Hedström Number

Drilling mud, mineral slurry and cement paste refuse to move until wall stress overcomes the yield stress. That pushes the laminar-to-turbulent transition well away from the Reynolds 2100 everyone memorised at university, and sizing a pump on the Newtonian criterion gets the pressure drop wrong in an awkward direction: designing for turbulent flow a line that still runs laminar, or the reverse. The Hedström number pins down where the transition actually landed.

He = ρ·τ₀·D² / μₚ², with ρ the fluid density in kg/m³, τ₀ the Bingham yield stress in Pa, D the pipe inner diameter in metres and μₚ the plastic viscosity in Pa·s. Output has no unit and climbs fast: doubling the diameter multiplies He by four. The figures already on screen, 1250 kg/m³ mud, 8.5 Pa yield stress, a 4-inch pipe at 0.1016 m and 35 cP plastic viscosity, return He = 89532.4. Under the Hanks correlation that value lifts the critical Bingham Reynolds number to roughly 6600, so the line stays laminar at flow rates where a Newtonian fluid would already have gone turbulent.

The expression assumes Bingham plastic rheology, a fair description of bentonite mud and dense slurry, yet wrong for anything following power-law or Herschel-Bulkley behaviour; those have a generalised Hedström number with a different definition. Unit slips cause most of the trouble: plastic viscosity in centipoise needs dividing by 1000, yield point read in lbf/100 ft² multiplies by 0.4788 to reach Pa, mud weight in lb/gal by 119.83, and diameter in inches by 0.0254. Leaving μₚ in cP inflates He by a factor of a million.