I never understood why the Froude number played a role in pneumatic conveying calculations and my questions on this forum (and explanations) were never leading to any respons

Reading an article about pneumatic conveying, I noticed that the Froude number was presented as

Fr=v^2/(g*D) which is wrong.

But I immediately recognized that the representation v^2/(g*D) is a part of the general resistance formula:

∆p=1/2**/g*v^2*L/D

Which can be rewritten as:

∆p=1/2***v^2/(g*D)*L

The term v^2/(g*D) is also known as the Froude number to the square

Fr^2=v^2/(g*D)

Fr=v/√(g*D)

The Froude number represents:

A dimensionless number defined as the ratio of the flow inertia to gravity.

Developed by William Froude a naval architect and hydro dynamic engineer.

The Froude number plays a role in the wave resistance of a partially submerged body (a ship, floating in water)

The moving ship is exiting a force to the water at the bow (flow inertia) which causes the water in front of the bow to rise (gravity), forming a ship’s wave.

However, in pneumatic conveying there is no partially submerged body.

In pneumatic conveying there is only a fully enclosed pipe and only fully submerged particles.

The conclusion is that using the Froude number in pneumatic conveying calculations is a

misinterpretation of the meaning of the Froude number.

The Froude number should not be used in pneumatic conveying calculations.

In flow calculations in closed flow tube, the law of Bernoulli is ruling.

p1+H1+1/2**/g*v1^2=p2+H2+1/2**/g*v2^2

Where =function(Re)

Nevertheless, in most pneumatic conveying calculations the Froude number is used.

Still the calculation results of those algorithms, using the Froude number, result in a usable outcome.

The reason for this “usability” can be explained by:

- The Froude number can be related to the Reynold number at equal velocity.

Fr=Re*viscosity/(D^1.5*√g)

- Relating the Froude number(s) (either pipe diameter or particle) to laboratory- and field tests, the exponent of the deduced Froude number were adapted to get the measured data with the calculation results

- The use of the adapted Froude numbers worked as a huge fudge factor, whereby it became totally unclear what the physical meaning of the formulas and the Froude number meant.

Therefore, it is impossible to understand pneumatic conveying, based on the wrong algorithms.

Calculations based on the flow dynamics based on the law of Bernoulli (flow in tubes) explains the physics completely.

- Which part of the pressure drop is due to wall friction

- Which part of the pressure drop is due to interparticle collisions

- The only left factor is the material resistance factor, which is a material property and depending on the Reynold number and the solid loading ratio.

- The calculation of material velocities is now possible.

I am very relieved now that I now found out how the Froude number got into the pneumatic conveying calculations, because it bothered me for a long time.

I never assume that re writing the general resistance formula could lead to such a misconception of physics by scientist and was never questioned over the years behind us.

This observation must be seriously researched by every engineer in the field of pneumatic conveying and implemented in the existing calculation algorithms.

My education in electro technics, mechanical engineering and ship building finally paid off. ■