Abrasion Rate Constant (ξm) Measurements in the 25- and 50-L High Shear-Mixer
In this study, two vertical axis high-shear mixers of different geometries (as described in Table ) were used to assess how the abrasion rate constants (ξm
) of brittle agglomerates scale with process variables in high-shear mixers. The bCTPs mass reduction (Mrel
) over time was determined. It obeys apparent first order kinetics, with mass reduction rate constant as described by Willemsz et al.
as the mass after blending time t
as the initial mass.
The purpose of the current experiments was to investigate the effects of abrasion rate constants of agglomerates with different porosities when process variables are varied. The results are depicted in Fig. .
Fig. 1 The effect of process settings on the mass based abrasion rate constant for the 25-L (black square) and 50-L (white square) high-shear mixers. Fr represents the Froude number, the degree of fill of the equipment, ξm the abrasion rate (more ...)
Figure shows that the abrasion rate constants (ξm
) of the agglomerates in the 50-L mixer are always lower than those obtained in the 25-L mixer when Froude numbers are identical. Abrasion rates increase with Froude number but decrease with increasing fill levels. These results are in line with findings discussed in previous papers (e.g.
It is reasonable to assume that a reduction in fill volume implies that the contribution of the impeller to the total rate of agglomerate abrasion will increase. To visualize this effect additional tests have been performed where the powder just covers the impeller. This corresponds with a relative fill level of 8% in the 25-L mixer. Figure shows the abrasion rates (ξm) of the bCTPs at different fill levels for two different Fr numbers. Figure clearly shows a considerable additional effect of the impeller at low fill levels on the abrasion rates of the particles. Moreover, there seems to be a step change in behavior: abrasions rates at fill levels above 16% are more or less in line, a low fill level gives much higher abrasion rates.
Fig. 2 The mass based abrasion rate constants (ξm) of agglomerates for the 25-L high-shear mixer at a (v/v) of 8% (white circle), 16% (black circle), 27% (black square), and 37% (black diamond)
Powder Surface Velocity and Abrasion
The powder surface velocity has been measured as previously described (10
). The powder velocities (vp
) were determined at the conditions described in Table , and Fig. depicts the results.
Figure shows decreasing powder surface velocity at increasing relative fill volume. This was observed in both mixers. The figure also shows that the powder velocities measured in the 50-L mixer are significantly lower compared to those in the 25-L mixer scale at comparable Fr numbers. Filler particle velocity is an important parameter in relation to the rate of abrasion of the agglomerates (8
). The effects were correlated using the Stokes abrasion number (StAbr
). Applying StAbr
numbers gives the possibility to assess how the abrasion rate constant (ξm)
scales with process variables in different types/scale of high-shear mixers.
The Stokes abrasion number (StAbr
) concept has been discussed in more detail earlier in our previous paper (9
compares the energy density during blending (
) with the work of fracture of an agglomerate (
bulk density of the filler, vp
powder surface velocity, Y
elastic moduli, and σc
The mechanical properties Y
(elastic modulus) and σc
(fracture stress) of the bCTPs have been calculated as previously described (9
) and are based on the porosity (ε
) values of the agglomerates. Table summarizes the fit parameters for elastic moduli and fracture stresses after performing a least square fit analysis assuming exponential relationships (14
Figure shows the relationship between the abrasion rates and the Stokes abrasion numbers in the blending experiments at different working conditions. Visually, three distinct relationships can be seen: with the largest group of tests at fill levels larger than 16% and two groups that both describe relationships when fill level is low.
Fig. 4 The relationship between the abrasion rate constants (ξm) and the Stokes abrasion number (StAbr) of various bCTPs at defined conditions. The two solid lines indicate the abrasion data deviating from the proposed regression model (dashed line) (more ...)
Model diagnostic plots of the data-set indicate that both variables (ξm and StAbr) should be log-transformed before analysis to fulfill the statistical requirements for normal distribution of the values to identify outliers. From these data sets several outlier diagnostics (studentized residuals, DFFITTS, leverage, and DFbetas) were used to identify outliers in the data-set of Fig. .
From the five curves depicted in Fig. , the analyses marked three observations as real outliers. These outliers correspond with bCTPs collected during tests using the 50-L high-shear mixer. This mixer is larger which implicates that larger amounts of filler had to be sieved to collect the model agglomerates. It is likely that this introduces additional errors. This was the rationale to remove the outliers from the data set. These data points are not shown in Fig. . After removing these three outliers, different regression models correlating ξm
have been produced:
Table lists the models produced.
Regression Models Between ξm and StAbr at Various Relative Fill Volumes for the Curves Depicted in Fig.
The results demonstrate a relationship between abrasion rate of agglomerates and the value of StAbr
. The R2
presented in Table indicates the extent that the values of StAbr
explain abrasion at various process conditions. Here, R2
approaching 100% indicates that abrasion is fully explained by the parameters that describe the Stokes abrasion (StAbr
) number of the system. The R2
presented in Table shows that the StAbr
number is a reasonable way to predict agglomerate abrasion while there is no clear difference between the fits of the results between the different blenders. This makes it possible to combine these results into one model. This model includes 78% of the variance when the fill level exceeds 16%. The regression analysis in our previous study (9
) included 84% of the variance using a smaller data set. The data-set in this study also covers the abrasion data for the 50 L high-shear mixer scale.
It is clear that a low fill level leads to much faster abrasion of the test particles (Fig. and Table ). There is apparently a transition where the impeller starts to dominate the abrasion. To study the impact of fill level, regression analysis has been performed separately for all fill levels. These results are depicted in Table .
Regression Models Between ξm and StAbr at Various Fill Degree for Two Different High-Shear Mixer Scales Depicted in Fig.
To study the effect of the impeller, the fill degree of the blender has been defined relative to the impeller height, the relative fill height (Δhpowder
height of the stationary powder and himpeller
the impeller height (Table ). Figure shows the relationship between the fit constants in Table and the relative fill height of the powder in the blenders.
Fig. 5 Relationships between the variables slope (α, upper figure) and intercept (β, lower figure) with the relative distance between impeller and powder height (hpowder) for the 25 L (black circle) and 50 L (letter x (more ...)
The slope (αi) of the fits is almost constant and has a value of around 1. This implicates that the relationships between abrasion rate and StAbr are almost linear relationships. As a consequence, the intercept β describes the slope of the (almost) linear relationships in Fig. . The value of β increases drastically when the relative fill height is low.
The intercept between the dotted and solid lines shown in Fig. has been calculated and gives a transition point at a
value of 3. This result shows that agglomerate abrasion is predominantly determined by the powder bed movements when the
value is larger than 3. Obviously, the presence of enough powder is a prerequisite for the applicability of the Stokes number approach. When insufficient powder is present, the impeller starts to dominate the process. Logically, direct contact between impeller and the bCTP’s yields a deviating abrasive phenomenon than the shear forces occurring when there is plenty of powder present.