Schmidt Number Equation
The Schmidt number compares momentum diffusivity (kinematic viscosity) to mass diffusivity. It is the mass-transfer counterpart of the Prandtl number. When Sc ≈ 1, velocity and concentration boundary layers grow at the same rate. When Sc >> 1, mass diffuses much more slowly than momentum.
Sc = ν / D
How It Works
The Schmidt number compares momentum diffusivity (kinematic viscosity) to mass diffusivity. It is the mass-transfer counterpart of the Prandtl number. When Sc ≈ 1, the velocity and concentration boundary layers grow at the same rate. When Sc >> 1, mass diffuses much more slowly than momentum, producing a very thin concentration boundary layer. Sc appears in Sherwood-number correlations used to predict mass-transfer coefficients in chemical reactors, absorbers, and separation columns, just as Pr appears in Nusselt-number correlations for heat transfer.
Example Problem
CO₂ diffuses in water at 25 °C. The kinematic viscosity of water is 8.9 × 10⁻⁷ m²/s and the diffusion coefficient of CO₂ in water is 1.92 × 10⁻⁹ m²/s. What is the Schmidt number?
- Identify the known values: kinematic viscosity ν = 8.9 × 10⁻⁷ m²/s, mass diffusivity D = 1.92 × 10⁻⁹ m²/s.
- Determine what we are solving for: the Schmidt number Sc, which quantifies the relative thickness of velocity and concentration boundary layers.
- Write the Schmidt number equation: Sc = ν / D.
- Substitute the known values: Sc = 8.9 × 10⁻⁷ / 1.92 × 10⁻⁹.
- Divide the exponential terms: 10⁻⁷ / 10⁻⁹ = 10² = 100.
- Compute the result: Sc = 8.9 / 1.92 × 100 = 463.5. The concentration boundary layer is much thinner than the velocity boundary layer.
A Schmidt number of 464 means the concentration boundary layer is much thinner than the velocity boundary layer, which is typical for dissolved gases in liquids. For gases at atmospheric pressure, Sc is usually between 0.5 and 2.
When to Use Each Variable
- Solve for Schmidt Number — when you know the kinematic viscosity and mass diffusivity, e.g., characterizing the boundary layer behavior for a dissolved gas in a liquid.
- Solve for Kinematic Viscosity — when you know the Schmidt number and diffusivity, e.g., back-calculating fluid viscosity from mass-transfer experiments.
- Solve for Mass Diffusivity — when you know the Schmidt number and kinematic viscosity, e.g., estimating diffusion coefficients from published Sc correlations.
Key Concepts
The Schmidt number is the ratio of momentum diffusivity (kinematic viscosity) to mass diffusivity, making it the mass-transfer analogue of the Prandtl number for heat transfer. When Sc is close to 1, velocity and concentration boundary layers have similar thicknesses. For dissolved species in liquids, Sc is typically 100-1,000 because molecular diffusion in liquids is much slower than momentum diffusion, resulting in very thin concentration boundary layers.
Applications
- Chemical reactor design: using Sc in Sherwood-number correlations to predict mass-transfer coefficients
- Gas absorption columns: sizing packed towers where Sc determines the rate of gas dissolution into liquid
- Environmental modeling: estimating pollutant dispersion rates in rivers and atmospheric boundary layers
- Membrane separation: characterizing concentration polarization at membrane surfaces
Common Mistakes
- Confusing kinematic viscosity with dynamic viscosity — Sc uses kinematic viscosity (ν = μ/ρ), not dynamic viscosity (μ)
- Using gas-phase Sc values for liquid-phase calculations — liquids have Sc of 100-1,000, while gases are typically 0.5-2
- Neglecting temperature dependence — both viscosity and diffusivity change with temperature, significantly altering Sc
- Mixing up Schmidt and Prandtl numbers — Sc involves mass diffusivity while Pr involves thermal diffusivity
Frequently Asked Questions
What does the Schmidt number tell you about mass transfer in a fluid?
The Schmidt number tells you how thick the concentration boundary layer is relative to the velocity boundary layer. When Sc >> 1 (typical for liquids), the concentration boundary layer is much thinner than the velocity boundary layer, meaning mass transfer resistance is concentrated in a very thin region near surfaces. This determines how much agitation or flow speed is needed to enhance mass transfer.
How does the Schmidt number compare to the Prandtl number?
Both are dimensionless ratios of momentum diffusivity to another diffusivity. Sc = ν/D uses mass diffusivity, while Pr = ν/α uses thermal diffusivity. The Lewis number connects them: Le = Sc/Pr = α/D. In gases, both Sc and Pr are near 1. In liquids, Sc is typically 100-1,000 while Pr is 1-10, because mass diffusion in liquids is much slower than heat diffusion.
