1. ## Froude number

The Froude number is defined as:

Fr=(Inertia forces)/(gravity forces)=(mass*acceleration)/(mass*gravity)=(m*a)/(m*g)=a/g

Introducing a time T:

Fr≡(v⁄T)/g≡v/(g*T)

Where:

T≡L⁄v

Resulting in:

Fr≡v/(g*T)≡v/(g*L⁄v)≡v^2/(g*L)

Redefining the Froude number as:

Fr≡v/√(g*L)

In flow dynamics, the Fr-number is related to wave energy, existing at the boundary surface between 2 mediums of different density.
(F.i. a moving ship at the surface between water and air)

When submerged in a medium, the existence of waves is not possible, as there is no boundary with a second medium.
( A submerged submarine makes no waves at the surface and therefore moves faster submerged than at the surface)

As this situation applies to a particle moving in a pipe, fully submerged in the conveying gas, waves cannot occur, causing the Fr-number becoming irrelevant.
Only friction forces can exist, which are governed by the Re-number.

In many publications, the Fr-number is used as a dimensionless variable to relate pneumatic conveying parameters to, which is to be considered as incorrect.

2. In my opinion use of Fr number is relevant in relating minimum suspension velocity to pipe diameter
when scaling up on pipe size. It is common knowledge the smaller bore pipes require lower pickup
velocties this is due to a phenomenon known as pipe wall effect yes it is related to RE number also
but if you find the minimum FR number in a small pipe the during scaling up you keep above that
FR number you will remain above suspension velocities.

3. Dear Dr. Mantoo,

I have some questions:

To which pipe wall effect are you referring?
The boundary layer effect, whereby the wall velocity is dependent on the Re-number according:

v_wall=(-0.0165+0.0369*ln⁡(Re) )*v_(gas average)

To which FR-numbers and Re-numbers are you referring?
Particle size or pipe diameter related or mixed?

How is the relevance of the particle Froude number explained?
Example cement:

〖Fr〗_particle=v_suspension/√(g*d)=1.78/√(9.81*0.00005)=80

When Fr>1, the particle travels faster than the created wave between two medium surfaces in which the particle must be moving.
Moreover, the particle is fully submerged in the conveying gas, where waves are not existing.

Furthermore, it cannot be neglected that the particle drag factor (determining the suspension velocity of particle) is a function of the Reynolds number and not of the Froude number.
See: https://www.princeton.edu/~asmits/Bi...web/blunt.html

I must admit, that I am not an expert in scaling technics, although it was a part of my study of ship resistance at school.

Interesting topic.

Have a nice day

4. Dear Mr Teus

I think you have never tried to convey in small diameter pipes i.e. 10 mm in these small diameter
pipes fully suspended flow is achieved at about 10 m/s for fine powders. If it was only related to
RE number and Particle properties then if we scale up keeping the velocities same in a 100 mm pipe
(For same air velocity Particle drag forces / RE numbers etc remain the same in all pipe diameters)
we should get fully suspended flow at 10 m/s velocities but this is not the case we need about
15 m/s velocities to get fully suspended flow. If you go 200 mm you will need much higher velocities
than 15m/s to achieve suspended flow! How do you explain this? I must admit i have never scaled
ships but i have scaled up a lot of conveying pipes based on empirical data using FR number.

The pipe wall effect is the turbulent region close to the pipe wall as we move the towards the
center flow becomes relatively less turbulent. I am referring to the ratio of the area of turbulent region with
the total pipe cross-sectional area. Since the turbulent area remains constant at given conditions
and in bigger pipe diameters the ratio between the two changes. If you keep the FR number constant
you will always achieve suspended flow while scaling up on pipe diameter.

It is indeed an interesting and very debatable topic and you don't have to agree with my opinion but i
think it is widely accepted in academia and I disagree with you statement about Fr number.

