Properly accounting for how bulk solids actually will flow in a vessel or overall process can be crucial for successful operations. So, in this article, we will look at two parameters — the Compressibility Index and the angle of repose (see sidebar) — that can help. While neither provides definitive answers about “flowability,” they do give rough guidance about how a material is likely to behave.

However, before we discuss these parameters, it is important to understand bulk density. It is probably one of the most common and widely used of the bulk characteristics. It is employed to determine wall loading in hopper design, to size volumetric feeders, such as screws and rotary valves, to estimate “flowability,” and in many other ways. It is rather unfortunate then that such a useful characteristic is not a constant for a given material. The bulk density of a material is simply the mass of material divided by the volume that it occupies. The density of the particles themselves can be taken as constant; however, the complication arises because the amount of “space” between the particles depends upon how the material has been handled before the measurement. The volume that a unit mass of product can occupy can differ by 50% between the material being in a compressed and a very loose state. Cement, for example, has a compacted bulk density of 1,400 kg/m3 and an aerated bulk density of 1,000 kg/m3. It is obviously important that the correct bulk density value is selected for any calculation.

Bulk density

The full expression for bulk density is:
         (1)
For dry bulk solids, the void spaces would usually contain air or some other gas, the density of which can be taken as negligible compared to the density of the solid particles; so, we can approximate:
          (2)
We can relate this to another common characteristic, voidage or void fraction, which is the percentage of the total volume not occupied by particles:
          (3)
Again, assuming air or gas in the void spaces and taking particle density as  , we can write:
          (4)
To illustrate the range of values that voidage can take, consider a static heap of mono-sized spheres. If the spheres are in a regular hexagonal packing (the classic “cannon ball” stack), the voidage would be 26%. In contrast, if they were in regular cubic packing, the voidage would increase to 48%. However, even this does not represent the loosest packing possible for large smooth identical spheres. The cannon ball stack gives each ball six contact points, but simple static mechanics requires only two contact points below the center of gravity of the ball for equilibrium. Therefore, it is possible to have a stable structure with far fewer contact points and a resulting increase in voidage [2]. If the particles are irregular in shape, have a size distribution and in some way cohere to one another, the packing arrangement can be very loose and so the voidage can be very large.

Measurement of bulk density is, in theory, quite simple; it only requires a knowledge of material mass and volume and is generally based on one of two techniques.

The first is to weigh out a quantity of material using a simple balance and put this into a calibrated cylinder in much the same way as you would a liquid. If the particulate material is poured into the cylinder, the volume taken up would be of the material in a loose or poured state; the associated bulk density is commonly described as “poured bulk density.” If this same cylinder is then tapped or dropped from a small height onto the bench several times, the volume would likely decrease and the new value is called the “tapped bulk density.” Similar techniques can be used to determine aerated bulk density from a fluidizing column or compacted bulk density from a material placed under load.

The second technique is to fix the volume of the bulk material by filling a cup-like vessel to overflowing and then leveling it with a straight edge. The vessel then is weighed on a balance and the bulk density calculated. This approach gets around some of the problems of trying to estimate the actual level of powder in a cylinder with a surface that typically is anything but flat and seeing through a glass that has become coated in powder. Table 1 lists typical bulk density values for a few common materials.

Click to enlarge Table 1.

One possible complication with bulk density measurements is the effect of the porosity of the particles themselves; so, to avoid ambiguity, it is worthwhile stating whether the bulk density value is inclusive or exclusive of closed pores. Confusion could arise if the method of determining particle density does not take account of internal voids. (Using a helium pycnometer, which determines particle density by a measure of displaced gas, may be advisable when porosity is a factor. The gas generally can penetrate open pores as long as these are not comparable in size to the gas molecule but obviously cannot penetrate closed pores.) These differences become important if, for example, we are concerned with surface area available for reaction or the total solids’ fraction available for reaction.

Flowability based on bulk density

Bulk density measurements have been used to give some qualitative prediction of the flowability or “handlability” of a bulk solid — that is, some estimate of the likely ease or difficulty in dealing with these materials. One such predictor is the often-quoted Hausner ratio: