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G. Fragnière et al.
Here n P is the number of particles per suspension volume, ϕ gm the grinding bead
filling degree and ∈ the grinding bead porosity. The collision frequency and, thus,
the load frequency are proportional to the stirrer speed and number of grinding
media. The capture probability is calculated using an “active volume” between two
grinding media during a contact. A further assumption is that in the actual size and
concentration range only one particle is trapped between two grinding media and
significantly stressed.
Using this approach, the breakage rate was determined for different grinding
media sizes and stirrer speeds as well as two different stirrer geometries and solids
concentrations. Figure 13 shows very clearly that the stress model well describes the
influence of changes in operating parameters on the breakage rate in batch operation
per stirrer geometry. However, a simple transfer to another stirrer geometry is not
possible, which is usually unproblematic since the stirrer geometry is normally not a
variable quantity in the process. Furthermore, the model approach for the breakage
rate according to the stress model does not allow predicting the absolute value of the
breakage rate from scratch. Few experiments with the used material in the respective
mill are necessary to determine the location parameters of the linear function of the
breakage rate.
Approach via mean values from CFD-DEM simulations: The collision energy
distribution and the collision number were calculated for various operating parameters using CFD-DEM simulations and the result was compared with the characteristic
values of the stress model [7]. It was shown that simulated grinding media contacts
and collision energies do not show exactly the same dependencies on the operating
parameters as defined for the stress model. For the product of the two parameters SE
and SF, however, the dependency is approximately the same. This is an indication of
why the combination of stress energy and stress frequency shows good correlations
with experimentally measured breakage rates. It should be noted that the capture
probability was not determined from the DEM-CFD simulations. A strong correlation between the simulation results and the experimental breakage rates could be
shown (Fig. 14). The breakage rate of a product particle size class can be described
by a linear function. The smaller particles exhibit higher strength; therefore, their
breakage rate is lower. No material parameter is included in the index in Fig. 14, so
two curves can be seen. The integration of material values into the model is shown
in the next section.
Approach via CFD-DEM simulation and material function: The CFD-DEM
simulations provide complete stress energy distributions that are characteristic for
different mills and operating parameters. Tavares and Carvalho [22] consider in their
approach for dry ball mills the stress energy distribution of the mill as well as the
particle breakage energy distribution through the convolution of the two distributions.
In combination with the collision frequency and the capture probability, this results
in the breakage rate. For validation of this model for stirred media mills, yeast cells
were used as test material [27]. The yeast cells have the advantage that the burst or
breakage energy can be adjusted by the osmotic conditions and that they are almost
monodisperse with a particle size of ~5 μm. In addition, yeast cells show no breakage
function after digestion and the burst or breakage rate can be clearly determined
G. Fragnière et al.
Here n P is the number of particles per suspension volume, ϕ gm the grinding bead
filling degree and ∈ the grinding bead porosity. The collision frequency and, thus,
the load frequency are proportional to the stirrer speed and number of grinding
media. The capture probability is calculated using an “active volume” between two
grinding media during a contact. A further assumption is that in the actual size and
concentration range only one particle is trapped between two grinding media and
significantly stressed.
Using this approach, the breakage rate was determined for different grinding
media sizes and stirrer speeds as well as two different stirrer geometries and solids
concentrations. Figure 13 shows very clearly that the stress model well describes the
influence of changes in operating parameters on the breakage rate in batch operation
per stirrer geometry. However, a simple transfer to another stirrer geometry is not
possible, which is usually unproblematic since the stirrer geometry is normally not a
variable quantity in the process. Furthermore, the model approach for the breakage
rate according to the stress model does not allow predicting the absolute value of the
breakage rate from scratch. Few experiments with the used material in the respective
mill are necessary to determine the location parameters of the linear function of the
breakage rate.
Approach via mean values from CFD-DEM simulations: The collision energy
distribution and the collision number were calculated for various operating parameters using CFD-DEM simulations and the result was compared with the characteristic
values of the stress model [7]. It was shown that simulated grinding media contacts
and collision energies do not show exactly the same dependencies on the operating
parameters as defined for the stress model. For the product of the two parameters SE
and SF, however, the dependency is approximately the same. This is an indication of
why the combination of stress energy and stress frequency shows good correlations
with experimentally measured breakage rates. It should be noted that the capture
probability was not determined from the DEM-CFD simulations. A strong correlation between the simulation results and the experimental breakage rates could be
shown (Fig. 14). The breakage rate of a product particle size class can be described
by a linear function. The smaller particles exhibit higher strength; therefore, their
breakage rate is lower. No material parameter is included in the index in Fig. 14, so
two curves can be seen. The integration of material values into the model is shown
in the next section.
Approach via CFD-DEM simulation and material function: The CFD-DEM
simulations provide complete stress energy distributions that are characteristic for
different mills and operating parameters. Tavares and Carvalho [22] consider in their
approach for dry ball mills the stress energy distribution of the mill as well as the
particle breakage energy distribution through the convolution of the two distributions.
In combination with the collision frequency and the capture probability, this results
in the breakage rate. For validation of this model for stirred media mills, yeast cells
were used as test material [27]. The yeast cells have the advantage that the burst or
breakage energy can be adjusted by the osmotic conditions and that they are almost
monodisperse with a particle size of ~5 μm. In addition, yeast cells show no breakage
function after digestion and the burst or breakage rate can be clearly determined