Regards

5. Dear Dr. Mantoo,

I think you have never tried to convey in small diameter pipes i.e. 10 mm
You are correct, the smallest pipe diameters, I worked with were 3”

in these small diameter pipes fully suspended flow is achieved at about 10 m/s for fine powders.
If it was only related to RE number and Particle properties then if we scale up keeping the velocities same in a 100 mm pipe (For same air velocity Particle drag forces / RE numbers etc remain the same in all pipe diameters) we should get fully suspended flow at 10 m/s velocities but this is not the case we need about 15 m/s velocities to get fully suspended flow.
If you go 200 mm you will need much higher velocities than 15m/s to achieve suspended flow!
How do you explain this?
The pipe Re-number at the same velocity increases with the diameter and thereby the wall velocity.
Another question is, why do we need higher drag forces to keep the particles in suspension.
Do you have any formulas or mathematical explanation for this observation?

I must admit i have never scaled
ships but i have scaled up a lot of conveying pipes based on empirical data using FR number.
A ship model is towed through a tank, whereby the Fr-number of the model is the same as the Fr-number of the real ship.
This implies that the model velocity in the tank is much lower than the real ship velocity.
The wave patterns around the model and the real ship are the comparable.
At the same time, the Re-number (governing the friction resistance) of the model is much lower than the real ship and therefore cannot be scaled up in this way.

The pipe wall effect is the turbulent region close to the pipe wall as we move the towards the center flow becomes relatively less turbulent. I am referring to the ratio of the area of turbulent region with the total pipe cross-sectional area. Since the turbulent area remains constant at given conditions and in bigger pipe diameters the ratio between the two changes.
If you keep the FR number constant you will always achieve suspended flow while scaling up on pipe diameter.
Are you referring to the pipe Froude number?

It is indeed an interesting and very debatable topic and you don't have to agree with my opinion but i think it is widely accepted in academia and I disagree with you statement about Fr number.
It is not a matter of agree or disagree but a matter of finding how things work.

All for now

6. Dear Dr. Mantoo,

On the 27th March 2009, Mr Agarwal wrote:

https://forum.bulk-online.com/showth...-Froude-Number

Froude Number
Froude No. gives the relationship between pipe line diameter and conveying velocity. As pipe diameter increases, conveying velocity also increases.

For example, when using a stepped pipeline Froude No. is kept constant to determine the conveying velocity at each step.

Froude Number is also used for scaling-up lab test data (typically small diameter conveying lines) to commercial size pipelines.
(For same air velocity Particle drag forces / RE numbers etc remain the same in all pipe diameters)
we should get fully suspended flow at 10 m/s velocities but this is not the case we need about 15 m/s velocities to get fully suspended flow.
Both statements indicate that higher velocities are required in larger bore pipelines.

If the Fr-number is kept constant at a pipe step, then the following equations are applied:

〖Fr〗_1=v_1/√(g*D_1 )

And

〖Fr〗_2=v_2/√(g*D_2 )

As the Fr-number is kept constant at the step:

〖Fr〗_1=v_1/√(g*D_1 )=〖Fr〗_2=v_2/√(g*D_2 )

Thus:

v_1/√(g*D_1 )=v_2/√(g*D_2 )

v_2=(√(D_2 ) )/√(D_1 )*v_1

As D_2>D_1 results in v_2>v_1

In other words: after a diameter increase, the velocity increases.

Conclusion:
When the Fr-number is kept constant at a diameter step, extra air volume is required.

This conclusion undermines the theory that a stepped pipeline is invented to reduce the velocity in order to minimize energy losses.

Still looking for a mathematical explanation why the use of Froude numbers in pneumatic conveying is justified.

Take care

7. ## inhomogeneous expansion

Originally Posted by Teus Tuinenburg
The Froude number is defined as:

Fr=(Inertia forces)/(gravity forces)=(mass*acceleration)/(mass*gravity)=(m*a)/(m*g)=a/g

Introducing a time T:

Fr≡(v⁄T)/g≡v/(g*T)

Where:

T≡L⁄v

Resulting in:

Fr≡v/(g*T)≡v/(g*L⁄v)≡v^2/(g*L)

Redefining the Froude number as:

Fr≡v/√(g*L)

In flow dynamics, the Fr-number is related to wave energy, existing at the boundary surface between 2 mediums of different density.
(F.i. a moving ship at the surface between water and air)

When submerged in a medium, the existence of waves is not possible, as there is no boundary with a second medium.
( A submerged submarine makes no waves at the surface and therefore moves faster submerged than at the surface)

As this situation applies to a particle moving in a pipe, fully submerged in the conveying gas, waves cannot occur, causing the Fr-number becoming irrelevant.
Only friction forces can exist, which are governed by the Re-number.

