There are several equations listed in

Handling of Bulk Solids,

Shamlou P A

Buterworths 1988,

ISBN 0-407-01180-3

Chapter 4 "Gravity flow of particulate solids" ■

The classic equation for flow of granular material through an orifice is the Beverloo equation. There are many similar equations, but the Beverloo equation is the standard for this calculation. You can google that .

If you look carefully, you will see that this equation can be reduced to a form where the density term can be eliminated. In that case, it reduces to a statement of constant Froude Number, where the linear dimension is carefully defined.

That is to say that it is the volume flow that is relevant, not the mass flow. Lead shot will assume a similar volumetric flow to round grain seeds.

The equation is not valid for particle sizes below around 200 microns, depending on the material, as you will see from various references, like Glastonbury et al. That is because the forces between particles finer than around 200 microns are dominated by surface interactions. ■

Beverloo equation is normally used for funnel / core flow for mass flow Johanson equation fits better.

For fine particles both will deviate as there are other dominating factors. ■

As Mantoo suggests, an equation like that of Johanson will be more suitable for a mass flow conical hopper, which will have steep sides. Materials that are not free-flowing will further complicate the analysis and there is much expertise available in the field of hopper design that will incorporate other variables.

It depends what sort of material you are considering and what kind of orifice it is. But, if we consider the simple case of a non-cohesive granular material, I offer some comments.

If the ratio of the orifice diameter to the particle diameter is less than around 30, you may want to consider using an effective diameter rather than the actual diameter. Similarly, if the orifice is not round, then you will want to use an expression for the effective diameter.

The Johanson formula can also be expressed in terms that eliminate the bulk density and therefore, like the Beverloo equation and many others, it becomes an expression for the Froude number for a given set of conditions.

Interestingly, neither the bulk density nor the height of the material are necessary variables in the flow formulas, which runs against your intuition in the original post. ■

For the case you describe, a splitter, it will probably be a fairly safe initial assumption that the relative flow rate will vary with aperture size to the power 2.5

It would be interesting to hear of your test results. ■

Unfortunately not many people comeback to share there experience after getting advice from this forum. ■

Originally Posted by Mantoo

Unfortunately not many people comeback to share there experience after getting advice from this forum.

That's true. But discussions between ourselves can be educational. I'd never come across the Beverloo equation till it was mentioned here. ■

Originally Posted by donecker

As Mantoo suggests, an equation like that of Johanson will be more suitable for a mass flow conical hopper, which will have steep sides. Materials that are not free-flowing will further complicate the analysis and there is much expertise available in the field of hopper design that will incorporate other variables.

It depends what sort of material you are considering and what kind of orifice it is. But, if we consider the simple case of a non-cohesive granular material, I offer some comments.

If the ratio of the orifice diameter to the particle diameter is less than around 30, you may want to consider using an effective diameter rather than the actual diameter. Similarly, if the orifice is not round, then you will want to use an expression for the effective diameter.

The Johanson formula can also be expressed in terms that eliminate the bulk density and therefore, like the Beverloo equation and many others, it becomes an expression for the Froude number for a given set of conditions.

Interestingly, neither the bulk density nor the height of the material are necessary variables in the flow formulas, which runs against your intuition in the original post.

What are the practical limitations for the Johanson Equation with respect to the semi included angle, noting that the equation goes to infinity as the angle approaches 0? Just playing with some rough math I used terminal velocity and density through the effective diameter to get an idea of where the equation diverges from reality. Solving for theta, with a particular set of properties, resulted in 1deg. This test looks promising as most of my applications are ~10deg, but I just want to make sure I am estimating with the limit of this equation. ■

This discussion highlights the tendency to look to apply formula without any concept of the fundamental principles involved. The rate of gravity flow through an orifice depends on many factors. The Beverloo equation, whilst a useful first approximation, is no more than a curve fitting exercise on a limited range of bulk materials, equipment and operating conditions. It takes no regard to whether the flow channel is mass or funnel flow, the wall contact friction or internal friction conditions, the particle shape or size, structural or void gas impeded flow, by how much the orifice size exceeds the critical arching dimension, the effect of initial bulk state or how the rate of discharge can be enhanced by hopper inserts or air injection. Bulk solids technology is an involved subject in which research is difficult and the results limited because of the number of interacting factors that may be involved. Equipment designers should at least know some fundamental characteristics to assess what may be influential in a specific situation.

Anyone interested in the mechanisms involved in flow through orifices should perhaps review the mechanics of the stressed arch over an outlet and the inclination of the reaction on the hopper walls to assess the unsupported region of material above the outlet that can disengage from the mass to accelerate through the opening. Some notes on this and related themes are available from lyn@ajax.co.uk.

Calibration is fine, provided the test condition replicate those of the operation of interest. This becomes awkward to do when dealing with large scale projects. In sensitive cases it is good practice to fit a feeder. ■

Originally Posted by JerryA

What are the practical limitations for the Johanson Equation with respect to the semi included angle, noting that the equation goes to infinity as the angle approaches 0? Just playing with some rough math I used terminal velocity and density through the effective diameter to get an idea of where the equation diverges from reality. Solving for theta, with a particular set of properties, resulted in 1deg. This test looks promising as most of my applications are ~10deg, but I just want to make sure I am estimating with the limit of this equation.

Johanson's experiments in 1965 went down as low as 10 degrees semi included angle for slotted hoppers and 15 degrees for conical hoppers, at the laboratory scale. Larger scale verification was done at 20 degrees. He does comment that the limiting case of a vertical tube does not yield a steady state solution. I doubt that anyone has systematically explored the region below 10 degrees experimentally as there would be few hoppers of industrial interest in that region.

There have been experiments done with apertures next to vertical walls on one side that showed increases in flow rate in comparison with symmetrical hoppers.

He does point out the assumptions for his derivation and notes that the equation yields a maximum value of discharge rate such as would be experienced for a free flowing material. The addition of moisture gave results that were less than the predictions of the equation. ■