Pile Shapes
Figure 1. Most piles of solids fall into one of these four basic shapes. 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.