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What is a Schmidt Number? | Process Engineering Glossary
What is a Schmidt Number?
In piping engineering and process engineering, the Schmidt number (Sc) is a dimensionless fluid property that quantifies the ratio of momentum diffusivity (kinematic viscosity) to mass diffusivity:
Sc = ν / D_AB = μ / (ρ × D_AB)
Where ν is the kinematic viscosity (m²/s), D_AB is the binary diffusion coefficient of species A in carrier B (m²/s), μ is the dynamic viscosity (Pa·s), and ρ is the fluid density (kg/m³). Named after the German engineer Ernst Heinrich Wilhelm Schmidt, the Schmidt number plays exactly the same role in mass transfer that the Prandtl number plays in heat transfer — it is the second dimensionless group alongside the Reynolds number in every Sherwood number correlation for convective mass transfer.
The physical meaning of Sc is direct: it describes the relative ease with which momentum and mass diffuse through the fluid. When Sc >> 1, momentum diffuses much faster than mass — the velocity boundary layer is much thicker than the concentration boundary layer. When Sc ≈ 1, both boundary layers have similar thicknesses. When Sc << 1, mass diffuses faster than momentum — a situation that arises only for liquid metals and rarely encountered in conventional process engineering applications.
Engineers rarely encounter the Schmidt number in isolation. It appears inside Sherwood number correlations of the form Sh = f(Re, Sc), and calculating the mass transfer coefficient k from a given Sh requires Sc to be evaluated at the fluid temperature and composition of interest. Knowing how Sc varies across fluid classes — and what it means physically — allows the engineer to anticipate mass transfer performance, compare candidate solvents or process configurations, and apply the correct correlations for each fluid type.
Applications of the Schmidt Number
Liquid-Liquid Extraction Column Design
In liquid-liquid extraction, mass transfer occurs across a liquid-liquid interface from the continuous phase to the dispersed droplet phase (or vice versa). The Schmidt number for the solute in each liquid phase feeds into the continuous-phase and dispersed-phase mass transfer correlations. For the continuous liquid phase surrounding droplets of the dispersed phase, the external Sherwood number for mass transfer to a circulating drop is:
Sh_external = (2 + 0.6 × Re_drop^(1/2) × Sc_continuous^(1/3))
The high Schmidt number of the continuous liquid phase (Sc ≈ 200 to 2,000 for typical extraction solvents) means the external mass transfer coefficient k_external is relatively low — the concentration boundary layer around each droplet is thin but diffusion is slow. Increasing the continuous phase velocity relative to the drops (by mechanical agitation or pulsing the column) improves k_external by thinning the concentration boundary layer further.
Soil and Groundwater Remediation
Contaminant transport in soil involves mass transfer from the soil matrix to the pore water (desorption), and from the pore water to flowing groundwater (advective-dispersive transport). The Schmidt number for organic contaminants in water (Sc ≈ 1,000 to 5,000 for typical small organic molecules in water) enters the mass transfer correlations for both desorption from soil particles and for dispersion in porous media. High Sc indicates slow molecular diffusion relative to viscous momentum transport — which means in low-velocity groundwater flow (low Re), the mass transfer coefficient k is very small and remediation timescales can be extremely long.
Benefits of Understanding the Schmidt Number
Selects the correct Sherwood number correlation. Sh = f(Re, Sc) correlations depend critically on Sc. Gas-phase correlations developed for Sc ≈ 1 cannot be applied to liquid-phase mass transfer where Sc ≈ 500 without accounting for the enormous Sc difference in the Sh expression. Recognising that Sc for the specific fluid is the key parameter that determines which Sh correlation to use prevents the common error of applying gas-phase mass transfer correlations to liquid-phase systems.
Explains the gap between gas-phase and liquid-phase film coefficients. The observed fact that gas-phase mass transfer coefficients k_G are typically 100 to 1,000 times higher than liquid-phase coefficients k_L in the same column at the same Reynolds number — despite the lower D_AB in liquids — is entirely explained by the Schmidt number. Both phases have similar Sh values at the same Re, but k = Sh × D_AB / L gives a much lower k for the liquid phase because D_AB,liquid << D_AB,gas more than compensates for Sc_liquid >> Sc_gas.
