Library / Dimensional analysis and similarity / DIM-14
Dimensional analysis and similarity
Schmidt number
Ratio of momentum diffusivity to the mass diffusivity of a species.
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- DIM-14
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- v1
- Field
- Dimensional analysis and similarity
- Result
- Sc
Sc = ν/DAB compares how fast momentum diffuses (the kinematic viscosity) with how fast a species diffuses. It is the mass-transfer counterpart of the Prandtl number. Gases have Sc of order 1 (about 0.6 for water vapour in air), while a solute in a liquid has Sc of order 10² to 10⁴, so the concentration boundary layer is much thinner than the velocity boundary layer, roughly by the factor Sc−1/3 in laminar flow. Take DAB for the species pair and fluid state of interest.
Ratio of momentum diffusivity to the mass diffusivity of a species. It is the mass-transfer counterpart of the Prandtl number: a large value means the concentration boundary layer is much thinner than the velocity boundary layer.
Calculator
- Sc1/37.937
- Concentration to velocity boundary-layer thickness, Sc−1/30.126
- Mass diffusivity0.002 mm²/s
- Sc if the diffusivity were doubled250
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Sc = 500: momentum diffuses 500 times faster than the species, so the concentration boundary layer is about 0.13 times as thick as the velocity boundary layer in laminar flow.
- Together with the Reynolds number, Sc gives the Sherwood number through a correlation such as Sh = f(Re, Sc), the mass-transfer counterpart of Nu = f(Re, Pr).
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ν Kinematic viscosity | 0.01 to 1,000 mm²/s | 5 | 500,000 | 500,000 | +10% | +10 % |
| DAB Binary mass diffusivity | 0.01 to 20,000 10⁻⁹ m²/s | 0.05 | 100,000 | 100,000 | +10% | −9.09 % |
Sweep of Binary mass diffusivity
| Binary mass diffusivity (10⁻⁹ m²/s) | Schmidt number |
|---|---|
| 0.01 | 100,000 |
| 0.06132 | 16,310 |
| 0.3761 | 2,659 |
| 2.306 | 433.6 |
| 14.14 | 70.71 |
| 86.72 | 11.53 |
| 531.8 | 1.88 |
| 3,261 | 0.3066 |
| 20,000 | 0.05 |
Sweep one variable
Variables and how each one contributes
Schmidt number
What the equation gives you. Everything else feeds it.
Kinematic viscosity
The kinematic viscosity of the fluid, in mm²/s (water about 1, air about 15). Raises the Schmidt number in direct proportion. Raising it by 10 % (to 1.1) moves the result by +10 %, from 500 to 550.
Binary mass diffusivity
The binary mass diffusivity of the species in the fluid, in 10⁻⁹ m²/s (a solute in water is about 1 to 2; a vapour in air is about 20000). Lowers the Schmidt number in inverse proportion. Raising it by 10 % (to 2.2) moves the result by −9.09 %, from 500 to 454.5.
Worked example with the values above
- Convert to m²/s: ν = 1 mm²/s = 1×10−6 m²/s; DAB = 2 × 10⁻⁹ m²/s.
- Divide: 1×10−6 / 2×10−9.
- Sc = 500.
Assumptions and limits
- A binary mixture or a dilute species, with the properties at the actual temperature and composition.
- The fluid viscosity is that of the carrying fluid.
Common mistakes
- Mixing the units: ν in mm²/s and DAB in 10⁻⁹ m²/s here.
- Using a gas-phase diffusivity for a liquid, which is about 10⁴ times larger.
- Confusing Sc with the Prandtl number: Pr is for heat, Sc is for a species.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (ν × 1, DAB × −1). Mass diffusivity usually has the largest uncertainty, often 10 % or more, and varies with temperature and concentration, so use a value for the actual conditions.
Where it is used
- Selecting and applying mass-transfer correlations for evaporation, dissolution, absorption and electrochemical processes.
- Estimating how thick the concentration boundary layer is relative to the velocity boundary layer.
Maths you need
- Kinematic viscosity: dynamic viscosity divided by density
- Binary diffusion: a species spreading through another fluid