Schmidt number
The Schmidt number (Sc) is a dimensionless quantity in fluid mechanics and mass transfer that represents the ratio of a fluid's momentum diffusivity (kinematic viscosity, ν) to its mass diffusivity (D), expressed as Sc = ν / D.[1] This parameter, named after the German engineer Ernst Heinrich Wilhelm Schmidt (1892–1975), quantifies the relative rates at which momentum and mass are transported through a fluid medium, thereby influencing the structure of velocity and concentration boundary layers in convective processes.[2] Physically, a high Schmidt number indicates that mass diffuses much more slowly than momentum, leading to a thinner concentration boundary layer compared to the velocity boundary layer, which is particularly relevant in liquids where molecular diffusion dominates over turbulent mixing.[1] It serves as the mass transfer analog to the Prandtl number in heat transfer, enabling similar scaling analyses for coupled transport phenomena.[3] Typical values of the Schmidt number vary by fluid and conditions: for gases such as air at standard temperature and pressure, Sc is approximately 0.7 to 1, reflecting comparable momentum and mass diffusion rates; in liquids like water, Sc ranges from 100 to 10,000 or higher, due to the much lower mass diffusivity relative to viscosity.[4] In practical applications, the Schmidt number is essential for predicting mass transfer rates in processes such as chemical reactions, pollutant dispersion in environmental flows, gas absorption in oceans, and corrosion in engineering systems, where it helps determine whether diffusion or convection controls solute transport.[5] For turbulent flows, an effective turbulent Schmidt number (often around 0.7–1.0) extends its utility in computational fluid dynamics models to account for enhanced mixing.[4]Definition and Properties
Mathematical Formulation
The Schmidt number (Sc) is defined as the ratio of the kinematic viscosity (ν) to the molecular mass diffusivity (D) of a species in a fluid, expressed mathematically as\mathrm{Sc} = \frac{\nu}{D}.
This dimensionless quantity characterizes the relative rates of momentum and mass diffusion in fluid flows.[6] The kinematic viscosity ν represents the diffusivity of momentum and is given by ν = μ / ρ, where μ is the dynamic viscosity (with units Pa·s) and ρ is the fluid density (with units kg/m³); thus, ν has units of m²/s. The molecular mass diffusivity D quantifies the rate at which mass (e.g., a chemical species) diffuses through the fluid due to concentration gradients and also carries units of m²/s.[6] A units analysis confirms the dimensionless nature of Sc, as both ν and D share the same dimensions (length²/time), making Sc dimensionless.[6] The Schmidt number emerges naturally during the nondimensionalization of the governing equations in fluid dynamics, specifically the incompressible Navier-Stokes equations for momentum conservation and the species conservation (diffusion) equation. To nondimensionalize, characteristic scales are introduced: a reference velocity V, length L, time L/V, concentration c₀, and the equations are scaled accordingly. In the nondimensional Navier-Stokes equation, the viscous diffusion term becomes (1/Re) ∇² u, where Re = V L / ν is the Reynolds number and ∇* is the nondimensional gradient operator. Similarly, in the nondimensional species conservation equation, ∂c*/∂t* + u* · ∇* c* = (1/Pe) ∇² c, where Pe = V L / D is the Péclet number for mass transfer and c* is the nondimensional concentration. The Schmidt number then appears as Sc = ν / D = Pe / Re, linking the relative strengths of the diffusive terms in the two equations.[6]
Physical Interpretation
The Schmidt number represents the ratio of the momentum diffusivity (kinematic viscosity, ν) to the mass diffusivity (D) in a fluid, quantifying how much faster momentum diffuses compared to mass or species in processes involving convective mass transfer.[2][7] This ratio indicates the relative rates at which velocity perturbations (momentum) spread through the fluid versus how quickly dissolved species or solutes disperse, providing insight into the dominance of viscous effects over molecular diffusion in mass transport phenomena.[8] In boundary layer flows, the Schmidt number governs the relative thicknesses of the velocity (hydrodynamic) boundary layer and the concentration boundary layer. A high Schmidt number implies that mass diffusivity is much lower than momentum diffusivity, resulting in a thinner concentration boundary layer compared to the velocity boundary layer, where species gradients are steeper and confined closer to the surface.