Chemical Engineering November 2013 - 25

Calculating Volumes
of Bulk Solids
Department Editor: Scott Jenkins
C
alculating volumes of solids in piles as they are poured into various
shaped containers can be important information for solids-handling
applications. The following set of equations represents guidance to
help determine solids volume in these cases.
Solids in containers
Solids fed through a nozzle into a cylindrical or rectangular silo normally
form a partial cone of a complex shape when the material is loaded
off-center (eccentrically). Few articles have been published to estimate the
volume of material in such cones, and those that have are limited [1,2]. In
Ref.1, a graph is given to find the volume in a cylindrical silo. The article
in Ref. 2 mentions shortcut methods for various types of containers to calculate
the volume ratio with imaginary reference cones. These equations
aim to provide more exact analytical methods to calculate the volume of
material of solids heaps, characterized by their angles of repose α.
Cylindrical silo
To find the effective volume of the silo, first the remaining cylindrical volume
of the silo should be calculated and the total added to the volume
of the bottom cone and of the heap.
In the case of concentric loading in a cylindrical silo with radius Rs,
the height of the heap, h, is equal to Rs × tan(α) and its volume, V, is
π/3 × Rs3 × tan(α).
If the filling nozzle is located at the circumference of the silo roof, h
equals 2Rs × tan(α) and its V = (2π - 32/9) × Rs3*tan(α).
In case of eccentric loading at a distance εRs (dimensionless eccentricity
ε) from the center, the height of the heap is Rs(1+ε) × tan(α).
The exact solution of the volume calculation is rather complex, containing
both complete and incomplete elliptic integrals of the first and second kind.
The final equation for the total volume of such a heap is given by:
V
total = Rs
3 * tan(α * ( + ) + 2i
)
modulus m =
= EllipticE
1
1


+
−
1
1
ε
ε
+
−
ε
ε






1
ε Π
(1− ε) + 2ε ) E i− 4
9
(7
and parameter z =
Equation (2) and (3) are:
E
F = EllipticK


V = Rs
V = Rs
1
1
)
+
−
ε
ε
3 * tan(α * ( + ε)Π −


or
3 * tan(α) *


Π
3
+ Πε +
6Π 32 2
ε
−
9


= Rs
3 * tan(α) * ( +1 3ε + 2ε ) 3


2 Π
−
32
9
ε
2


(5)
This polynomial is not the best regression function because it overestimates
the volume of the heap over the whole range 0<ε<1, with a maximum
error of about 1.15% at ε=0.7. However, the result is identical at

Chemical Engineering November 2013

Table of Contents for the Digital Edition of Chemical Engineering November 2013

Contents
Chemical Engineering November 2013 - Cover1
Chemical Engineering November 2013 - Cover2
Chemical Engineering November 2013 - Contents
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Chemical Engineering November 2013 - Cover3
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