Plant InSites: Come Up With a Solid Estimate

Large-scale solids-handling operations often rely on storing material in piles. The pile layout and solid's properties set the volume storable in a given area. Piles most often approximate one of four common shapes: a cone: a frustum of a cone; a wedge; or a frustum of a wedge (Figure 1) — a frustum essentially is the underlying shape with its top cut off.

Angle of repose, Θ, which is the stable angle between the surface of the pile and the horizontal, is the major bulk material property affecting volume. Angle of repose varies with base material, particle size, particle size distribution (or fines content), particle shape and other factors such as moisture content. Table 1 (at the bottom of this page) lists reasonable values of Θ for some common materials; many references also provide such information. Values for specific cases can vary considerably. So, always try to get data from industrial applications similar to yours; in critical cases, you may need to test your actual materials.

For a cone, the volume is:
v = πhr²/3        (1)

where h is the height and r is the radius.
For a frustum of a cone, the volume is:

v = πh(r₁² + r₂² + r₁r₂)/3           (2)
where r₂ = r₁ – h/tan Θ.

For a wedge, the volume is:

v = hb(2a₁ + a₂)/6 = hb[3a₁ – 2h/tan Θ]/6   (3)
where a₂ = a₁ – 2h/tan Θ₁.

For a frustum of a wedge, the volume is:

v = h[a₁b₁ + (a₁ + a₂)(b₁ + b₂) + a₂b₂]/6      (4)
where b₂ = b₁ – 2h/tan Θ₂.

You can use these equations, which apply both to unconstrained piles and bulk solids flowing in constrained spaces such as hoppers, with those for other standard shapes to determine a variety of volumes.

For example, let's find the volume of grain remaining in a 10-ft.-dia. 50-ft.-high feed hopper when it's nominally half full (Figure 2). We break the hopper into three zones: a frustum of a cone on the bottom, a cylinder in the middle, and a cylinder minus a cone on the top.

The first section has 1 ft. of height cut from the cone. We determine r₂ as 5 - 4/tan 45° or 1 ft. This gives a volume of the bottom section of π4[5² + 1² + (5 x 1)]/3 or 130 ft³.

The second section has a cylinder height of 25 - 5 tan 28° or 22.3 ft. This gives a cylinder volume of 1,753 ft³.

The third section has the volume of a cylinder 25 - 22.3 or 2.7 ft. tall minus the frustum height. Volume is π5² × 2.7 - π5² × 2.7/3 or 141 ft³.

So, total volume in a half-full feed hopper is 130 + 1,753 + 141 or 2,024 ft³.

This relatively simple example illustrates how you can estimate bulk solid volumes.

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Andrew Sloley is a Chemical Processing contributing editor. You can e-mail him at ASloley@putman.net.