[8][9] Conversely, a low Schmidt number leads to broader concentration boundary layers, as mass diffuses more readily relative to momentum.[2] Typical Schmidt number values vary significantly between gases and liquids due to differences in molecular diffusion rates. For gases like air, Sc is approximately 0.7 under standard conditions, reflecting comparable momentum and mass diffusion.[7] In liquids such as water, Sc is much higher, often around 1000 for common solutes, owing to the substantially lower mass diffusivity in denser fluids.[2] These values highlight that Sc >> 1 for most liquids, emphasizing slower species spreading relative to momentum.[4] In simple laminar or turbulent flows, the Schmidt number qualitatively influences mass transfer coefficients by scaling the Sherwood number in correlations like Sh = f(Re, Sc), where higher Sc typically reduces the effective mass transfer rate for a given flow regime, as the limited molecular diffusion hinders solute transport across boundary layers.[5][10]Related Dimensionless Numbers
Comparison to Prandtl Number
The Prandtl number, denoted as Pr, is defined as the ratio of momentum diffusivity (kinematic viscosity, ν) to thermal diffusivity (α), expressed mathematically as \Pr = \frac{\nu}{\alpha}.[11] The Schmidt number (Sc) and Prandtl number share a fundamental analogy in transport phenomena, as both quantify the relative rates of momentum diffusion to another diffusive process—mass diffusion for Sc and thermal diffusion for Pr—thereby indicating the relative thicknesses of velocity and concentration/thermal boundary layers in convective flows. This similarity underpins the Reynolds analogy, which assumes equal turbulent transport rates for momentum, heat, and mass when Pr ≈ Sc ≈ 1, allowing simplified predictions of transfer coefficients in high-Reynolds-number flows.[12] Key differences arise in their applications: Sc governs mass transfer processes involving species concentration gradients, whereas Pr pertains to heat transfer driven by temperature gradients.[8] Typical values reflect these distinctions; for gases, Pr and Sc are both approximately 0.7–1.0, leading to comparable boundary layer behaviors, while in liquids, Pr is often around 1–10 (e.g., ~7 for water at room temperature), but Sc is significantly higher at 200–1500 due to slower mass diffusion relative to momentum.[2][11] This analogy extends to the Chilton-Colburn j-factor method, which correlates friction factors (f/2) with heat and mass transfer Stanton numbers via j_H = \St \Pr^{2/3} and j_m = \St_m \Sc^{2/3}, enabling prediction of mass or heat transfer coefficients from momentum transfer data across a wide range of Pr and Sc values (typically 0.6–10,000).[12]Relation to Lewis Number
The Lewis number (Le) is a dimensionless parameter that quantifies the relative diffusion rates of heat and mass in fluid systems undergoing simultaneous thermal and solutal transport. It is defined as the ratio of thermal diffusivity \alpha to species mass diffusivity D, \mathrm{Le} = \frac{\alpha}{D}. This relation connects directly to the Schmidt number (Sc) and Prandtl number (Pr) through \mathrm{Le} = \frac{\mathrm{Sc}}{\mathrm{Pr}}, where Sc measures momentum-to-mass diffusivity and Pr measures momentum-to-thermal diffusivity; in certain contexts, such as combustion modeling, the reciprocal form \mathrm{Le} = \mathrm{Pr}/\mathrm{Sc} is employed to emphasize mass diffusion dominance.[13][14] Physically, the Lewis number represents the comparative scales of thermal diffusion versus molecular diffusion of a species, influencing how temperature and concentration profiles evolve in coupled transport scenarios. For many gases, such as air-water vapor mixtures, Le is approximately 1 due to similar orders of magnitude for \alpha and D (typically $10^{-5} m²/s), implying aligned heat and mass boundary layers. In liquids, however, D is orders of magnitude smaller than \alpha (e.g., D \sim 10^{-9} m²/s versus \alpha \sim 10^{-7} m²/s for water), yielding Le ≫ 1 and indicating faster thermal equilibration relative to solute spreading.[15][16] In applications involving concurrent heat and mass transfer, such as evaporation from liquid surfaces or premixed combustion, the Lewis number governs the alignment of thermal and concentration fields; when Le ≈ 1, these fields scale similarly, enabling analogies between heat and mass transfer coefficients, whereas Le ≠ 1 introduces differential diffusion effects that can alter flame stability or evaporation rates. For instance, in combustion, Le > 1 stabilizes flames by promoting heat release over fuel diffusion losses, while Le < 1 (common for light fuels like hydrogen in air) can induce cellular instabilities. These insights are critical for designing processes like spray drying or gas turbine combustors where coupled transport dictates efficiency.[17][18] The Lewis number is named after Warren K. Lewis (1882–1975), the founding head of chemical engineering at MIT, whose seminal 1920s analyses of evaporation and drying processes first highlighted the interplay of heat and mass transfer, influencing fields like psychrometrics (e.g., wet-bulb thermometry assuming Le ≈ 1 for air-water systems) and industrial drying operations.[19][20]Turbulent Schmidt Number
