Power Law Fluid
Non-Newtonian fluids occur commonly in our world. Power law model is applicable for time independent non-Newtonian fluids and can be written as
τ = K γ^{n}
where, τ is shear stress, γ is shear rate, K is called flow consistency index and n is called flow behavior index.
For n < 1, the apparent viscosity decreases with increasing shear rate and fluid is called pseudoplastic or shear-thinning. A majority of non-Newtonian fluids like polymer solutions, pulp suspensions, pigments and food materials can be found in this category.
For n > 1, the apparent viscosity increases with shear rate increase and fluid is termed dilatant or shear-thickening. Examples are starch and clay suspensions in water.
For n = 1, Newtonian flow behavior is expected.
Reynolds number for the power law fluid is defined as
where, D is pipe inside diameter, V is fluid velocity and ρ is fluid density.
For Re < 2100, flow is laminar and friction factor is calculated as
f = 16 / Re
For turbulent flow, following relationship was developed by Dodge and Metzner.
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 Power Law Fluid Flow
References
- Fluid friction at Petrowiki
- Chemical Engineering Fluid Mechanics, Ron Darby, 2^{nd} Edition