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.

**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 = πhr*

^{2}/3**(1)**

where

*h*is the height and

*r*is the radius.For a frustum of a cone, the volume is:

*v = πh(r*

_{1}^{2}+ r_{2}^{2}+ r_{1}r_{2})/3**(2)**

where

*r*

_{2}=

*r*

_{1}–

*h*/tan Θ.For a wedge, the volume is:

*v = hb(2a _{1} + a_{2})/6 = hb[3a_{1} – 2h/tan Θ]/6*

**(3)**

where

*a*

_{2}=

*a*

_{1}– 2

*h*/tan Θ

_{1}.

For a frustum of a wedge, the volume is:

*v = h[a _{1}b_{1} + (a_{1} + a_{2})(b_{1} + b_{2}) + a_{2}b_{2}]/6 *

**(4)**

where

*b*

_{2}=

*b*

_{1}– 2

*h*/tan Θ

_{2}.

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.

**Feed Hopper**

**Figure 2.**Figuring out the volume of material remaining requires splitting the hopper into three sections.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

*r*

_{2}as 5 - 4/tan 45° or 1 ft. This gives a volume of the bottom section of π4[5

^{2}+ 1

^{2}+ (5 x 1)]/3 or 130 ft

^{3}. The second section has a cylinder height of 25 - 5 tan 28° or 22.3 ft. This gives a cylinder volume of 1,753 ft

^{3}. The third section has the volume of a cylinder 25 - 22.3 or 2.7 ft. tall minus the frustum height. Volume is π5

^{2}× 2.7 - π5

^{2}× 2.7/3 or 141 ft

^{3}. So, total volume in a half-full feed hopper is 130 + 1,753 + 141 or 2,024 ft

^{3}. This relatively simple example illustrates how you can estimate bulk solid volumes.

*Andrew Sloley is a* Chemical Processing *contributing editor. You can e-mail him at* [email protected].

**Angle of Repose for Common Bulk Materials**

**Table 1.** This list presents reasonable values for the materials but actual value can vary markedly depending upon application.

Material |
Θ, degrees |

Activated carbon | 20-29 |

Alfalfa, meal | 45 |

Alfalfa, pellets | 9 |

Alumina | 22 |

Anthracite ash | 45 |

Barley | 30 |

Bauxite | 31 |

Carbon black | 25 |

Caustic soda | 29-43 |

Cement | 37-40 |

Cinders, bituminous coal | 25-41 |

Clay, soft | 10 |

Clay, compacted | 10-25 |

Coal, anthracite | 27 |

Coal, bituminous | 35 |

Coke, lose pile | 30-45 |

Corn | 23 |

Diatomaceous earth, calcined | 30-44 |

Flaxseed | 21 |

Gravel, round | 30 |

Gravel, sharp | 40 |

Gypsum | 30 |

Iron Ore | 35-45 |

Limestone, pulverized | 30-44 |

Lime, hydrated | 40 |

Linseed | 25 |

Millet | 23-25 |

Phosphate rock, crushed | 25-29 |

Phosphate rock, ground | 40 |

Plastic resin grains, cellulose acetate | 16 |

Plastic resin grains, fluorocarbon | 6 |

Plastic resin grains, nylon | 9 |

Plastic resin grains, polyethylene pellets | 23 |

Plastic resin grains, vinyl | 10 |

Plastic resin grains, viscose | 10 |

Oats, rolled | 30-44 |

Quartz, ½-in. screenings | 20-29 |

Quartz, 1.5-3 in. | 30-44 |

Rapeseed | 18-22 |

Rice | 30-44 |

Safflower | 24 |

Salt | 36 |

Sand, dry | 25-35 |

Sand, moist | 30-45 |

Sand, wet | 20-40 |

Scrap metal | 35-45 |

Silt, compacted | 25-40 |

Silt, loose | 20-30 |

Sand-clay, compacted | 40-50 |

Soda ash, dense | 32-42 |

Soda ash, light | 59 |

Sulfur, granules | 35 |

Sunflower seed | 23 |

Wheat, whole | 28 |

Wheat, cracked | 30-44 |

Wood chips | 45 |