Schmidt Number Formula
The Schmidt number relates momentum diffusivity to mass diffusivity.
The mass-transfer analog of the Prandtl number, used in mass transfer correlations.
The Formula
The Schmidt number (Sc) is a dimensionless quantity that characterizes the relative thickness of the velocity boundary layer to the concentration boundary layer in mass transfer. It is the mass-transfer analog of the Prandtl number in heat transfer. Named after German engineer Ernst Heinrich Wilhelm Schmidt, it appears in mass transfer correlations just as Pr appears in heat transfer correlations.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| Sc | Schmidt number | dimensionless |
| ν | Kinematic viscosity (momentum diffusivity) | m²/s |
| D | Mass diffusivity (diffusion coefficient of species in fluid) | m²/s |
| μ | Dynamic viscosity | Pa·s |
| ρ | Fluid density | kg/m³ |
Typical Schmidt numbers:
- Gases: Sc ≈ 1 (e.g., O&sub2; in air: Sc ≈ 0.83) — momentum and mass diffuse at similar rates
- Liquids: Sc ≈ 100–10,000 (e.g., O&sub2; in water: Sc ≈ 500) — viscosity dominates mass diffusion
- Very viscous liquids: Sc > 10,000
The Schmidt number relates to the Prandtl and Lewis numbers: Le = Pr/Sc = D_thermal/D_mass. Le ≈ 1 for gases (thermal and mass diffusivity are similar), Le << 1 for liquids.
Example 1 — Oxygen in Air
Calculate Sc for O&sub2; diffusing in air at 25°C: νair = 1.516 × 10−5 m²/s, DO2,air = 1.82 × 10−5 m²/s.
Sc = ν/D = 1.516 × 10−5 / 1.82 × 10−5
Sc ≈ 0.83 — in air, O&sub2; diffuses slightly faster than momentum is transported. Sc ≈ 1 confirms gases behave similarly for heat and mass transfer.
Example 2 — In Mass Transfer Correlations
The Sherwood number (Sh, mass-transfer analog of Nu) for pipe flow: Sh = 0.023 Re^0.8 Sc^(1/3). For Re = 10,000, Sc = 500 (O&sub2; in water): find Sh.
Sh = 0.023 × (10000)^0.8 × (500)^(1/3)
= 0.023 × 1585 × 7.937
Sh ≈ 289 — much higher than in air because Sc for liquids is much larger than for gases
When to Use It
The Schmidt number is used when:
- Calculating mass transfer coefficients using Sherwood number correlations (Sh = f(Re, Sc))
- Designing chemical reactors, packed beds, and membrane separators
- Analyzing electrochemical reactions at electrode surfaces
- Environmental engineering — modeling pollutant diffusion in rivers and the atmosphere
- Biomedical engineering — oxygen transfer in blood and tissue