## Bingham Plastic Fluid

Tue, 25 Aug 2015

Bingham plastic is a material that behaves as rigid body at low stresses but flows as a viscous fluid at high stress. This behaviour is exhibited by slurries, suspensions of solids in liquids, paints, emulsions, foams, etc.

Bingham model is described by following relation.

``τ = τo + μp γ``

where, τ is shear stress, γ is shear rate, τo is called minimum yield stress and μp is called plastic viscosity.

Reynolds number for Bingham plastic fluid is defined as

``Re = D V ρ / μp``

where, D is pipe inside diameter, V is fluid velocity and ρ is fluid density.

For Laminar flow, friction factor is provided by Buckingham Reiner equation.

where, He is Hedstrom number and is calculated as

``He = D²ρτo / μp²``

For turbulent flow, an empirical relationship was developed by Darby and Melson.

The friction factor for a Bingham plastic can be calculated for any Reynolds number from the equation.

``f = ( fLm + fTm ) 1/m``

where, fL is laminar flow friction factor and fT is turbulent flow friction factor. Factor m is calculated from following equation.

``m = 1.7 + 40,000 / Re``

Pressure drop is calculated as

``ΔP = 2fρV² (L'/D)``
``L' = L + Le``

where, L is the pipe length and Le is equivalent length due to loss in pipe fittings and is calculated as

``Le = kD / 4f``

### References

1. Bingham plastic at Wikipedia
2. Fluid friction at Petrowiki
3. Chemical Engineering Fluid Mechanics, Ron Darby, 2nd Edition

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