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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

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# Particulate matter in a cylindrical bin

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