsilon_A) \overline{F_{adh}}}{2\pi \cdot d_p^2} \tag{11}$$ | + | $$\tau_{Sch} = \frac{1}{2} \sigma_z = \frac{(1 - \varepsilon_A) \overline{F_{adh}}}{2\pi \cdot d_p^2} \tag{24}$$ |
| Inserting the agglomerate strength according to Rumpf (eqn. (11) results in the following differential equation for the agglomerate rupture: | 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, | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | **Figure:** Schematic representation of erosion | ||
| + | |||
| + | The power required to detach individual particles from an agglomerate results, as with fragmentation, | ||
| + | |||
| + | $$P = \sigma \frac{\partial A}{\partial t} \tag{26}$$ | ||
| + | |||
| + | 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, | ||
| + | |||
| + | $$n_0 \pi \frac{d_p^2}{2} \cong (1 - \psi_F) \pi \bar{d}^2 \tag{28}$$ | ||
| + | |||
| + | $$n_0 = \frac{2(1 - \psi_F) \bar{d}^2}{d_p^2} \tag{29}$$ | ||
| + | |||
| + | 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, | ||
| + | $$\frac{5}{32} \tau_{Fluid} \dot{\gamma} \bar{d} = 2n_0 k \overline{F_H} (1 - \psi_F) \frac{\partial \bar{d}}{\partial t} \tag{33}$$ | ||
| + | |||
| + | 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 ===== | ===== 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: | 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: | ||
| - | $$S_{üb} = \frac{\pi}{4} (d_M + d)^2$$ | + | $$S_{üb} = \frac{\pi}{4} (d_M + d)^2 \tag{36}$$ |
| {{ : | {{ : | ||
| 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: | 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, | In relation to the monitoring cross-section, | ||
| - | $$\partial n_M = \frac{\partial n_A S_{mon}}{S_D}$$ | + | $$\partial n_M = \frac{\partial n_A S_{mon}}{S_D} \tag{38}$$ |
| Substituting the equations, we obtain: | 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 | By substituting the solid concentration | ||
| - | $$c_F = \frac{V_F}{V}$$ | + | $$c_F = \frac{V_F}{V} \tag{40}$$ |
| and the solid volume | 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: | we obtain an expression for the rate of change of the number of particles in the volume element: | ||
| - | $$\frac{\partial n_M}{\partial t} = \frac{\partial}{\partial t} \left(\frac{n_A \pi \left(\bar{d}_0 + \bar{d}\right)^2}{n_{A0} 4 V_{A0}} c_F \partial z\right)$$ | + | $$\frac{\partial n_M}{\partial t} = \frac{\partial}{\partial t} \left(\frac{n_A \pi \left(\bar{d}_0 + \bar{d}\right)^2}{n_{A0} 4 V_{A0}} c_F \partial z\right) |
| The number $n_a$ of particles participating in the agglomeration is given by the ratio of the particle volume at time $t > t_0$ to the particle volume at time $t = t_0$: | The number $n_a$ of particles participating in the agglomeration is given by the ratio of the particle volume at time $t > t_0$ to the particle volume at time $t = t_0$: | ||
| - | $$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}$$ |
| Since the essential process in the agglomeration of particles is collision transfer, the change in the number of particles undergoing a collision must be equal to the change in the number of particles participating in the agglomeration. This yields: | Since the essential process in the agglomeration of particles is collision transfer, the change in the number of particles undergoing a collision must be equal to the change in the number of particles participating in the agglomeration. This yields: | ||
| - | $$\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}$$ |
| Substituting | Substituting | ||
| - | $$\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}$$ |
| we obtain: | we obtain: | ||
| - | $$\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}$$ |
| Since the solid mass in the volume element does not change over time as the agglomerates coalesce, the ratio of the number of particles can be calculated from a mass balance: | Since the solid mass in the volume element does not change over time as the agglomerates coalesce, the ratio of the number of particles can be calculated from a mass balance: | ||
| - | $$\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}$$ |
