Characterization of the Performance of Bin Blenders Part 1 of 3: Methodology
The other sampling technique uses a core sampler (a hollow tube filed to a thin edge at one end) to gather samples. The tube is thrust into the mixture and retrieved, leaving a core of material in the sampler that is held in place by static friction forces, and is then extruded in a last-in-first-out manner (see Figure 3). The use and accuracy of this sampler has been described extensively (20). This sampler has proven to give more-accurate representations than typical thief probes of mixture distributions while simultaneously causing less disturbance of the powder bed.
The splitting of variance is established by first defining an axial zone j as
is the local (core) mean, xij is a given sample, and Ni is the number of samples within that zone/core. The standard definition of variance is
is the mean composition. Substituting equation  into equation  and rearranging, leads to
In equation , the first term is a measure of axial variance and the second term radial variance.
To quantitatively compare mixing efficiency, it is useful to determine the rate of variance (or RSD) decrease at different processing conditions. A quantitative measure of the rate of variance decrease can be obtained by assuming exponential decay (7, 8) and defining a mixing constant k such that
is the variance at revolution M
is variance at revolution N, and (N-M) is the elapsed number of revolutions (RSD can be substituted for σ2 in equation ). The mixing constant k yields a quantitative measure that can be used to evaluate how changes in process parameters or mixture composition affect mixing rates in the blender.
Lower fill levels induce faster mixing rates. When the amount of material in the blender is reduced, mixing should be faster. However, very low fill levels (<~25%) interfere with natural mixing mechanisms and hinder mixing rates. Filling the blender to more than 60% of its capacity can lead to dead zones in the middle of the mixture that do not interact with the rest of the mixture.
A free-flowing mixture is one for which flow is determined by the dynamics of individual particles. In essence, each particle can be tracked and accounted for separately and mixing mechanisms arise from the time-averaged flow of these individual particles. A cohesive mixture is one for which single particles do not flow independently; rather, groups of particles act in concert when force is applied to the entire mass (avalanching). Hence, mixing mechanisms derive from the motions of groups of particles rather than single particles and there may not be any meaningful time-averaged flow field.
A commonly used particle dynamics method for the modeling of granular flow is the discrete element method (DEM). DEM uses Newtonian physics to determine the velocity, angular momentum, and position of particles. Each particle is tracked in the system and particle-particle and particle-boundary interactions are computed. DEM simulations are often thought of as a macroscopic equivalent of short-range molecular dynamics in which the inelastic nature of particle collisions is taken into account. In recent years, the use of particle dynamic simulations has proliferated (26-29), and currently is being applied to certain complex industrial problems (30-32).
An example A preliminary example of how to dissect and analyze mixing data is presented: the mixing of 1% w/w% diphenhydramine HCl with an excipient matrix in a 56-L Gallay bin blender run at 10 rpm (see Figure 1a). The excipient matrix is composed of microcrystalline cellulose (PH102 Avicel, FMC Corp., Philadelphia, PA), granulated lactose (Fast-Flo, Foremost Farms, Baraboo, WI) and magnesium stearate (nonbovine, Mallinckrodt, Hobart, NY) with a formulation of 39%, 60%, and 1% w/w%, respectively. Core sampling was used to gather samples; nine cores from throughout the blender surface (i.e., total sampling) were taken for every time-point and each core yielded 15-25 samples of 0.8g. In this study, UV spectroscopy was used to determine the composition of samples extracted from the blender. To use UV spectroscopy, a linear calibration curve of absorbance versus active concentration was obtained for a series of 1:500 dilutions in de-ionized water. A sharp peak was observed at 215 nm and shown to correspond solely to the active and not the excipients.
The evolution of the RSD for experiments run at 50%, 65%, and 85% of total blender capacity is shown in Figure 9a. In the early phases of the mixing process, the 50% case had the highest mixture variance, which is contrary to the expected results. At later times, the 50% case "caught up" and eventually became the lowest RSD (best-mixed) mixture after 200 revolutions. At 65% and 85% fill, the RSD was nearly constant throughout the mixing process, which appears to indicate that mixing was complete after only 4 revolutions. This curious result at short mixing times can be better understood by analyzing other computed statistics rather than conjecturing solely based on interpreting the decay of the RSD. In this case, tracking the change in the minimum and maximum sample concentrations provides more insight into mixture distributions (see Figure 9b). At early mixing times (<64 revolutions), the difference between the maximum and minimum values was greater for the 50% case than for either the 65% or 85% case. This distribution implies that the 50% case was radially mixed more poorly than the other cases, which is confirmed by examining the evolution of radial variance (see Figure 9c).
