# Bingham Plastic Fluid

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 = τ`_{o} + μ_{p} 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 / μ`_{p}

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 = D²ρτ`_{o} / μ_{p}²

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 = ( f`_{L}^{m} + f_{T}^{m} ) ^{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`

Spreadsheet for Bingham Plastic Fluid

### References

- Bingham plastic at Wikipedia
- Fluid friction at Petrowiki
- Chemical Engineering Fluid Mechanics, Ron Darby, 2
^{nd}Edition