Schmidt Number Calculator (Sc = ν/D)
Calculate the Schmidt number from kinematic viscosity and mass diffusivity.
Classifies the flow regime and compares against common gas-liquid systems.
What the Schmidt number tells you
The Schmidt number (Sc) is a dimensionless ratio that compares how fast momentum diffuses through a fluid to how fast a dissolved species (mass) diffuses through the same fluid. It is the mass-transfer cousin of the Prandtl number (which compares momentum to heat). Together they appear everywhere in transport phenomena: distillation columns, gas absorbers, polluted-air models, even biology.
The formula
Sc = ν / D = μ / (ρ × D)
Where:
- ν = kinematic viscosity (m²/s)
- D = mass diffusivity of the species through the fluid (m²/s)
- μ = dynamic viscosity (Pa·s)
- ρ = fluid density (kg/m³)
Both ν and D have the same dimensions (m²/s), so Sc is unitless, the same in any system of units.
Reading the value
| Sc range | Physical meaning | Examples |
|---|---|---|
| Sc ≈ 1 | Momentum and mass diffuse at similar rates | Gases (air-water vapor, gases in air) |
| 1 < Sc < 100 | Mass diffusion is moderately slower than momentum | Dissolved gases in water, dilute solutions |
| 100 < Sc < 1000 | Mass diffusion much slower than momentum | Organic vapors in water |
| Sc > 1000 | Mass diffusion extremely slow vs momentum | Liquid metals at low temperature, polymers |
For gases at normal conditions, Sc is near unity because gas molecules transfer momentum and mass by the same mechanism, random thermal motion of similarly sized molecules. For liquids, momentum transfer relies on molecular interactions over much shorter distances than mass transfer over many molecular distances, so Sc grows by orders of magnitude.
Reference values for common systems
| System | ν (m²/s) | D (m²/s) | Sc |
|---|---|---|---|
| Water vapor in air, 25°C | 1.55 × 10⁻⁵ | 2.42 × 10⁻⁵ | 0.64 |
| CO₂ in air, 25°C | 1.55 × 10⁻⁵ | 1.60 × 10⁻⁵ | 0.97 |
| O₂ in air, 25°C | 1.55 × 10⁻⁵ | 2.10 × 10⁻⁵ | 0.74 |
| CO₂ in water, 25°C | 8.93 × 10⁻⁷ | 1.91 × 10⁻⁹ | 467 |
| O₂ in water, 25°C | 8.93 × 10⁻⁷ | 2.10 × 10⁻⁹ | 425 |
| NaCl in water, 25°C | 8.93 × 10⁻⁷ | 1.61 × 10⁻⁹ | 555 |
| Glucose in water, 25°C | 8.93 × 10⁻⁷ | 6.73 × 10⁻¹⁰ | 1327 |
| Benzene in water, 25°C | 8.93 × 10⁻⁷ | 1.02 × 10⁻⁹ | 875 |
Notice the gap: Sc near 1 for gas-gas systems jumps to hundreds or thousands for dissolved species in liquids. The bigger the dissolved molecule, the higher the Sc.
Where Sc shows up
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Mass transfer correlations. The Sherwood number (Sh, the mass-transfer Nusselt analog) is correlated against Reynolds (Re) and Schmidt (Sc) in dimensionless form. The Chilton-Colburn analogy says Sh/(Re × Sc^(1/3)) is roughly constant for similar geometries, which is how engineers predict gas absorption rates without rebuilding wind tunnels.
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Distillation column design. Tray and packing efficiency depend on Sc. Higher Sc means slower mass transfer per unit height of column, so you need more theoretical plates.
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Atmospheric pollution modeling. How fast does a tracer (CO₂, methane, aerosol) spread through the boundary layer? Sc shows up in the eddy diffusivity scaling.
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Biotech and bioreactors. Oxygen transfer rate from sparger bubbles to bacterial culture depends on Sc for O₂ in the broth.
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Electrochemistry. Mass transfer of redox species to an electrode surface depends on Sc; this sets fundamental limits on electrolysis and battery rates.
The Lewis number connection
Le = Sc / Pr = α / D
The Lewis number compares thermal to mass diffusivity directly. It tells you whether a hot plume spreads faster than a chemical plume. Combined with Sc and Pr, the three dimensionless groups fully characterize coupled heat and mass transfer.
A common confusion
People sometimes write Sc with the kinematic and dynamic viscosities mixed up. Both expressions are valid:
Sc = ν / D = μ / (ρ × D)
But you must use ν (m²/s) with D (m²/s), or μ (Pa·s = kg/m·s) with ρ × D (kg/m·s after multiplication). Mixing units is the easiest mistake here. This calculator takes ν and D both in m²/s, the cleanest combination.
Why the Schmidt name
Ernst Heinrich Wilhelm Schmidt was a German engineer who in the 1920s worked on heat and mass transfer, including the boundary layer theory that Prandtl had developed for momentum. Schmidt’s name attached to the mass-transfer dimensionless number, paralleling Prandtl’s role in heat transfer. The convention has stuck for a century.