Hedstrom Number (He): Characterising Bingham Fluid Dynamics

Definition, Units, and Measurement

Hedstrom number is defined as:

He = ρ D² τ_y / μ_p².

In this expression ρ denotes density (kg·m⁻³), D is the characteristic diameter or hydraulic dimension (m), τ_y is yield stress (Pa), and μ_p is plastic viscosity (Pa·s). Because Pa equals kg·m⁻¹·s⁻², substituting yields dimensionless units, aligning with similarity analysis requirements. Experimentally, τ_y and μ_p are obtained by fitting shear stress versus shear rate data from rotational rheometers to the Bingham equation τ = τ_y + μ_p γ̇. Engineers often employ controlled stress tests to determine τ_y precisely and complement them with capillary or pipe viscometer measurements for validation.

Accurate Hedstrom evaluation requires temperature control because both τ_y and μ_p can vary significantly with thermal history. Additionally, materials containing suspended solids demand careful sampling to prevent settling, as slight changes in solids volume fraction alter rheological parameters and thus He.

Historical Origins and Development

Gunnar Hedstrom introduced the number in the 1950s while studying laminar pipe flow of drilling muds. Oilfield engineers sought reliable scale-up methods for mud circulation during rotary drilling, where maintaining cuttings suspension without excessive pump power is critical. Hedstrom’s analysis compared yield stress forces to viscous forces, leading to the dimensionless He term that, combined with Reynolds number, predicts friction factors for viscoplastic fluids. Subsequent researchers, including Buckingham and Metzner–Reed, refined correlations to incorporate turbulent transition and non-circular conduit shapes.

Beyond petroleum engineering, food scientists adopted the Hedstrom framework to analyse chocolate, tomato paste, and starch slurries. The number provided a bridge between laboratory rheology and equipment design, enabling accurate sizing of pumps, extruders, and filling lines. Today, standards organisations such as ISO and ASTM reference Hedstrom-based correlations when specifying test procedures for drilling fluids and cement slurries.

Conceptual Relationships and Mathematical Behaviour

Hedstrom number often appears alongside the Reynolds number for Bingham plastics, defined as Re_B = ρ U D / μ_p, where U is mean velocity. Friction factor correlations typically take the form f = 16/Re_B + 0.5/He for laminar flow, illustrating how yield stress inflates pressure drop even when inertial effects are small. In fully turbulent regimes, empirical models use composite parameters such as the Bingham number Bn = τ_y D/(μ_p U) and He to adjust the smooth-pipe Moody chart.

Plotting friction factor versus Reynolds number on log–log scales reveals that increasing He shifts the laminar-to-turbulent transition to higher Re, reflecting the stabilising influence of yield stress. When He < 10, yield stress is minor and the fluid behaves nearly Newtonian; as He exceeds 10³, plug flow regions dominate and additional pressure gradient is required to initiate motion. Computational fluid dynamics codes incorporate He implicitly by solving viscoplastic constitutive equations with regularisation schemes, but non-dimensionalisation still aids interpretation of scaling laws.

In annular geometries, such as drilling between casing and borehole, modified Hedstrom numbers incorporate hydraulic diameter based on gap width. Designers must account for eccentricity and rotation, which can locally reduce effective He and trigger undesirable cuttings beds or gelled zones.

Applications in Industry and Research

In petroleum drilling, He guides selection of mud formulations to balance suspension capability with pump limits. High Hedstrom numbers ensure cuttings remain suspended during circulation pauses, preventing stuck pipe, yet require higher pressures; optimisation involves adjusting clay concentration, polymer additives, and temperature to target He between 200 and 800 depending on well depth.

Civil engineers leverage He when designing grouting operations and concrete pumping. Cement slurries exhibit yield stress behaviour; He informs the selection of superplasticisers and pumping speeds that keep flow laminar enough for placement but energetic enough to avoid segregation. In tunnelling projects, He-based calculations ensure that bentonite slurries maintain face support without overloading slurry circulation pumps.

Food and pharmaceutical processing also benefit from Hedstrom analysis. Manufacturers of yogurts, creams, and nutraceutical pastes tune solids content and homogenisation conditions to achieve target He values that maintain texture while enabling efficient filling. In pharmaceutical hot-melt extrusion, He helps to predict die pressure and avoid over-shearing sensitive actives.

Environmental engineers employ Hedstrom number when transporting sludges or biomass slurries in wastewater treatment. Maintaining He within safe limits prevents pipeline blockages and allows accurate modelling of residence time in digesters and dewatering equipment.

Importance, Best Practices, and Future Outlook

Hedstrom number enables consistent comparison of viscoplastic fluids across scales and processes. Best practice involves coupling rheometer measurements with inline sensors that monitor temperature and solids content, ensuring that He calculations reflect real-time conditions. Engineers should propagate measurement uncertainty to determine safety margins and avoid operating near thresholds where small property changes dramatically alter pressure drop.

Future developments include machine-learning-assisted rheology models that update Hedstrom predictions as formulations or shear histories evolve. Additive manufacturing of viscoplastic pastes, energy transition slurries (such as redox flow battery electrolytes), and carbon capture sorbents will continue to rely on He to balance pumping efficiency with stability. By embedding Hedstrom-based diagnostics into digital twins, operators can anticipate deviations, adjust additives, and maintain product quality in real time.