Particulate matter in a cylindrical bin

[majin_andrew]

Hello :)
I am currently confused about the forces acting on particulate matter in a cylindrical bin.

It is apparently widely accepted that the vertical forces acting on an elemental slice of particulate solids in a cylindrical bin are:

• The pressure difference between the bottom and the top of the slice (upwards force)
• The weight of the slice (downwards force)
• Friction from the side of the wall (upwards force)

As the thickness of the slice (dh) approaches zero, the pressure difference between the top and the bottom of the slice also approaches zero, implying that the friction force is a reaction to the weight of the slice.

However, the function for the maximum friction force is
Wall friction = [tex]\mu_{w}p_{r}\pi D dh[/tex]
Where [tex]\mu_{w}=[/tex] coefficient of friction
[tex]p_{r}=[/tex] Pressure from wall of container directed inwards
D = diameter of cylindrical bin
dh = slice thickness.

According to the well known Janssen formula, the function for [tex]p_{r}[/tex] is
[tex]p_{r}=\frac{\rho_{b} g D}{4 \mu_{w}}(1-e^{-\frac{4\mu k h}{D}})[/tex]
where [tex]\rho_{b}=[/tex]density of particulate solid
g = acceleration of gravity
k = ratio of lateral pressure to vertical pressure
h = height below surface of the bin.

The issue I have is, the value for [tex]p_{r}[/tex], and thus for the maximum frictional force, starts at zero when h=0, and increase until a certain maximum value is approached. This contrasts with the weight of the slice, which is independant of h. Therefore, at low values of h, the particulate material cannot be at equilibrium.

Surely this is incorrect. What am I missing?

Thanks
Andrew

 

[Navin A S]

It is important to note that the friction force described in the Janssen formula is the maximum frictional force. The actual frictional force experienced by the particulate solids in the cylindrical bin will be less than this maximum value due to the angle of repose of the particulate material. The angle of repose is the angle at which the particulate material will remain in place without any external forces acting upon it. This angle is typically between 30-40 degrees and can be used to calculate the actual frictional force acting on the particulate material. Additionally, it is also possible for the pressure difference between the top and bottom of the slice to not be zero even at low values of h. The pressure difference will depend on the density of the particulate material and the angle of repose, as well as the coefficient of friction and other factors.