Definition and Formulation
The turbulent Schmidt number, denoted Sc_t, quantifies the ratio of turbulent momentum diffusivity to turbulent mass diffusivity in fluid flows dominated by turbulence. It is mathematically formulated as Sc_t = \frac{\nu_t}{D_t} = \frac{\varepsilon}{\varepsilon_M}, where \nu_t represents the eddy kinematic viscosity, responsible for the turbulent transport of momentum, and D_t denotes the eddy diffusivity for mass, governing the turbulent dispersion of scalar species. This definition arises from the need to close the transport equations for both momentum and mass in turbulent flows, assuming analogous gradient-diffusion mechanisms for Reynolds stresses and scalar fluxes.[21][22] In the framework of Reynolds-averaged Navier-Stokes (RANS) equations, the turbulent Schmidt number emerges during the modeling of turbulent correlations. The RANS momentum equation incorporates the Boussinesq approximation for Reynolds stresses, -\overline{u_i' u_j'} \approx \nu_t \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3} k \delta_{ij}, where U_i is the mean velocity, k is turbulent kinetic energy, and primes denote fluctuations. Similarly, the RANS species transport equation models the turbulent scalar flux as -\overline{u_i' c'} = D_t \frac{\partial C}{\partial x_i}, with C as the mean species concentration. The relation D_t = \nu_t / Sc_t thus provides the necessary closure, linking mass transfer to the established eddy viscosity from turbulence models like k-ε. This formulation preserves the dimensionless character of Sc_t, analogous to the molecular Schmidt number, but shifts its determination from molecular properties (viscosity and diffusivity ratios) to the structural features of the turbulent flow, such as length and time scales of eddies.[23][24] Unlike the molecular Schmidt number, which varies widely (e.g., approximately 0.7 for gases like air but 600–1000 for liquids like water at room temperature), the turbulent Schmidt number typically assumes values between 0.7 and 1.0 in standard engineering models to reflect near-equivalent turbulent transport efficiencies for momentum and mass. These values are empirically calibrated for simplicity in simulations, though actual Sc_t can deviate based on flow regime, buoyancy effects, or near-wall behavior, highlighting its dependence on turbulence intensity rather than intrinsic fluid properties.[22][4]Modeling Approaches
In Reynolds-Averaged Navier-Stokes (RANS) simulations, particularly those employing the standard k-ε turbulence model, the turbulent Schmidt number (Sc_t) is commonly assumed to be a constant value, typically in the range of 0.7 to 0.9, to close the scalar transport equations under the eddy viscosity approximation.[25] This assumption simplifies the modeling of turbulent mass diffusivity by relating it directly to the turbulent viscosity, with many commercial CFD packages defaulting to Sc_t = 0.7 for passive scalar transport in free shear flows.[26] However, this fixed value often underperforms in complex geometries or non-canonical flows, where sensitivity analyses reveal optimal constants varying by up to 50% to match experimental spreading rates.[25] Advanced modeling approaches rely on the gradient diffusion hypothesis, which posits that the turbulent scalar flux is proportional to the mean scalar gradient, analogous to molecular diffusion but scaled by an eddy diffusivity. Under this hypothesis, Sc_t is defined as the ratio of turbulent momentum diffusivity (eddy viscosity) to turbulent mass diffusivity, frequently set equal to the turbulent Prandtl number (Pr_t) for simplicity, as both quantify analogous non-dimensional ratios in momentum and scalar transport.[25] In large eddy simulations (LES), variations of this approach incorporate subgrid-scale models where Sc_t is dynamically adjusted based on local flow resolvability, often yielding values closer to 0.4-0.7 in resolved high-Reynolds-number turbulence to account for anisotropic subgrid fluxes.