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
Resources
- Web based calculation available at checalc.com
- Spreadsheet for Bingham Plastic Fluid
References
- Bingham plastic at Wikipedia
- Fluid friction at Petrowiki
- Chemical Engineering Fluid Mechanics, Ron Darby, 2^{nd} Edition