| For two agglomerates to collide, it must also be ensured that there is a relative velocity between the two particles: | For two agglomerates to collide, it must also be ensured that there is a relative velocity between the two particles: | ||
| - | $$\bar{v}_{z, | + | $$\bar{v}_{z, |
| If we now derive an expression for the shear velocity of the surrounding fluid | If we now derive an expression for the shear velocity of the surrounding fluid | ||
| - | $$\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}$$ |
| and substitute this, we obtain the average relative velocity: | and substitute this, we obtain the average relative velocity: | ||
| - | $$\frac{\partial z}{\partial t}|_{avg} = \bar{v}_{z, | + | $$\frac{\partial z}{\partial t}|_{avg} = \bar{v}_{z, |
| Substituting gives: | Substituting gives: | ||
| - | $$\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 |
| - | If, analogous to the dispersion analyses by Bolen and Colwell [[grundlagenhandbuch: | + | If, analogous to the dispersion analyses by Bolen and Colwell [[en:grundlagenhandbuch: |
| - | $$\bar{d} = \sqrt[3]{2} \bar{d}_0$$ | + | $$\bar{d} = \sqrt[3]{2} \bar{d}_0 |
| Finally, introducing the simplification yields the differential equation describing the cluster formation process: | Finally, introducing the simplification yields the differential equation describing the cluster formation process: | ||
| - | $$\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}$$ |
| - | + | ===== Superimposition | |
| - | ==== Agglomerate rupture | + | |
| - | + | ||
| - | Assuming, analogous to Bolen and Colwell [[en: | + | |
| - | + | ||
| - | The process of agglomerate rupture involves the creation of new interfaces between the agglomerate fragments and the surrounding polymer (see figure). | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | **Figure:** Schematic diagram of the agglomerate rupture. | + | |
| - | + | ||
| - | The work needed to increase the interface, can generally be defined for this process as: | + | |
| - | + | ||
| - | $$W = \int \sigma dA \tag{1}$$ | + | |
| - | + | ||
| - | where σ is the interfacial energy. The power needed to rupture the agglomerates follows from the differentiation of the required rupture work with the time: | + | |
| - | + | ||
| - | $$P = \frac{\partial W}{\partial t} = 2\sigma \pi \bar{d} \frac{\partial \bar{d}}{\partial t} \tag{2}$$ | + | |
| - | + | ||
| - | where $\bar{d}$ is the mean agglomerate diameter. | + | |
| - | + | ||
| - | The force necessary for the break down of an agglomerate in this fracture plane, must exceed the sum of the acting adhesion forces. | + | |
| - | + | ||
| - | Equation (3) shall define the interfacial tension as the sum of acting adhesive forces related to the particle diameter analogous to [[en: | + | |
| - | + | ||
| - | $$\sigma = \sum_{n=1}^{n_p} \sum_{k=1}^k \frac{F_{adh}}{\bar{d}} = n_p k \frac{\overline{F_{adh}}}{\bar{d}} \tag{3}$$ | + | |
| - | + | ||
| - | with np being the number of primary particles in the fracture plane and k the number of contact points of the primary particles in the fracture plane. | + | |
| - | + | ||
| - | Inserting eqn. (3) into eqn. (2) results therefore in a description of the minimum power required to rupture agglomerates: | + | |
| - | + | ||
| - | $$P_{min} = \pi n_p k \overline{F_{adh}} \frac{\partial \bar{d}}{\partial t} \tag{4}$$ | + | |
| - | + | ||
| - | Equating (4) and (13) results in: | + | |
| - | + | ||
| - | $$\frac{5}{32} \cdot \tau_{Fluid} \cdot \dot{\gamma} \cdot \pi \cdot \bar{d}^3 = \pi \cdot n_p k \overline{F_{adh}} \frac{\partial \bar{d}}{\partial t} \tag{5}$$ | + | |
| - | + | ||
| - | Looking at the fracture plane of the agglomerate, | + | |
| - | + | ||
| - | $$\varepsilon_A = \frac{A_V}{A} = 1 - \frac{n_p \cdot A_p}{A} \tag{6}$$ | + | |
| - | + | ||
| - | Equation (6) allows the formulation of an equation to calculate the number of particles in the fracture plane which are dependent on the area void fraction: | + | |
| - | + | ||
| - | $$n_p = (1 - \varepsilon_A) \cdot \frac{A}{A_p} = (1 - \varepsilon_A) \cdot \frac{\bar{d}^2}{d_p^2} \tag{7}$$ | + | |
| - | + | ||
