Considerations In The Determination Of Bulk Density

As a consultant in the bulk material industry, I am frequently asked to provide a single value. This number is often one of the starting points for the engineering of equipment. This number is sought after to use in specifying what is to be built or bought. The value is the bulk density of a material.

What is density though? It has units of mass per volume, indicating that bulk solids within a certain space have so much stuff. Absent from this, is any defining of what conditions achieve that value.

In the age of the internet, there is accessibility to tables and databases brimming with published values for the bulk density of all types of materials. These values are taken and used in quite a few ways, which is where many well intended designs can go astray.

Three common uses for bulk density in solids handling application are for estimates of capacity, design and operation of feeders, and determination of silo loads. Materials have a plethora of variations.

Particle size, moisture content, chemical composition, and other factors can result in two materials bearing the same name having very different bulk densities (1).

Published values for density often do not provide information on these factors making their use risky. If we wish to look beyond the published values, we must perform laboratory tests.

Testing density allows a direct measurement of how much mass the solid occupies in a given volume as a result of exposing the material to some force or condition. In silos and hoppers our interest is in the compacted density and how that changes with varying pressures.

Certain solids are considered incompressible and may have a fairly constant bulk density, but many materials will compress with pressure. The typical testing involves a cell with a known volume being filled with the solid, and then applying a series of increasing pressures, measuring the change in volume at each pressure level (2).

Another key aspect is that density changes with particle size. Therefore density measurements should be made on the particle size range that will be handled.

Academic literature presents different equations to model the relationship between bulk density and consolidation pressure (3). However almost all abide by these three principles:

• Bulk density varies as a function of pressure.
• There is a diminishing increase in density as a function of pressure, until even very large pressure increases make little to no change in the bulk density.
• There is either a minimum bulk density or an understanding that the equation cannot be used to predict density at very low pressure levels.

Once testing has been performed and the relationship between pressure and bulk density have been established, we must determine which values to use.

Storage vessel capacity:

The required capacity is usually process driven. Continuous processes may require storage to supply material for some length of time, while batch operations may require a storage capacity corresponding to a certain number of batches.

Selection of the silo or bin will often be constrained by space and cost. A given bin will have a total volume, and then it will also have an available storage volume, which is usually less than the total volume. During filling, solids form an angle of repose, which is why available storage capacity is less than the total vessel volume.

In many storage vessels the majority of the capacity occurs in the vertical section. Pressure follows an asymptotic increase in pressure, which can be approximated with the Janssen equation (4).

As the pressure varies, so will the bulk density. A good approach is taking a midway point for the pressure, and then using the density at this pressure to provide an estimate of the capacity. An even more conservative approach is to look at the minimum compacted bulk density and use that value to determine capacity.

When solids discharge from a hopper they will flow in one of two ways, called flow patterns. Mass flow is characterised with all material being in motion during discharge.

The alternative, core (or funnel) flow, involves some degree of stagnant material within the bin. For more cohesive solids stagnant material may significantly reduce the live storage capacity.

Feeder Output:

Feeders meter material in a number of ways. Typically either the weight or volume of material discharging is controlled. Volumetric feeding devices allow for the control of discharge rate based on volume. The most common volumetric feeding devices are rotary valves, screw feeders, and vibratory feeders.

The control of these devices allows the user to adjust the rate of volume over time that is removed from the bin. If the density varies at the outlet this may make control difficult. Mass flow offers an advantage here, as the pressure at the outlet remains consistent.

Meanwhile, in core flow the outlet pressure is a function of material level. This feature led to one plant who handled a fine powder to control their process feedrate not by changing the speed of their rotary airlock, but by adjusting their silo level.

In core flow it may be difficult to select a density to design for because outlet pressure will vary. Therefore a range of densities should be taken into consideration. In mass flow the density at the outlet will typically be closer to those of the material at low levels of consolidation pressure.

Structural Integrity:

Any time a bin or silo is being built a structural analysis should be performed by a qualified structural engineer in accordance with good engineering practices and the relevant standards.

In the analysis, the engineer will likely request the density of the material. It is best to provide the entire relationship of density as a function of consolidation pressure. If a single value is used, it should be a higher density based on the material under greater consolidation pressure.

The following table briefly summarises some of the consequences of selecting a density value both below and above the actual value.

What is the potential consequence of….

Figure 3. Table with implications of density selection

In selecting a bulk density, we must implement an appropriate level of simplicity. The published values for bulk solids often contain no descriptors for particle size, moisture content, or other variations within products. Several products may all be grouped under the same general bulk solid name, yet may vary in their density.

These items may seem inconsequential, but solids are great respecters of detail. Published tables have worth, but an awareness and caution should be maintained in their use. Whenever possible, testing should be performed to determine the actual bulk density range of the specific material in question.

With many bulk materials, we cannot treat density as a singular constant. We must treat it as a range of values, understanding that density can vary as a function of pressure. This matters because density determines silo capacity, influences feeder output, and is necessary for structural analysis.

When a single value is selected in the design of any of these, a conservative value should be used. Design often requires compromises, a balancing of competing priorities, and simplifications.

However these actions should be performed appropriately; requiring a level of nuance. Such an approach applies even in the selection of the bulk density.

References:

1. Cabrejos, F.J.: Total Storage Capacity of Stockpiles Handling Compressible Materials; bulk solids handling Vol. 20 (2000) No. 4, pp. 421-427

2. ASTM D6683-19, Standard Test Method for Measuring Bulk Density Values of Powders and Other Bulk Solids as Function of Compressive Stress, ASTM International, West Conshohocken, PA, 2019, www.astm.org

3. Mehos, G.: Hopper Design Principles for Chemical Engineers. 2020

4. Schulze, D. Stresses in Silos. Retrieved March 25, 2020, from https://www.dietmar-schulze.de/spanne.html