[27] These methods improve accuracy over constant assumptions by capturing intermittency in scalar mixing, though they increase computational demands. Empirical correlations for Sc_t often express its dependence on the molecular Schmidt number (Sc) or Reynolds number (Re), particularly in high-Sc flows where molecular effects dominate near interfaces. For instance, direct numerical simulations across wide parameter ranges (Re_λ from 8 to 650, Sc from 1/2048 to 1024) indicate that Sc_t exhibits a unique functional form when plotted against the molecular Péclet number (Pe = Re Sc), collapsing scatter observed in individual Re or Sc dependencies and typically yielding Sc_t ≈ 0.6 for isotropic turbulence. In practical engineering contexts, such as jet-in-crossflow configurations, correlations show Sc_t increasing mildly with momentum flux ratios (e.g., from 0.7 to 1.0) or decreasing with Sc in boundary layers, enabling tuned predictions for mass transfer coefficients without full DNS. Modeling Sc_t remains challenging in near-wall regions, where low-Reynolds-number effects and damping of turbulence lead to significant uncertainties, often requiring wall functions or hybrid models to avoid overprediction of scalar gradients. In multiphase flows, such as sediment-laden or particle-dispersed systems, Sc_t deviates from single-phase assumptions due to interphase momentum transfer, with values exceeding 1.0 in non-dilute suspensions from reduced effective diffusivity. Typical literature ranges for Sc_t span 0.4 to 1.5 across environmental and engineering applications, reflecting these sensitivities and underscoring the need for flow-specific calibrations.[28]Applications
Laminar Flow Mass Transfer
In laminar flow mass transfer, the Schmidt number (Sc) plays a central role in the Sherwood number (Sh), which quantifies the ratio of convective to diffusive mass transfer and is typically expressed as a function of the Reynolds number (Re) and Sc, such as Sh = f(Re, Sc).[29] For mass transfer from a flat plate in a laminar boundary layer, the average Sherwood number over the plate length L is given bySh_{L,avg} = 0.664 \, Re_L^{1/2} \, Sc^{1/3},
where Re_L is based on the plate length and free-stream velocity, valid for dilute solutions and Sc > 0.6.[29] This correlation arises from solving the convective diffusion equation using similarity transformations analogous to the Blasius solution for momentum boundary layers, highlighting Sc's influence on the concentration boundary layer thickness. Higher values of Sc result in steeper concentration gradients near the surface because the molecular diffusivity is low relative to momentum diffusivity, leading to thinner concentration boundary layers and consequently lower mass transfer coefficients for a given flow rate. In external flows like the flat plate, this manifests as a weaker dependence of Sh on Re compared to low-Sc cases, where diffusion dominates more broadly.[29] For internal flows, such as in ducts, the effect is pronounced in developing regions, where high Sc confines mass transfer to a narrow layer adjacent to the wall, reducing overall rates unless enhanced by flow development.[30] Analytical solutions for laminar mass transfer in developing flows, such as the Graetz problem for circular ducts, provide exact expressions for the Sherwood number as a function of the Graetz number (Gz = Re Sc d/L, where d is the diameter and L the axial length). The Graetz series solution yields Sh approaching 3.66 for fully developed concentration profiles under constant wall concentration, but in the entrance region (high Gz), it scales with Sc through the diffusion equation eigenvalues.[31] For high Sc, the Lévêque approximation simplifies this by assuming a linear velocity profile near the wall, leading to the local Sherwood number scaling as Sh_x ∝ (Re Sc / (x/d))^{1/3}, where x is the axial distance.[32] Experimental validations of these correlations often employ electrochemical techniques to measure mass transfer in pipes and channels at high Sc (e.g., Sc ≈ 1000 for aqueous electrolytes), confirming the Lévêque form for entry lengths where Gz > 100.[33] A widely adopted average Sherwood number correlation for laminar developing flow in circular tubes is
Sh_{avg} = 1.62 \, (Re \, Sc \, d/L)^{1/3},
accurate to within 5-10% for short tubes (L/d < 0.1) and high Sc, as verified against dissolution and limiting current density data.[34] In parallel-plate channels, similar forms hold with adjusted constants, emphasizing Sc's role in amplifying entrance effects for low-diffusivity solutes.[35]