| - | Inserting (7) in (5) leads to the following equation: | + | |
| - | + | ||
| - | $$\frac{5}{32} \tau_{Fluid} \cdot \dot{\gamma} \cdot \bar{d} = (1 - \varepsilon_A) \cdot \frac{1}{d_p^2} \cdot k \cdot \overline{F_{adh}} \cdot \frac{\partial \bar{d}}{\partial t} \tag{8}$$ | + | |
| - | + | ||
| - | The model is so far only valid for cleavages in simple shear flow. Usually the flow in an extruder can not be described using the assumption of a simple shear flow. | + | |
| - | + | ||
| - | The problem is finding a material parameter to describe the strength of an agglomerate. Tensile strengths can only be used for cleavages. For cleavages occurring normal to the principle stress direction, the maximum principle stress criterion can be used. This criterion can not be used here, since the pressures occurring within a continuous mixer are usually greater than the agglomerate strength. We propose to apply Mohr's criterion. It assumes different critical strengths and uses the enveloping lines, Mohr's circles that correspond to the limiting stresses. From this criterion follows: | + | |
| - | + | ||
| - | $$\tau = f(\sigma) \tag{9}$$ [[en: | + | |
| - | + | ||
| - | + | ||
| - | Since there is usually insufficient data, the enveloping lines are replaced by three lines (see figure) | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | **Figure:** Limiting agglomerate strength according to Mohr [[en: | + | |
| - | + | ||
| - | Due to the high pressures occurring in continuous mixers, shear failures are more than likely to occur. From this follows: | + | |
| - | + | ||
| - | $$\tau_{Sch} = \frac{1}{2} \sigma_v = \sigma_3 - \sigma_1 \tag{10}$$ | + | |
| - | + | ||
| - | With σ1 and σ3 being the maximum principle stresses. We assume the hoop stress σv to be almost equal to the tensile strength σz, which we predict using Rumpf' | + | |
| - | + | ||
| - | $$\tau_{Sch} = \frac{1}{2} \sigma_z = \frac{(1 - \varepsilon_A) \overline{F_{adh}}}{2\pi \cdot d_p^2} \tag{11}$$ | + | |
| - | + | ||
| - | Inserting the agglomerate strength according to Rumpf (eqn. (11) results in the following differential equation for the agglomerate rupture: | + | |
| - | + | ||
| - | $$\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}$$ | + | |
| - | + | ||
| - | ==== Erosion ==== | + | |
| - | + | ||
| - | In order for primary particles to separate from the agglomerate, | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | **Figure:** Schematic diagram of erosion | + | |
| - | + | ||
| - | The work necessary to separate individual particles from an agglomerate, | + | |
| - | + | ||
| - | $$P = \sigma \frac{\partial A}{\partial t} \tag{1}$$ | + | |
| - | + | ||
| - | With the surface of an agglomerate consisting of primary particles with the diameter dP: | + | |
| - | + | ||
| - | $$A = n_0 \pi \frac{d_p^2}{2} \tag{2}$$ | + | |
| - | + | ||
| - | For the calculation of the number of primary particles n0 on the surface of the agglomerates, | + | |
| - | + | ||
| - | $$n_0 \pi \frac{d_p^2}{2} \cong (1 - \psi_F) \pi \bar{d}^2 \tag{3}$$ | + | |
| - | + | ||
| - | $$n_0 = \frac{2(1 - \psi_F) \bar{d}^2}{d_p^2} \tag{4}$$ | + | |
| - | + | ||
| - | It therefore follows for the time-dependent change of the surface of an agglomerate, | + | |
| - | + | ||
| - | $$\frac{\partial A}{\partial t} = \pi (1 - \psi_F) \bar{d} \frac{\partial \bar{d}}{\partial t} \tag{5}$$ | + | |
| - | + | ||
| - | The minimum power required to erode particles is calculated using: | + | |
| - | + | ||
| - | $$P_{min} = \sigma \pi (1 - \psi_F) \bar{d} \frac{\partial \bar{d}}{\partial t} \tag{6}$$ | + | |
| - | + | ||
| - | The minimum power needed to separate all primary particles from the outer shell of the surface of an agglomerate can therefore be calculated by equating (13) and (6): | + | |
| - | + | ||
| - | $$\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{7}$$ | + | |
| - | + | ||
| - | If, analogous to the rupture, equation (3) is inserted in equation (8), the result is: | + | |
| - | + | ||
| - | $$\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{8}$$ | + | |
| - | + | ||
| - | With the number of agglomerates (eqn. 4) the result is: | + | |