Early in the mixing process, the mixture became radially well-mixed at both 65% and 85%, but remained radially unmixed at 50% fill. These results contradict the expected outcomes that lower fill levels would mix faster than higher fill levels. Evaluating the evolution of sample mean provides more insight into the cause of this quandary (see Figure 9d). For all three fill levels, the data in Figure 9d, indicate that early in the mixing process the sample mean is much higher than the expected mean, which appears to indicate that the blend is superpotent in the sampled region. In time, the sample mean decreases for all three fill levels, but only at the 50% fill level does the sample mean approach the true mean.
The loading procedure involved the use of a hopper to deposit material in the center of the blender. The active was loaded last and, hence, initially was concentrated in the middle of the blender. Sampling was limited to the middle ~40% of the mixture because of limited access from the opening of the blender to the mixture. This loading and sampling methodology placed a premium on axial transport to the edges of the blender for achieving a uniformly well-mixed product. Apparently, at higher fill levels, this axial transport was diminished and the active remained trapped in the middle of the blender, which rapidly led to a well-mixed but superpotent blend in the middle of the mixture while leaving the blender extremes deficient in active (leading to fast declines in radial variance but high sample means). In the 50% case, however, axial transport was more efficient at moving the active to all portions of the mixture, but led to increased radial variances because active concentrations were constantly in flux as higher potent material from the center intermingled with subpotent material from the edges. This example illustrates that using multiple means of determining mixture quality can provide important insight into the mixing mechanisms within the blender. In addition, it serves as a warning that relying on a single measure of mixture quality can lead to misleading and erroneous conclusions about the mixing process in the blender. Finally, this example also shows that loading and sampling methods must be included in the analysis of any mixing process.
Discharge Mixing materials in a tumbling blender does not end the processing of that mixture. At some point, the mixture has to be discharged from the mixer into a conveyer, a larger mixer, a tablet press, etc. Experiments were run that compared the measured variance in the blender (in situ) to that in a container that collected the discharged material from the blender. Two mixtures were used: a 50/50 mixture of 450-μm of sand of two colors and a 3% mixture of sodium chloride with 96% microcrystalline cellulose and 1% magnesium stearate. Figure 10 shows the results of sampling the blender before discharging and then sampling the discharged matter in a bucket for both mixtures. For the cohesive salt mixture, discharging into a secondary container had a mixing effect and the RSD declined. For the sand mixture however, the RSD increased slightly after the mixture was discharged.
Logically, there is little apparent reason for the mixture to separate when discharged unless the mixture had strong segregation tendencies, which was not the case. Examining the difference between radial and axial variances for the sand mixture gives some clues (see Figure 11). On discharge the measured radial variance increased while the axial variance decreased; the RSD increased because the rise in radial variance was greater than the drop in axial variance. There are two likely causes of the increasing RSD with discharge: initial conditions and sampling bias. The blender was loaded top-to-bottom to produce an axially symmetrical initial condition. However, because the blender is not symmetric in a top-to-bottom sense, the initial conditions produced a gradient in concentration from the middle of the blender outward (see Figure 12). As sampling in situ was limited to the middle of the blender, this axial gradient was overlooked and the mixture may have appeared to be better blended than it really was. After discharge, the mixture was more thoroughly sampled which results in this apparent separation during discharge that was actually the result of better sampling techniques. During discharge, the axial variability was transformed into radial variability. Although in this case (a nonsegregating mixture) the quality of the mixture was expected to improve upon discharge, the apparent decrease in quality was solely a function of improved sampling of the mixture. The difference between the axial and radial variances in situ and post-discharge virtually disappeared as the mixture approaches a well-mixed state (32 revolutions). Thus, when discharging a nonsegregating mixture from a bin blender, it can be expected that well-mixed products will not be affected but that poorly mixed blends may actually improve in mixture quality. Any results that contradict these general expectations are likely caused by sampling biases or segregation.
Conclusion Bin blenders continue to play an increasingly important role in the processing of granular and powdered materials. The dearth of specific information about the performance of bin blenders (and other tumbling blenders) makes it important to determine the performance of these devices in a variety of processing situations.
In this article, the basic approaches to gathering and analyzing performance data have been addressed along with a summary of basic operational guidelines both in terms of blender parameters and mixture types. An example of mixing analysis has been presented and the effects of discharge have been discussed, along with some cautionary information about mixture sampling. The next two articles in this series will describe, in detail, bin blender performance using free-flowing and cohesive materials.