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Section summary |
---|

1. Volume of bulk
solids contained in a hopper or a silo |

2. Total volume of
a conical hopper |

3. Actual solid
inventory in a silo |

Solids have a different behavior than liquid when they are poured in a storage vessel : they do not form a flat surface but form a heap, the heap being more or less steep depending on the nature of the solid. As a consequence, there is a strong difference in between the total volume of a hopper (what we could call the "water volume") and the volume that can actually be filled by the solid, what we would call the useful volume.

Such calculations are critical in order to design a new hopper. If one is just using the bulk density of the solids without considering the heap, there are high risks that the vessel will not be large enough. Such calculations are also very useful for existing installation when an engineer wants to check the actual volume of material in a hopper.

The method presented in paragraph 3 below was published years ago, it is approximate but can be already helpful, especially in the 2nd case mentioned above, when someone wants to check the capacity of an existing hopper.

Many hoppers and silos are using a conical design. It is the case studied in this page.

Such hoppers are made of :

- A conical bottom
- A cylindrical shell
- A top cover, let's say elliptical

**Conical bottom volume calculation**

The volume of a truncated cone can be calculated thanks to the following formula [aqua-calc.com]:

With

V_{cone} = volume of the conical bottom of the hopper in
m^{3}

D = inside diameter of the shell of the hopper in m

h_{cone} = total height of the conical discharge of the
hopper in m

D_{o} = diameter of the outlet of the hopper in m

h_{tip} = (virtual) height of the cone located below the
discharge valve

**Cylindrical shell volume calculation**

With

V_{shell} = volume of the cylindrical part of the hopper
in m^{3}

D = inside diameter of the shell of the hopper in m

h_{shell} = total height of the cylindrical part of the
hopper in m

**Top cover volume calculation - considering it is an elliptical
cover** [webcalc]

With

V_{ellip} = volume of the elliptical top of the hopper in
m^{3}

D = inside diameter of the shell of the hopper in m

h_{ellip} = height of the elliptical cover in m

Note : if the cover of the hopper is not elliptical, please adapt
accordingly.

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Paragraph 2 allows to calculate the total volume (or water volume) of a hopper, but what about the actual volume of bulk solids contained in this silo ?

The method [Koehler] presented allows to calculate the actual volume of material contained in the heap of product knowing :

- The diameter of the hopper
- The position of the heap (where is the material loaded in the silo)
- The angle of repose of the material

The top of the heap is located at a distance a of the center of
the silo. The silo has a diameter r_{0}. The ratio a/r_{0}
can then be calculated

This calculation requires to know the angle of repose γ of the material.

The height of the heap b can the be calculated by b = (r_{0}+a).tan(γ)

The volume of the heap is equal to **V _{heap} = (V_{heap}
/ V_{cyl}).π.r_{0}^{2}.b**

With V_{heap} / Vcyl being determined thanks to the
following abacus :

The total volume is then equal to :

V_{solid} = V_{cone} + V_{cyl} + V_{heap}

With :

V_{cone} = the volume of the cone filled by the solid in
m3

V_{shell} = the volume of the cylindrical shell filled by
the solid in m3, whose height is h_{total} - b

V_{heap} = the volume of the heap of solid after loading
in m3

**This method is applicable if the cone of the silo is entirely
filled and if the method of loading is allowing the formation of
the heap of product, which is typically the case when the hopper
is loaded by gravity.**

Note : at design stage, if an accurate calculation cannot be
done, the approximation

(max volume of product) / (total volume of hopper) = 0.8 can give
a 1st idea of what a hopper / silo can really hold in regards to
the total volume.

**Source**

[Koehler] Estimate the solids inventory in a silo, Frank H. Koehler, Chemical Engineering, 1982

[webcalc] http://www.webcalc.com.br/blog/tank_volume.pdf (page 7, case a = h)

[aqua-calc.com] https://www.aqua-calc.com/calculate/volume-truncated-cone