Thermal and mass diffusion: the distinct roles of Pr, Sc, Pe and Nu

One Reynolds number does not determine every field

Momentum, heat and a dilute solute can diffuse at different rates within the same velocity field. Assume a Newtonian fluid, constant isotropic properties, no reaction and no internal heat source. Strong dependence of viscosity or diffusivity on temperature or concentration requires a more coupled model.

First compare diffusion over a shared exposure time, then distinguish that estimate from a convective boundary layer. The thermal diffusivity α and species diffusivity D are different inputs, even though both have units of m²/s.

Variables and units

Here D denotes diffusivity; in the pipe-energy guide, D denotes diameter. Check units as well as symbols when moving between sources.

Obtaining a diffusion scale

Scale the stationary-medium equation ∂T/∂t = α∂²T/∂y² as ΔT/t ≈ αΔT/δ². It gives δ ≈ √(αt). Analogously, δν ≈ √(νt) and δD ≈ √(Dt). These are order-of-magnitude lengths, not a precise 99% boundary-layer thickness. Numerical coefficients depend on boundary conditions and the definition of thickness.

The square-root ratios apply to this shared-time diffusion comparison. They are not universal thickness ratios in convective boundary layers, where near-wall speed varies with position and couples to the thermal or concentration field.

An advection-time cross-check

Take illustrative ν = 10⁻⁶, α = 2×10⁻⁷ and D = 10⁻⁹ m²/s with v = 0.2 m/s and L = 0.05 m. The SI inputs ρ = 1000, μ = 0.001, k = 0.6 and cₚ = 3000 give the specified ν and α. This constructed set is not a measured fluid property table.

The identity δ/L ≈ 1/√Pe checks the same time-scale argument. Large Pe means advection crosses the chosen length faster than diffusion crosses it. Diffusion can still control transfer across a much shorter distance normal to a wall.

Nusselt number requires a heat-transfer coefficient

Define wall heat flux by q″ = h(Twall − Treference) and connect it to Fourier conduction, q″ = −k(∂T/∂n)wall. The coefficient h contains information about the wall gradient. For a defined linear thermal layer of effective thickness δeff, h = k/δeff and Nu = hL/k = L/δeff. Do not equate that effective length with the earlier pure-diffusion estimate or a 99% thickness.

The value 120 is an input, not a prediction from Pe. Predicting h needs a solution or correlation with stated geometry, flow regime, local-versus-average convention, wall condition and validity range.

Conditions to keep in a comparison

• Use the same length, speed and property conditions for Re–Pe checks.
• Use α for heat and the appropriate species D for mass transfer.
• Distinguish diffusion estimates from boundary-layer thickness definitions.
• Record the reference temperature and whether h is local or averaged.
• Matching Pr or Sc alone does not match geometry, wall conditions or transfer rates.