One Reynolds number does not determine every field
Momentum, heat and a dilute solute diffuse at different rates in the same velocity field. Assume a Newtonian fluid, constant isotropic properties, no reaction and no heat source. Strongly temperature- or concentration-dependent properties need a coupled model.
First compare diffusion over a shared exposure time, then separate that estimate from a convective boundary layer. α and D share units of m²/s but are different inputs. The MIT 3.185 notes state that Schmidt numbers in liquids are uniformly much greater than one, while heat-transfer Prandtl numbers are smaller and fall well below one for liquid metals.
Variables and units
| Variable | SI unit | Meaning |
|---|---|---|
| ρ, μ | kg/m³, Pa·s | Density and dynamic viscosity |
| ν = μ/ρ | m²/s | Momentum diffusivity |
| k, cₚ | W/(m·K), J/(kg·K) | Thermal conductivity and specific heat |
| α = k/(ρcₚ) | m²/s | Thermal diffusivity |
| D | m²/s | Species diffusivity |
| v, L, t | m/s, m, s | Transport speed, length and exposure time |
| h, q″ | W/(m²·K), W/m² | Heat-transfer coefficient and heat flux |
Here D denotes diffusivity; in the pipe-energy guide it denotes diameter, so Example 3 writes the tube diameter as d. Check units as well as symbols.
Obtaining a diffusion scale
Scaling ∂T/∂t = α∂²T/∂y² in a stationary medium as ΔT/t ≈ αΔT/δ² gives δ ≈ √(αt); likewise δν ≈ √(νt) and δD ≈ √(Dt). These are order-of-magnitude lengths, not a 99% thickness; coefficients depend on boundary conditions and the thickness definition.
δν/δα ≈ √Pr; δν/δD ≈ √Sc
The square-root ratios belong to this shared-time comparison, not to convective boundary layers, where near-wall speed varies and couples to the scalar field. For a laminar flat-plate layer the MIT 3.185 notes give δu/δT = 0.975 Pr1/3 for Pr above about 0.6; the exponent 1/3 reflects a nearly linear velocity profile inside a thin thermal layer. Example 2 compares the two.
An advection-time cross-check
Example 1. One velocity, three diffusion lengths
Given (round numbers, not a measured fluid): ν = 10⁻⁶, α = 2×10⁻⁷ and D = 10⁻⁹ m²/s, v = 0.2 m/s, L = 0.05 m. Calculator inputs ρ = 1000, μ = 0.001, k = 0.6 and cₚ = 3000 reproduce this ν and α.
tadv = L/v = 0.25 s
Diffusion scales over tadv: δν = 0.500 mm, δα = 0.2236 mm, δD = 0.01581 mm
Re = vL/ν = 10000, Pr = 5, Sc = 1000
Thermal Pe = vL/α = 50000 = Re × Pr
Species Pe = vL/D = 10000000 = Re × Sc
Check: δ/L ≈ 1/√Pe, and 1/√50000 × 50 mm = 0.2236 mm. Large Pe means advection crosses L faster than diffusion does; diffusion can still control transfer over the short wall-normal distance where the 0.016 mm solute layer sits.
Real properties: water and dissolved oxygen
Round numbers hide how strongly Sc depends on temperature. The table uses IAEA saturated-liquid water data as an approximation for low-pressure water, and Cornell's Wilke–Chang estimates of O₂ diffusivity in water (about ±10% for dilute solutions).
| Property | 20 °C | 60 °C | Source |
|---|---|---|---|
| ρ (kg/m³) | 998.2 | 983.2 | IAEA Table 3.3 |
| ν (m²/s) | 1.0020 × 10⁻⁶ | 4.757 × 10⁻⁷ | IAEA Table 3.3 |
| k (W/(m·K)) | 0.5995 | 0.6510 | IAEA Table 3.3 |
| cₚ (J/(kg·K)) | 4185 | 4183 | IAEA Table 3.3 |
| Pr (table) | 6.981 | 3.005 | IAEA Table 3.3 |
| D of O₂ (m²/s) | 1.97 × 10⁻⁹ | 4.82 × 10⁻⁹ | Cornell, Table 2 |
Example 2. Oxygen absorbed into still water for one second
Given: 20 °C properties from the table and an assumed exposure time t = 1 s, such as a brief contact between a bubble and the liquid.
