Hartmann Number (Ha): Magnetohydrodynamic Flow Scaling
The Hartmann number Ha characterises the relative strength of electromagnetic forces to viscous forces in electrically conducting fluids. Defined as Ha = B L √(σ / (ρ ν)), it combines magnetic flux density B, characteristic length L, electrical conductivity σ, fluid density ρ, and kinematic viscosity ν.
Engineers use Hartmann number analyses to design liquid-metal cooling loops, magnetohydrodynamic generators, and electromagnetic pumps. Pair this explainer with the NTU-effectiveness calculator to quantify how magnetically damped flow regimes influence heat transfer coefficients.
Definition and Scaling
Hartmann number emerges from nondimensionalising the Navier–Stokes equations coupled with Maxwell’s equations under the quasi-static approximation. It reflects the ratio of Lorentz forces to viscous forces; high Ha indicates strong electromagnetic damping and flatter velocity profiles.
The square root term arises from combining electrical conductivity, density, and viscosity to form the Hartmann layer thickness, the boundary region where magnetic forces alter velocity gradients. Expressing inputs in SI units (tesla, metre, siemens per metre, kilogram per cubic metre, square metre per second) keeps Ha dimensionless and comparable across studies.
Historical Context
Danish physicist Julius Hartmann studied mercury flows in magnetic fields during the 1930s while investigating magnetically controlled measuring instruments. His experiments revealed boundary layers now called Hartmann layers, where shear is concentrated near walls to satisfy no-slip conditions despite magnetic damping in the core flow.
Subsequent research generalised the concept to liquid metals such as sodium, potassium, and lead–bismuth eutectic, all relevant to nuclear reactor cooling and space power systems. Modern magnetohydrodynamics retains Hartmann’s nomenclature when analysing stability, turbulence suppression, and electric power generation.
Key Relationships
Interaction Parameter
The interaction parameter N = Ha² / Re compares electromagnetic forces to inertial forces by incorporating the Reynolds number Re. High interaction parameters signify laminarisation and suppression of turbulence, critical for predicting pressure drop in ducts and channels.
Hartmann Layer Thickness
The characteristic thickness δ = L / Ha delineates the region near walls where velocity gradients persist. Designers evaluate δ to select sensor placements, electrode spacing, and wall materials that withstand concentrated shear and Joule heating.
Magnetic Reynolds Number
While Hartmann number assumes low magnetic Reynolds number (induced fields negligible), high-speed or large-scale flows may require accounting for induced magnetic fields. Cross-referencing Ha with magnetic Reynolds number ensures modelling assumptions remain valid.
Applications
Fusion energy systems. Liquid metal blankets surrounding tokamak plasmas rely on Hartmann number analysis to balance electromagnetic damping with heat removal. Selecting appropriate channel dimensions and magnetic field strengths mitigates pressure drop while maintaining adequate flow.
Electromagnetic pumps. Induction and conduction pumps move molten metals without mechanical parts. Hartmann number helps tune magnetic field intensity and channel geometry to deliver required volumetric flow while limiting Joule heating.
Metallurgical processing. Continuous casting and alloy refining use magnetic brakes to stabilise flow. Engineers adjust Hartmann number targets to suppress turbulence, reduce inclusions, and improve surface quality.
Data and Reporting Practices
Document magnetic field configurations (uniform vs. gradient), electrode placement, and wall conductivity when publishing Hartmann number studies. These parameters influence boundary conditions and measurement uncertainty.
Provide property tables for conductivity, viscosity, and density as functions of temperature, especially for liquid metals with significant thermal variation. Linking to calculators such as the liquid cooling load tool or electrolyzer energy calculator helps teams translate dimensionless analysis into equipment sizing and energy balances.