| - | + | ||
| - | $$\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{9}$$ | + | |
| - | + | ||
| - | Inserting the defined tensile strength (eqn. 11), according to Rumpf [[en: | + | |
| - | + | ||
| - | $$\frac{\partial \bar{d}}{\partial t} = \frac{5}{256 \pi (1 - \psi_F) \tau_{Sch}} \cdot \tau_{Fluid} \dot{\gamma} \cdot \bar{d} \tag{10}$$ | + | |
| - | + | ||
| - | ===== Clustering | + | |
| - | + | ||
| - | Clustering is essentially the collision of particles. In order for particles to collide, two conditions must be satisfied: | + | |
| - | - The particles must move at different velocities. | + | |
| - | - Assuming that all the particles are approximately spherical, the distance between the particles must be smaller than or the same as the sum of their radii. | + | |
| - | + | ||
| - | Assuming that the agglomerates have similar mean diameters, the second condition mentioned above results in a monitoring cross-section, | + | |
| - | + | ||
| - | $$S_{mon} = \frac{\pi}{4} \left(\bar{d}_M + \bar{d}\right)^2 \tag{1}$$ | + | |
| - | + | ||
| - | {{ : | + | |
| - | + | ||
| - | **Figure:** Definition of the monitoring cross section | + | |
| - | + | ||
| - | For the number of agglomerates ¶nA it is still possible to define the following equation for a differential volume element (see figure): | + | |
| - | + | ||
| - | $$\partial n_A = n_A \cdot \frac{S_D \cdot \partial z}{V} = n_A \cdot \frac{\left(\bar{d}_M - \bar{d}\right)^2}{V} \cdot d_z \tag{2}$$ | + | |
| - | + | ||
| - | Related to the monitoring cross section we can get a differential number of agglomerates ¶nM that can experience a collision: | + | |
| - | + | ||
| - | $$\partial n_M = \frac{\partial n_A \cdot S_{mon}}{S_D} = n_A \cdot \frac{\left(\bar{d}_M - \bar{d}\right)^2}{V} \cdot d_z \tag{3}$$ | + | |
| - | + | ||
| - | with the solids volume concentration: | + | |
| - | + | ||
| - | $$c_{so} = \frac{V_F}{V} \tag{4}$$ | + | |
| - | + | ||
| - | and the total volume of the filler: | + | |
| - | + | ||
| - | $$V_F = n_{A,0} \cdot V_{A,0} \tag{5}$$ | + | |
| - | + | ||
| - | We get a description for the change of the differential number of particles experiencing a collision in a differentially small time period ∂t. | + | |
| - | + | ||
| - | $$\frac{\partial n_M}{\partial t} = \frac{\partial n_M}{\partial t} \left(\frac{n_A}{n_{A, | + | |
| - | + | ||
| - | Since the solid matter in the volume element does not change over the time during the clustering of the agglomerates, | + | |
| - | + | ||
| - | $$\frac{n_A}{n_{A, | + | |
| - | + | ||
| - | The number of particles involved in the clustering process, is derived from the relationship between the particle volume at the time t > to and the particle volume at the time t = to to: | + | |
| - | + | ||
| - | $$n_a = \frac{\nabla_A}{\nabla_{A, | + | |
| - | + | ||
| - | Differentiating eqn (8) over the time we now reach: | + | |
| - | + | ||
| - | $$\frac{\partial n_a}{\partial t} = \left(\frac{\pi}{6} \cdot \frac{1}{\nabla_{A, | + | |
| - | + | ||
| - | Since the clustering of particles is essentially combined with an impact transmission, | + | |
| - | + | ||
| - | $$\frac{\partial n_M}{\partial t} = \frac{\partial n_a}{\partial t} \tag{10}$$ | + | |
| - | + | ||
| - | Inserting eqn (6) and (9) in (10) we get: | + | |
| - | + | ||
| - | $$\frac{\partial \bar{d}^3}{\partial t} = \frac{3}{2} \cdot \left(\frac{\bar{d}}{\bar{d}_0}\right) \cdot \left(\bar{d} - \bar{d}_0\right) \cdot c_{so} \cdot \frac{\partial z}{\partial t} \tag{11}$$ | + | |
| - | + | ||
| - | Where ¶z/¶t is the average relative velocity vz,rel between the agglomerates: | + | |
| - | + | ||
| - | $$\bar{v}_{z, | + | |
| - | + | ||
| - | With eqn. (12) and (11) we get: | + | |
| - | + | ||
| - | $$\frac{\partial \bar{d}^3}{\partial t} = \frac{1}{8} \frac{\bar{d}^3}{\bar{d}_0^3} c_F \dot{\gamma} \left(\bar{d}_0 + \bar{d}\right)^3 \tag{13}$$ | + | |
| - | + | ||
| - | Assuming now, analogous to the rupture investigations by Bolen and Colwell [[en: | + | |
| - | + | ||
| - | $$\bar{d} | + | |
| - | + | ||
| - | Finally, introducing the simplification in equation (14) into equation (13), results in a differential equation which describes the agglomeration process: | + | |