Guides solvent selection for extraction and absorption. Comparing Schmidt numbers across candidate solvents at the process temperature provides a preliminary ranking of their mass transfer performance: lower Sc (higher D_AB and/or lower ν) gives higher k_L and better column performance at the same column geometry and operating conditions. Solvents with lower viscosity (lower ν) and smaller solute-solvent interactions (higher D_AB, lower Sc) are preferred from a mass transfer perspective, all else being equal.
Limitations to Consider
Sc^(1/3) simplification has limits. The Sc^(1/3) exponent in turbulent flow Sherwood correlations is empirically derived and is most accurate for Sc in the range 0.6 to about 100. For very high Sc fluids (Sc >> 1,000, such as polymer solutions or viscous oils), the exponent may differ from 1/3 and dedicated high-Sc correlations or direct measurement is needed. For Sc < 0.1 (liquid metals, for the mass transfer analogue), different asymptotic expressions based on the Péclet number Pe = Re × Sc apply.
Binary diffusion coefficient D_AB is composition-dependent. The Schmidt number is only constant for dilute solutions where D_AB is the tracer diffusivity of A in pure B. At finite concentrations, D_AB changes with composition and Sc therefore varies along the column or reactor, particularly in gas absorption near the feed where the absorbed species is concentrated. Full mass transfer models for concentrated systems require integrating over the varying Sc profile rather than using a single Sc value.
Sc assumes Newtonian fluid behaviour. For non-Newtonian fluids (polymer solutions, slurries, biological broths), the effective viscosity used in Sc must be the apparent viscosity at the representative shear rate near the mass transfer interface, not the bulk viscosity. In a stirred bioreactor with a shear-thinning culture broth, using the zero-shear viscosity to compute Sc overestimates the true Sc at the high-shear gas-liquid interface and underestimates k_L.
Schmidt Number FAQ
What is the Schmidt number and how does it relate to mass transfer in process engineering?
The Schmidt number Sc = ν/D_AB is the ratio of momentum diffusivity (kinematic viscosity ν) to mass diffusivity (binary diffusion coefficient D_AB). Process engineering uses it in Sherwood number correlations Sh = f(Re, Sc) to calculate convective mass transfer coefficients k = Sh × D_AB / L. It plays the identical role in mass transfer that the Prandtl number plays in heat transfer. Sc ≈ 0.6 to 3 for gas-phase diffusion, meaning the concentration and velocity boundary layers are similar in thickness. Sc ≈ 200 to 2,000 for dissolved species in water, meaning the concentration boundary layer is approximately (Sc)^(1/3) ≈ 6 to 12 times thinner than the velocity boundary layer, making liquid-phase mass transfer far more sensitive to surface renewal and turbulent mixing near the interface than gas-phase transport.
How does the Schmidt number determine film coefficients in absorption and packed column design?
In an absorption column or packed column, the gas-phase Schmidt number (Sc_gas ≈ 0.6 to 2 for most absorbed species in air or nitrogen) and the liquid-phase Schmidt number (Sc_liquid ≈ 200 to 2,000 for dissolved gases in aqueous solvents) independently determine the gas-phase and liquid-phase film coefficients through their respective Sherwood number correlations. The resulting k_G and k_L — when combined with the Henry’s Law equilibrium constant — give the overall mass transfer coefficient that governs column height. For highly soluble gases (low Henry’s Law constant), the liquid-film coefficient k_L controls the overall rate and the liquid-phase Sc is the dominant fluid property. For sparingly soluble gases (high Henry’s constant), k_G controls and the gas-phase Sc governs performance.
How does the Schmidt number compare to the Nusselt number and kinematic viscosity in mass transfer correlations?
The Schmidt number Sc = ν/D_AB and the Prandtl number Pr = ν/α have the same mathematical form — both are the ratio of kinematic viscosity ν to a diffusivity. The Sc-Pr heat-mass transfer analogy means every Nusselt number correlation Nu = f(Re, Pr) has a direct mass transfer counterpart Sh = f(Re, Sc), with Sc substituted for Pr. The Chilton-Colburn analogy extends this to allow heat transfer measurements to predict mass transfer coefficients: k/h_conv ≈ (D_AB/α) × (Pr/Sc)^(2/3), giving the mass transfer coefficient from a known heat transfer coefficient and the ratio of diffusivities. Sc >> Pr for most liquids (because D_AB,liquid << α_liquid) while Sc ≈ Pr for gases (because molecular diffusion and thermal diffusion have similar mechanisms), which explains why the heat-mass transfer analogy is more accurate for gases than for liquids.
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