In many publications, the Fr-number is used as a dimensionless variable to relate pneumatic conveying parameters to, which is to be considered as incorrect.
Dear Teus,
in this statement the inhomogeneous expansion, for instance of fluidized beds, is not taken into account.

Regards

8. Originally Posted by Teus Tuinenburg
Dear Dr. Mantoo,

You are correct, the smallest pipe diameters, I worked with were 3”

The pipe Re-number at the same velocity increases with the diameter and thereby the wall velocity.
Another question is, why do we need higher drag forces to keep the particles in suspension.
Do you have any formulas or mathematical explanation for this observation?

A ship model is towed through a tank, whereby the Fr-number of the model is the same as the Fr-number of the real ship.
This implies that the model velocity in the tank is much lower than the real ship velocity.
The wave patterns around the model and the real ship are the comparable.
At the same time, the Re-number (governing the friction resistance) of the model is much lower than the real ship and therefore cannot be scaled up in this way.

Are you referring to the pipe Froude number?

It is not a matter of agree or disagree but a matter of finding how things work.

All for now
Dear Sirs,
Before making any comment, i want to say that i'm a flour miller. We deal with pneumatic conveying both in vacuum and overpressure. The size of the pneumatic conveying plants used in flour milling are by far much smaller in capacity that the plants you are using in different industries as for cement.
But the important thing is that the phenomenon is the same, i.e. having an air flow for to transport a solid fraction.
Due to the lack of the information of the depth understanding of the process, i started to make some research.
And as it is mentioned here, for the same particle, the increasing of the pipe diameter is affecting by increasing the floating speed of the same particle.
I tried to figure out what is really happening there, because starting with the standard equations there is nothing included as for the ratio between the diameter of the pipe and the diameter of the particle.
I have written a research paper and, in my humble opinion, the effect of increasing the floating speed for the particle by increasing the pipe diameter, is due to the increasing of the free flowing area determined by the sum of the cross section area of all particles at one specific plane and the cross section area of the pipe. It means that the "amount" of the cross section of the pipe "occupied" by the sum of the cross section of the particles, increases by the pipe diameter decrease.
Paper available at: http://tecnocereal.com/wp-content/up...-2016-0005.pdf

Hope that my comments are not too late.

Sincerely,
Tanase TANASE

9. Dear Mr. Tanase,

I read your very interesting paper and agree to the influence of the presence of particles in a pipeline on the resulting air velocity.

The air velocity between the particles in a pipe is higher than the same air flow under the same conditions in the same pipe without particles.

For the definition of the floating velocity (suspension velocity or terminal velocity) of a particle, the formula only uses the particle properties and standardized gas conditions and the relative velocity between particles and gas medium.
The relative gas velocity generates the drag forces a.s.o.
The pipe diameter is no part of the suspension velocity definition.

From the definition, the local suspension velocity can be calculated by entering the local gas conditions.

The calculated velocity I your paper is We, which is the air velocity between particles and pipe wall. The velocity We increases with higher pipe loads, which leads to the conclusion that at higher air loads, the gas flow could be reduced, while still maintain suspension.

Reasoning the other way, a loaded pipe diameter (high SLR) operates properly with the particles in suspension, but when the SLR reduces, the gas velocity decreases because of less occupation of the pipe cross section by the particles and the remaining gas velocity can become too low for suspension and sedimentation will start.
This only happens under special circumstances, which was at one point revealed by my calculation software.

I noticed, the Froude number plays no role in your paper and this thread is about the question, whether it is valid to use the Froude number in pneumatic conveying.

10. Dear Mr. Teus,

Thank you very much for the appreciative words for my paper.
I referred to the floating speed since here above were discussed the mathematical model for the influence of the ratio pipe diameter/particle diameter on the floating speed.

On the other side, in my opinion, the Froude criterion can be used as a tool for to assess the regime in pneumatic conveying. Not alone, but combined with some other parameters and especially for to assess the clogging regime. That is actually the important point for to size and energy efficient system.

My point is presented in document attached to the present message.
Your valuable opinion would help for to further better understand this extraordinary complex process of pneumatic conveying.

With great esteem and respect,
Yours sincerely,
Tanase TANASE

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