What is a typical Schmidt number for gases vs. liquids?
For gases at atmospheric pressure, Sc is usually between 0.5 and 2 (e.g., Sc ≈ 0.6 for H₂ in air, Sc ≈ 1.0 for CO in air). For dissolved species in liquids, Sc ranges from about 100 to over 1,000 because liquid-phase diffusion is much slower.
How is the Schmidt number used in mass transfer calculations?
It enters Sherwood-number correlations such as Sh = 0.023 Re⁰⋅⁸ Sc⁰⋅⁴ (analogous to Dittus-Boelter for heat transfer). From Sh, engineers calculate the mass-transfer coefficient needed to size absorbers, strippers, and other separation equipment.
What is the relationship between Schmidt, Prandtl, and Lewis numbers?
Le = Sc / Pr. If you know any two of these three dimensionless numbers, you can compute the third. This relationship is particularly useful in combustion modeling where all three appear together.
Why is the Schmidt number important in electrochemistry?
In electrochemical cells, the Schmidt number determines the thickness of the mass-transfer boundary layer at electrode surfaces. With Sc often exceeding 1,000 for ions in solution, the concentration boundary layer is extremely thin. This limits the maximum current density and is why techniques like rotating-disk electrodes are used to control and enhance mass transfer.
How does temperature affect the Schmidt number?
Temperature affects both kinematic viscosity and mass diffusivity, often in opposite directions. In liquids, viscosity decreases with temperature (reducing Sc) while diffusivity increases (also reducing Sc), so Sc drops significantly as temperature rises. In gases, both increase with temperature, and the net effect on Sc is smaller.
Schmidt Number Formula
The Schmidt number is a dimensionless ratio of momentum diffusivity to mass diffusivity:
Sc = ν / D
Where:
- Sc — Schmidt number (dimensionless)
- ν — kinematic viscosity, in m²/s
- D — mass diffusivity (diffusion coefficient), in m²/s
Sc is the mass-transfer analogue of the Prandtl number (Pr = ν/α). The Lewis number connects them: Le = Sc/Pr = α/D. In gases, ν and D are similar, so Sc ≈ 1. In liquids, D is much smaller, so Sc ranges from 100 to over 10,000.
Worked Examples
Environmental Engineering
What is the Schmidt number for CO&sub2; absorption in a packed column?
CO&sub2; absorbs into water at 25 °C. The kinematic viscosity of water is 8.9 × 10¹²&sup7; m²/s and the diffusivity of CO&sub2; in water is 1.92 × 10¹²&sup9; m²/s.
- Sc = ν / D = 8.9 × 10¹²&sup7; / 1.92 × 10¹²&sup9;
- Sc = 463.5
Sc >> 1 means the concentration boundary layer is much thinner than the velocity boundary layer. Column designers must account for this when sizing packing height.
Electrochemistry
What Schmidt number governs copper ion transport at an electrode?
CuSO&sub4; in water at 25 °C: ν = 1.0 × 10¹²&sup6; m²/s, D(Cu²³) = 7.2 × 10¹²¹&sup0; m²/s.
- Sc = ν / D = 1.0 × 10¹²&sup6; / 7.2 × 10¹²¹&sup0;
- Sc = 1,389
Very high Sc values are typical in electrochemical systems. The thin mass-transfer boundary layer limits the current density at electrodes, which is why rotating-disk electrodes are used to enhance transport.
Bioprocess Engineering
What is the Schmidt number for oxygen transfer in a bioreactor broth?
Fermentation broth at 37 °C: ν = 7.0 × 10¹²&sup7; m²/s, D(O&sub2;) = 2.1 × 10¹²&sup9; m²/s.
- Sc = ν / D = 7.0 × 10¹²&sup7; / 2.1 × 10¹²&sup9;
- Sc = 333.3
Oxygen is often the growth-limiting nutrient in aerobic fermentation. The high Sc means the mass-transfer boundary layer at bubble surfaces is thin, so agitation speed and sparger design strongly affect oxygen supply rate.
Related Calculators
- Prandtl Number Calculator — the heat-transfer analogue of the Schmidt number.
- Lewis Number Calculator — ratio of thermal to mass diffusivity (Sc/Pr).
- Sherwood Number Calculator — convective mass transfer ratio that depends on Sc.
- Reynolds Number Calculator — characterize flow regime for mass transfer correlations.
- Viscosity Converter — convert kinematic and dynamic viscosity units used in Sc calculations.
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