| - | + | ||
| - | $$\frac{\partial \bar{d}}{\partial t} = \frac{1}{8} \left(1 + \sqrt[3]{2}\right) c_F \dot{\gamma} \bar{d} \tag{15}$$ | + | |
| Which of the three cases discussed above occurs in a given section of a processing machine depends essentially on the shear stress level and the strength of the agglomerates. Three cases must be distinguished here: | Which of the three cases discussed above occurs in a given section of a processing machine depends essentially on the shear stress level and the strength of the agglomerates. Three cases must be distinguished here: | ||
| Zeile 416: | Zeile 272: | ||
| A closed-form solution incorporating all three cases can now be obtained from the superposition of the individual processes. If the differential equations describing cluster formation and agglomerate disintegration are superimposed, | A closed-form solution incorporating all three cases can now be obtained from the superposition of the individual processes. If the differential equations describing cluster formation and agglomerate disintegration are superimposed, | ||
| - | $$\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}$$ |
| To solve the equation, the following dimensionless parameters are defined, which result from the individual processes: | To solve the equation, the following dimensionless parameters are defined, which result from the individual processes: | ||
| Zeile 422: | Zeile 278: | ||
| **Dimensionless mean agglomerate diameter:** | **Dimensionless mean agglomerate diameter:** | ||
| - | $$d^x = \frac{\bar{d}}{\bar{d}_0}$$ | + | $$d^x = \frac{\bar{d}}{\bar{d}_0} \tag{55}$$ |
| **Dimensionless exposure time:** | **Dimensionless exposure time:** | ||
| - | $$t^x = \dot{\gamma} t$$ | + | $$t^x = \dot{\gamma} t \tag{56}$$ |
| **Disintegration constant:** | **Disintegration constant:** | ||
| - | $$C_{Ze} = \frac{5}{128 \pi}$$ | + | $$C_{Ze} = \frac{5}{128 \pi} \tag{57}$$ |
| **Cluster formation constant:** | **Cluster formation constant:** | ||
| - | $$C_{Cl} = \frac{1}{8} \left(1 + \sqrt[3]{2}\right)$$ | + | $$C_{Cl} = \frac{1}{8} \left(1 + \sqrt[3]{2}\right) |
| **Erosion constant:** | **Erosion constant:** | ||
| - | $$C_{Er} = \frac{5}{256 (1 - \psi_F) \pi}$$ | + | $$C_{Er} = \frac{5}{256 (1 - \psi_F) \pi} \tag{59}$$ |
| **Dimensionless stress:** | **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: | 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 | 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 | 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, | + | When applying this relationship, |
| - | ===== Superimposition of Clustering and Breakdown Models ===== | + | |
| - | A complete solution, including all three cases, is achieved by a superimposition of the individual processes. | ||
| - | The level of shear stress, the residence time and the tensile strength of the agglomerates determine which of the three cases (see above) take place in an designated section of a continuous mixer. The three cases must be distinguished: | ||
| - | * If the observed shear stresses are lower than the critical shear stress for erosion, there will be no deagglomeration. Agglomerates will cluster. The mean agglomerate diameter will increase over time. This will be the case, if the shear stresses are very low or if the tensile strength of the agglomerates is very high. | ||
| - | * If the shear stress in the observed area is higher than the critical shear stress of erosion, but lower than the critical shear stress for rupture, single particles will erode from the agglomerate surface. However, there will be no rupturing of the agglomerate. The erosion process superimposes the clustering process. | ||
| - | * If the shear stress is higher than the critical shear stress for the agglomerate rupture, one can expect that the agglomerates will rupture and that particles will erode from the surface of the agglomerate. Here one can also expect that the deagglomeration process is superimposed by a clustering process. | ||
