06 / CHAPTER 11
Dimensionless thermal groups
Use Reynolds, Prandtl, and Nusselt numbers as regime clues.
ANCHOR EQUATION
Nusselt number
In symbols: Nu = hL/k
A dimensionless measure of convection relative to conduction.
Dimensionless groups compare competing mechanisms
A dimensionless number is a ratio of effects, not a decorative label. Reynolds compares inertia with viscosity; Prandtl compares momentum diffusion with thermal diffusion; Nusselt compares convection with conduction.
Cancel units to expose a regime
For flow speed U over length L, inertial stress scales as ρU² and viscous stress as μU/L. Their ratio is Re = ρUL/μ = UL/ν, where ν = μ/ρ. For heat transfer, Pr = ν/α compares momentum diffusivity ν with thermal diffusivity α; Nu = hL/k compares convective to conductive heat transfer.
Water at U = 1 m/s, L = 0.01 m, ρ = 1000 kg/m³, μ = 0.001 Pa·s has Re = 10,000. Inertia dominates viscosity at that scale, so laminar assumptions need evidence rather than habit.
A group predicts a tendency, not every detail
Transition thresholds depend on geometry and disturbances. Matching Re between a model nozzle and full-scale nozzle improves dynamic similarity, but surface roughness, gravity, and free-surface effects may introduce additional groups.
Work it: Calculate Re for water at U = 0.2 m/s in 2 mm and 20 mm tubes. Which is more likely to show inertial effects, and why does diameter change the result?