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# Hoppers and silos bulk discharge rates

## How to calculate the flowrate of bulk solids discharging from a hopper

Section summary
1. What are the causes of blockages in silos
2. How to prevent blockages in silos
3. How to improve flow in silos : discharging aids

## 1. Definition

Process Engineers must often estimate the flow of powder, or more generally bulk solids, that they can get out of a hopper. Indeed, the calculation of the mass flowrate allows to size the outlet of hoppers or silos, calculate cycle times or make sure that the discharge capacity is enough for the downstream process. However, the calculation of a free flow of solids is not easy and depends on many parameters. This page presents different methods allowing to estimate the flow of powder from an existing bin, or size a new hopper / silo to get a required discharge flow rate.

## 2. Hopper discharge rate calculation method 1 : using shear cells data

In order to estimate the discharge rate of a silo one of the most reliable method is 1st to have assessed the flowability of the material that will be stored in the hopper. Different methods are available for evaluating the flowability but one of the most reliable, which allows to have quantitative data and not only a relative assessment is to use shear cells. This method requires a lot of testing but gives the basis for the design of hoppers and the estimation of discharge flow. The method is making the distinction in between coarse and fine powders.

These formulas are reported in the article : Using fundamental powder properties
to optimize flowability, Tablets and Capsules, Mehos et al, 2017

### 2.1 Coarse powder discharge rate

The following formula can be used for assessing the discharge rate of coarse powders :

Equation 1 : Hopper discharge rate for coarse solids

With :

ms = hopper discharge rate in kg/s
B = outlet diameter of the hopper in m
ρbo = powder bulk density at outlet conditions in kg/m3
θ' =mass flow hopper angle in deg

### 2.2 Fine powder discharge rate

Fine powder flow is generally lower than the flow of coarse powder. The fluidization and air balancing - flow of air from downstream to top - being detrimental to the mass flowrate of powder.

The following formula can be used to assess the discharge rate of fine powders.

Equation 2 : Hopper discharge rate for fine powders according to K Johansson

With :
ms = hopper discharge rate in kg/s
B = outlet diameter of the hopper in m
ρbo = powder bulk density at outlet conditions, flowing in kg/m3
ρbmax = powder bulk density at the major consolidation stress in the hopper in kg/m3
Ko =permeability of the powder at outlet conditions in m/s

It is thus necessary to have defined the bulk density of the material as a function of the stress.

The major consolidation stress can be calculated with the Janssen equation :

Equation 3 : Janssen equation

With :
D = cylinder diameter - for the shear cell experiment - in m
h = depth of powder in the cylinder section in m
k = Janssen coefficient, if unknown can be assumed at 0.4 as 1st approximation
Φ' is the wall friction angle in deg
σ1 = major consolidation stress
ρb = bulk density at hopper outlet, not flowing

## 3. Hopper discharge rate calculation method 2 : empirical methods

These formulas are reported in the Perry, 8th edition

### 3.1 Coarse particles (>400 microns)

2 types of equations are usually found in the litterature : the Johanson equation and the Berverloo equation. To be noted that these equations will allow to estimate the flow but in no case to have an accurate value.

Beverloo equation is the most direct expression, although different "lump" parameters are used. It is important to note that, for fine particles, the Beverloo equation will overestimate the discharge rate (actually, whne discharging fine particles, air fluidization happen which is detrimental to the discharge rate compared to large particles).

Beverloo Equation

Equation 4 : Beverloo equation (discharge rate through outlet for coarse particles)

W discharge rate in kg/s
C empirical discharge coefficient
k empirical shape coefficient
ρb is the bulk density in kg/m3
g is the acceleration of gravity 9.81 ms-2
dp is the particle diameter in m
d0 is the discharge diameter in m (note for no circular outlet, use hydraulic diameter 4*(cross sectional area)/(outlet perimeter)

C=f(ρb) and is in the range 0.55<C<0.65
k=f(particle shape, hopper angle) and is in the range 1<k<2 except for sand where it is 2.9

If acknown, consider C=0.58 and k=1.6

The Johanson equation has the following form :

Johanson Equation

Equation 5 : Johanson equation (discharge rate through outlet for coarse particles)

m_discharge discharge rate in kg/s
θ angle of hopper deg
ρb bulk density in kg/m3
g is the acceleration of gravity 9.81 ms-2

Table 1 : Parameters for Johanson equation

Parameter Conical hopper Wedge hopper
B D, diameter of outlet W
A Pi*D^2/4 WL
m 1 0

### 3.2 Fine particles (<400 microns)

As mentionned above, the flow of fine particles will be sensitive to the flow of air returning from the discharge point and opposing the flow of materials. The discharge rate can then be 100 times less than what is predicted by Beverloo or Johanson equations. Carleton proposes an equation to estimate the discharge rate of fine particles.

Carleton Equation

Equation 6 : Carleton equation (discharge rate through outlet for fine particles)

V0 average velocity of solids dicharging
A,B given above
ρp particle density

The very 1st action to take is actually to prevent this issue through a good design. The silo slope and outlet diameter can be calculated. The silo calculation is 1st based on measuring properly the flow properties of the bulk solids to be stored. The following page explains the calculation procedure to design a silo : link.

Note that it is sometimes possible, although costly, to modify a silo after installation to improve the flow.

If the design by itself cannot ensure a free flow, or if it is necessary to improve the flow of powder after installation, the use of discharging aids should be considered.