| - | By superimposing the differential equations for the description of the clustering, erosion and the rupture process, the result is a differential change of the agglomerate diameter over the time as a sum of the diameter changes: | ||
| - | $$\frac{\partial \bar{d}}{\partial t} = \left(\frac{\partial \bar{d}}{\partial t}\right)_{cluster formation} - \left(\frac{\partial \bar{d}}{\partial t}\right)_{dispersive} - \left(\frac{\partial \bar{d}}{\partial t}\right)_{erosion} \tag{1}$$ | ||
| - | For the solution of the differential equation, the following dimensionless characteristics, | ||
| - | |||
| - | Dimensionless mean agglomerate diameter: | ||
| - | |||
| - | $$d^x = \frac{\bar{d}}{\bar{d}_0} \tag{2}$$ | ||
| - | |||
| - | Dimensionless stress exposure time: | ||
| - | |||
| - | $$t^x = \dot{\gamma} t \tag{3}$$ | ||
| - | |||
| - | Rupture constant: | ||
| - | |||
| - | $$C_{Ze} = \frac{5}{128 \pi} \tag{4}$$ | ||
| - | |||
| - | Agglomeration constant: | ||
| - | |||
| - | $$C_{Cl} = \frac{1}{8} \left(1 + \sqrt[3]{2}\right) \tag{5}$$ | ||
| - | |||
| - | Erosion constant: | ||
| - | |||
| - | $$C_{Er} = \frac{5}{256 (1 - \psi_F) \pi} \tag{6}$$ | ||
| - | |||
| - | Dimensionless stress: | ||
| - | |||
| - | $$\pi_B = \frac{\tau_{Fluid}}{\tau_{Sch}} \tag{7}$$ | ||
| - | |||
| - | By applying the dimensionless characteristics, | ||
| - | |||
| - | $$\frac{\partial d^x}{\partial t^x} = C_{Cl} c_F d^x - (C_{Er} + C_{Ze}) \pi_B d^x \tag{8}$$ | ||
| - | |||
| - | Under the boundary condition that the initial agglomerate diameter is: | ||
| - | |||
| - | $$d^x (t = t_0 = 0) = 1 \tag{9}$$ | ||
| - | |||
| - | the solution of the differential equation results to: | ||
| - | |||
| - | $$d^x(t^x) = \exp\{(C_{Cl} c_F - (C_{Er} + C_{Ze}) \pi_B) t^x\} \tag{10}$$ | ||
| - | |||
| - | For the application of the relationship, | ||
| ===== Discussion and Results ===== | ===== Discussion and Results ===== | ||
| Zeile 511: | Zeile 320: | ||
| The figure shows the dimensionless mean diameter versus the dimensionless stress and the dimensionless exposure time for three different solids volume concentration. | The figure shows the dimensionless mean diameter versus the dimensionless stress and the dimensionless exposure time for three different solids volume concentration. | ||
| - | {{ : | ||
| - | |||
| - | **Figure:** Dimensionless agglomerate diameter vs. dimensionless time and dimensionless stress for different solid volume concentrations. | ||
| When looking at a particular case (see case 1) of the three diagrams, in which no agglomerate dispersion takes place, but the agglomerates form clusters, it becomes evident that the speed, at which the agglomerates form clusters is dependent solely on the solids volume concentration and the stress exposure time. | When looking at a particular case (see case 1) of the three diagrams, in which no agglomerate dispersion takes place, but the agglomerates form clusters, it becomes evident that the speed, at which the agglomerates form clusters is dependent solely on the solids volume concentration and the stress exposure time. | ||
| Zeile 520: | Zeile 326: | ||
| In the case of shear stresses above the critical shear stress for agglomerate rupture (see case 3), the agglomeration, | In the case of shear stresses above the critical shear stress for agglomerate rupture (see case 3), the agglomeration, | ||
| + | |||
| + | {{ : | ||
| + | |||
| + | **Figure:** Dimensionless agglomerate diameter vs. dimensionless time and dimensionless stress for different solid volume concentrations. | ||
| 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, | 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, | ||
| - | $$C_{F, | + | $$C_{F, |
| 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. | 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 531: | Zeile 341: | ||
| **Figure:** Critical solid volume concentration vs. dimensionless stress. | **Figure:** Critical solid volume concentration vs. dimensionless stress. | ||
| - | Taking | + | 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 |
| - | The model describes | + | The physical |
| ===== References ===== | ===== References ===== | ||