A short, boring, but somewhat informative article on flow channels inside flat-bottomed bins and other funnel flow hoppers

Funnel flow hoppers are pretty common. Funnel flow occurs when the walls of a hopper are not steep enough or have low enough friction to allow the powder to flow along them. In funnel flow, only a fraction of the solids will flow when the gate or valve to the outlet is opened. The moving material is called the flow channel, which diverges at an angle called the flow channel angle θ??. It should be obvious that if you have a flat-bottomed hopper, you gonna have funnel flow.

Funnel flow isn't necessarily a problem, especially if your powder isn't too cohesive and you aren't worried about segregation or having stagnant powder inside your hopper. Funnel flow will lead to the formation of a rathole, and not all of the rathole may collapse. And if the flow channel angle is steep, there can be a lot of material left behind, especially if it's cohesive.

If your funnel flow hopper has a round outlet, the flow channel is gonna be fightenly steep. The flow channel will diverge only slightly, and for the most part, the diameter of the flow channel won't be too much larger than the diameter of the outlet. You can see this from the adjacent photograph where powder was discharged through a round outlet.

You can estimate the flow channel angle from an equation that I found in a paper written by TUNRA (Arnold, Bulk Solids Handling, 5, 1 (February 1985)) that was attributed to <genuflect> Andrew Jenike </genuflect>:

where δ is the effective angle of friction, which can be determined by shear cell testing. For an effective angle of friction equal to 40°, the flow channel angle, referenced from vertical, is 8°, i.e., über steep.

Suppose we instead have a funnel flow hopper with an elongated outlet, one whose width is one third its length. Then the flow channel will open up dramatically, as seen in the adjacent photograph. Same powder as before, but an oval outlet in place of round outlet.

I haven't found a good equation to predict the flow channel angle in planar flow. Let's assume that for this case, the flow channel angle can be described by an equation of the form:

where α needs to be determined. The appearance is similar to the equation I use to predict the flow channel angle above round outlets except that that it's missing a 2. For reasons above my pay grade, this is frequently the case with analyses of axisymmetric solids flow and planar flow. The derivations frequently include an n+1 term where n equals 1 for a round outlet and n equals 0 for a slotted outlet. Based on the little bit of data that I have, a value of α fits well, so for funnel flow hoppers with slotted outlets, I claim that the flow channel angle can be described by

For an effective angle of friction equal to 40°, the flow channel angle that develops when solids flow in a funnel flow pattern above a slotted outlet is approximately 37° from vertical. That's more tolerable.

In general I can express an equation for determining the flow channel in both axisymmetric and planar flow hoppers (i.e., hoppers with round and slotted outlets, respectively) as:

And if you are too lazy to plug in the numbers and would rather use a graph, here you are:

Of course, you'll want to make sure that the hopper outlet is large enough to prevent bridging or arching. You should visit my website www.mehos.net - there are lots of potentially useful documents on powder handling and hopper design there that will guide you. Or you can ask me to provide a workshop. My email is [email protected].