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Indeed, the critical strength (fc) for arching is com- puted from the intersection of these two lines, and that strength is used—along with the bulk density (g) of the material—to compute the arching dimension as given by Equation 1. This seems like a straight-forward method. So why does it over-predict arches in some cases? One issue is the flow factor line. It approximates the expected stresses acting on material flowing in a conical hopper, but it only considers gravity as a force for break- ing an arch, which is supported at its abutments. Thus, the flow factor line assumes a linear relationship between the span of the hopper and the stress. If this linear behav- ior does not exist, then the analysis is faulty. Also, because the flow factor analysis assumes that only gravity is acting to break the arch, it ignores air-pressure gradi- ents, acceleration gradients, or vibrational gradients that would apply additional breaking force to the arch. In many cases, such simplification is valid, but in other cases—such as during dynamic operation with gas flows—the assumption is wrong, and the arching analysis fails to predict reality. However, as long as a contact bed exists, the basic requirement—that the local stress must be larger than the local strength to break the arch—is valid. Thus, if we can compute local stress in the process equipment and include all the dynamic effects in calculat- ing the stress, then we have a better indication of the stress needed to break an arch. To that end, Equation 1 must be revised to include the effect of external forces. Thus, a more robust method of computing arching in process equipment would include the stress level in the equipment due to gravity flow, gas pressures, vibrations, etc. The strength at any point in the process equipment can also be determined, just as Equation 1 can be used to predict the arching tendency (AI) at any point in the process. This arching-tendency value could be divided by the local span (D) of the con- verging geometry to yield a dimensionless number (arch- ing ratio (AR)). The arching ratio quantifies the potential of the material to arch in the process: If it's greater than 1.0, aching occurs at that point in the process equipment. This method resolves the problems associated with using Jenike methods to compute critical arching dimensions in small process equipment. Before presenting an example of the method, let's examine yet another potential problem with using the typical Jenike method to describe arching in small hop- pers. A close look at Figure 1b reveals that the lines inter- sect at a stress level that is almost an order of magnitude below the lowest measured strength value. That poses a problem because applying a test method limited to using high-stress strength values forces engineers to extrapolate a tremendous amount of strength data, increasing the risk of poor process prediction. Adapting the Jenike method However, recent developments in measurement tech- niques allow us to gauge the strength at much lower val- ues than traditional testing methods. As a result, we can measure strength to stress levels as low as 100 pascals (Pa), which is much closer to the intersection point in Figure 1b. The exact methodology for these measure- ments is not discussed here, but it involves using an instrument, and the data it generates are presented to point how they affect process predictions. Consider the case of a 50 percent ibuprofen mixture that also includes microcrystalline cellulose, lactose, mag- nesium stearate, and sodium starch glycolate and the results of measuring it at both high and low pressures (Figure 2). If we concentrate on the low-pressure region and per- form a Jenike arching analysis of these two curves, we get two very different arching values. Low-pressure strength data yields an arching dimension of 14.3 mil- limeters (mm) while using just high-pressure data cou- pled with extrapolation produces an arching dimension of 142 mm (Figure 3). The actual arching dimension was found by placing the ibuprofen mixture in a hopper and varying the size of its opening; it was found to be about 38 mm. It is obvious from this analysis that much of the arching prediction problem stems from a misunderstand- ing of or poor characterization of stress in the process equipment. To clarify the source of off-target predictions, we computed the stress that would be expected in a typical feed hopper above a tablet press (Figure 4), which is a small conical hopper typically about 450 mm in diameter that necks down to a hopper outlet of 50 to 75 mm in diameter. From there, the material flows down a chute of 50 to 75 mm in diameter and empties into the feed frame on the die table. From there the material flows into the die cavities. But if the material doesn't flow by gravity into the dies, the feed frame includes a rotating paddle that pushes the material into them. In addition, as the lower punch descends to accept the powder, it may cre- ate a gas pressure gradient, which helps suck material into the die. 30 May 2014 Tablets & Capsules Figure 2 Comparison of strength measurements of 50 percent ibuprofen mixture made at high and low stress levels 0 1600 1400 1200 1000 800 600 400 200 0 5000 50% ibuprofen mixture High pressure tests 10000 15000 20000 25000 30000 35000 Major principle stress (Pa) Unconfined yield strength (Pa) h-Johanart_28-32_Masters 5/14/14 10:17 AM Page 30