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How Sc differs from Pr
Sc = μ/(ρD) = ν/D compares how fast momentum spreads with how fast the concentration of a species spreads. It is the Prandtl number with mass diffusivity D in place of thermal diffusivity α, which is why it is sometimes called the mass-transfer Prandtl number. Sc above 1 means the concentration boundary layer tends to be thinner than the velocity boundary layer.
Variables and SI units
| Symbol | Meaning | SI unit | Example |
|---|---|---|---|
| μ | Dynamic viscosity of the carrier fluid | Pa·s | 0.001 |
| ρ | Density of the carrier fluid | kg/m³ | 1000 |
| D | Molecular diffusivity of the species in that fluid | m²/s | 1e-9 |
Choosing a value of D
- D is not a property of one substance; it belongs to a pair, a given species diffusing through a given fluid. Oxygen in water and a large protein in water have different D.
- Molecular diffusivities are typically of order 10⁻⁹ m²/s in liquids and 10⁻⁵ m²/s in gases. A value many orders away from these is a cue to check units.
- Data in cm²/s convert to m²/s by multiplying by 10⁻⁴.
- Entering a turbulent (eddy) diffusivity as D no longer gives a Schmidt number for molecular diffusion.
- μ and ρ belong to the carrier fluid (the solvent) and should be taken at the same temperature and pressure as D.
Reading the result
The example assumes a solute in water: μ = 0.001 Pa·s, ρ = 1000 kg/m³ and D = 1×10⁻⁹ m²/s, so Sc = 1000. Gas mixtures often have Sc around 1, while solutes in liquids commonly reach hundreds or more. In the same flow, heat can spread widely while a solute stays in a thin layer.
If the result lands far outside these ranges, at 0.01 or 10⁶ for example, suspect the units of D first. Entering a cm²/s value without converting makes Sc exactly 10⁴ times too small.
Oceanographers use temperature-dependent Schmidt numbers to estimate how fast gases cross the sea surface; the reference manual of the R package marelac describes this use.
Link to the Péclet number
With the same length and speed, the mass Péclet number is Re × Sc. The thermal and mass diffusion note uses v = 0.2 m/s and L = 0.05 m to get Re = 10000 and Sc = 1000, so the mass Péclet number is 10⁷. The same note shows that over 0.25 s the diffusion length √(Dt) is about 0.01581 mm, far shorter than the thermal length of 0.2236 mm. Entering D in the α field of the Péclet calculator gives the mass Péclet number directly.
Moving table data into the calculator
Example 1. Oxygen in 40 °C water, and the cm²/s slip
Given: Water at 40 °C has ρ = 992.2 kg/m³, μ = 6.529×10⁻⁴ Pa·s, ν = 0.6581×10⁻⁶ m²/s and Pr = 4.341 (IAEA Table 3.3, saturated liquid used as a low-pressure approximation). Cornell lists the oxygen diffusivity at 40 °C as 0.0000324 cm²/s (a Wilke–Chang estimate).
Unit conversion: 0.0000324 cm²/s × 10⁻⁴ = 3.24×10⁻⁹ m²/s.
Calculator input: μ = 6.529e-4, ρ = 992.2 and D = 3.24e-9 give Sc = 203.1. Checking with ν/D = 0.6581×10⁻⁶ ÷ 3.24×10⁻⁹ also gives 203.1. Sc/Pr = 46.79, so at this temperature oxygen still spreads about 47 times more slowly than heat.
Common slip: Skipping the conversion and entering D = 3.24e-5 gives Sc = 0.0203, smaller than for a gas. A liquid-phase solute with Sc below 1 is a reason to recheck units.
Example 2. Oxygen in 60 °C air: matching the pressure
Given: IAEA Table 3.1 gives dry air at 98.1 kPa and 60 °C as ρ = 1.026 kg/m³ and μ = 20.10×10⁻⁶ Pa·s. Cornell's gas-diffusion page lists D = 0.274 cm²/s for the oxygen-nitrogen pair at atmospheric pressure (1 atm) and 60 °C. Air is approximated as nitrogen.
Pressure correction: Treating gas diffusivity as inversely proportional to pressure at fixed temperature, move D to 98.1 kPa: 2.74×10⁻⁵ × 101.325 / 98.1 = 2.830×10⁻⁵ m²/s.
Calculator input: μ = 20.10e-6, ρ = 1.026 and D = 2.830e-5 give Sc = 0.692. Mixing the uncorrected 1 atm D with the 98.1 kPa density gives 0.715, about 3.3% high.
Judgment: In a gas both ν and D scale inversely with pressure, so Sc itself barely depends on pressure; the error comes from mixing data at different pressures. The guide's example gives Sc = 0.690 for oxygen in 20 °C air, so a 40 K rise hardly changes Sc in the gas, whereas the same rise cuts Sc for oxygen in water by more than a factor of five.
Schmidt calculator input questions
I only know ν. How do I fill the μ and ρ fields?
Since Sc = μ/(ρD) = ν/D, entering ν in the μ field and 1 in the ρ field gives the same result. With ν = 0.6581×10⁻⁶ m²/s from Example 1 and ρ = 1, Sc = 203.1. If you will reuse μ and ρ in other calculations, entering the real values avoids confusion.
Which D do I use for a solution with several species?
Compute Sc separately for each species of interest. Each species has its own D in the same solvent, and therefore its own Sc. For dilute solutions the solvent properties μ and ρ can be approximated by those of the pure solvent.
What if I need a value between the temperatures in the Cornell table?
Linear interpolation gives a rough value, but liquid-phase D is temperature-sensitive enough to more than double between 20 °C and 60 °C. Pair the interpolated D with μ and ρ at the same temperature, and expect the ±10% uncertainty of the underlying estimate on top.
Related guides and calculators
Guide
Calculation notes
- Thermal and mass diffusion: the distinct roles of Pr, Sc, Pe and Nu: Multiplies the Sc from this calculator by Re to get a mass Péclet number, then sets the diffusion length beside the thermal one.
Related calculators
- Prandtl number (Pr): Add cp and k for the same solvent to get Pr; Sc/Pr is then the Lewis number.
- Péclet number (Pe): Put the D used here in the α field to get the mass Péclet number.
- Nusselt number (Nu): Entering a mass transfer coefficient and D in place of h and k gives the Sherwood-number equivalent in the same form.
References
- IAEA: Thermophysical Properties of Materials for Nuclear Engineering (2008). Table 3.3 saturated water at 40 °C, Table 3.1 dry air at 98.1 kPa and 60 °C
- Cornell University: Calculating the Oxygen Diffusion Coefficient in Water. Oxygen D in water at 40 °C, 0.0000324 cm²/s, and the ±10% Wilke–Chang accuracy
- Cornell University: Calculating the Oxygen Diffusion Coefficient in Air. Oxygen-nitrogen D of 0.274 cm²/s at 60 °C and atmospheric pressure, and the decrease of D with pressure
- CRAN: marelac reference manual. The gas_schmidt function, which supplies seawater Schmidt numbers for gas transfer velocity estimates