α = k/(ρcₚ) = 0.5995/(998.2 × 4185) = 1.4351 × 10⁻⁷ m²/s
Pr = ν/α = 6.982 (the table lists 6.981; the difference is rounding in k and cₚ)
Sc = ν/D = 1.0020 × 10⁻⁶ / 1.97 × 10⁻⁹ = 508.6
After 1 s: δν = 1.001 mm, δα = 0.3788 mm, δD = 0.0444 mm
Shared-time ratios: √Pr = 2.642, √Sc = 22.55; laminar flat-plate form 0.975 Pr1/3 = 1.863
At 60 °C: Sc = 4.757 × 10⁻⁷ / 4.82 × 10⁻⁹ = 98.69
Oxygen penetrates about 44 µm in a second, one twenty-third of the momentum length, so resolving the liquid-side concentration layer in CFD needs much finer near-interface cells than the velocity field. From 20 °C to 60 °C, ν falls and D rises, lowering Sc about 5.2-fold; a Sc taken at the wrong temperature misjudges √Sc by more than a factor of two. The gap between 2.642 and 1.863 shows that the shared-time ratio is not a boundary-layer result.
Nusselt number requires a heat-transfer coefficient
Wall heat flux q″ = h(Twall − Treference) equals Fourier conduction −k(∂T/∂n)wall, so h carries the wall gradient. Only for a linear layer of effective thickness δeff is h = k/δeff and Nu = hL/k = L/δeff; δeff is neither the diffusion estimate nor a 99% thickness.
Example 3. From a supplied h to Nu, and from a known Nu back to h
Part A. With a supplied h = 120 W/(m²·K), k = 0.6 W/(m·K) and L = 0.05 m, Nu = 10; a 20 K difference gives q″ = 2400 W/m². The 120 is an input, not a prediction from Pe.
Part B. Water at 20 °C flows at an assumed 0.1 m/s through a smooth tube of assumed d = 10 mm, far from the inlet, with uniform wall temperature. For fully developed laminar tube flow the INL RELAP-7 manual uses Nu = 3.66 (constant wall temperature) and 4.36 (constant heat flux).
Re = ρvd/μ = 998.2 × 0.1 × 0.01 / 1.0000 × 10⁻³ = 998.2 (laminar)
Pe = Re × Pr = 6968
h = Nu k/d = 3.66 × 0.5995 / 0.01 = 219.4 W/(m²·K); with constant heat flux, 261.4 W/(m²·K)
Effective conduction length δeff = d/Nu = 2.732 mm
Commonly quoted thermal entry estimate 0.05 Re Pr d ≈ 3.48 m
Nu = 3.66 holds only after the temperature profile develops, which here takes metres of tube. A 1 m heated section lies mostly in the entry region, where local Nu exceeds 3.66. The wall condition alone changes h by 19%, so state it.
Predicting h needs a solution or correlation with stated geometry, regime, local or average convention, wall condition and validity range.
Conditions to keep in a comparison
- Use the same length, speed and property conditions for Re and Pe checks.
- Use α for heat and the appropriate species D for mass transfer, evaluated at the same temperature as ν.
- 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.
- Check that a fully developed Nu applies at the stations you evaluate, using an entry-length estimate.
Questions about diffusion scales and Nu
Why is √(αt) not the thermal boundary-layer thickness?
It is a stationary-medium scale with no boundary condition or thickness definition. A convective layer depends on the velocity profile, distance from the leading edge and wall condition. Use √(αt) for orders of magnitude and a boundary-layer solution or correlation for a thickness or h.
At what temperature should I evaluate Sc?
Evaluate ν and D at the same temperature, usually the film or bulk temperature your correlation specifies. In Example 2, Sc falls from 508.6 at 20 °C to 98.69 at 60 °C; mixing the two states describes neither.
Can I estimate h from the Peclet number alone?
No. Pe compares advection with diffusion over a length; h also depends on geometry, regime, wall condition and development length. In Example 3 the wall condition alone gives 219.4 or 261.4 W/(m²·K).
Why does my computed Pr differ slightly from the property table?
Tables round k, cₚ and μ independently: for IAEA 20 °C water, μcₚ/k gives 6.981 and ν/α gives 6.982. For identity checks such as Pe = Re × Pr, compute Pr from the same μ, cₚ and k used elsewhere.
References
- MIT OpenCourseWare (3.185): Lectures on Fluid Flow and Kinetics, §§3–4. Pr and Sc as diffusivity ratios, large Sc in liquids, Nu definition and δu/δT = 0.975 Pr1/3 for laminar flow at Pr above about 0.6.
- IAEA (2008): Thermophysical Properties of Materials for Nuclear Engineering. Table 3.3, saturated water at 20 °C and 60 °C.
- Cornell University: Calculating the Oxygen Diffusion Coefficient in Water. Wilke–Chang estimates of 1.97 × 10⁻⁹ and 4.82 × 10⁻⁹ m²/s at 20 °C and 60 °C, about ±10%.
- Idaho National Laboratory: RELAP-7 Theory Manual. Laminar Nu = 3.66 (constant wall temperature) and 4.36 (constant heat flux), equations (45) and (591).
- Michigan Technological University: CM3120 Module 2 Lecture IV. Pe listed as the comparison between the convective time scale and the thermal diffusion time scale.
The derivation, the Example 1 inputs and the assumed speed, diameter and exposure time are ours; printed results are recomputed by the site tests. Checked 2026-10-22.