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What is a Sherwood Number? | Process Engineering Glossary
What is a Sherwood Number?
In piping engineering and process engineering, the Sherwood number (Sh) is the dimensionless parameter that quantifies the ratio of convective mass transfer to diffusive mass transfer across a fluid boundary layer. Named after Thomas Kilgore Sherwood, a pioneer of mass transfer research at MIT, it is defined as:
Sh = k × L / D_AB
Where k is the convective mass transfer coefficient (m/s), L is the characteristic length of the geometry (m), and D_AB is the binary diffusion coefficient of the transferred species A through the carrier fluid B (m²/s). Rearranging gives the engineering application of the Sherwood number directly:
k = Sh × D_AB / L
This is precisely how the Sherwood number is used in practice: Sh is first calculated from an empirical correlation as a function of the Reynolds number Re and the Schmidt number Sc for the specific geometry and flow conditions. The mass transfer coefficient k — which is what the engineer actually needs to calculate the column height, reactor performance, or membrane area — is then extracted from the Sherwood number using the diffusion coefficient and the characteristic length.
The Sherwood number is the exact mass transfer analogue of the Nusselt number in heat transfer. The Nusselt number Nu = h × L / k_fluid is the dimensionless heat transfer coefficient; the Sherwood number Sh = k × L / D_AB is the dimensionless mass transfer coefficient. Every Nusselt number correlation Nu = f(Re, Pr) has a direct Sherwood number analogue Sh = f(Re, Sc), with the Prandtl number Pr replaced by the Schmidt number Sc and Nu replaced by Sh. This mathematical equivalence — the heat-mass transfer analogy — is one of the most useful and extensively exploited relationships in engineering science, allowing mass transfer performance to be estimated from heat transfer measurements and vice versa.
Applications of the Sherwood Number
Corrosion Mass Transfer in Pipelines
Flow-accelerated corrosion (FAC) of carbon steel in flowing water depends on the rate at which corrosive species (dissolved oxygen, H⁺) are transported to the pipe wall and corrosion products (Fe²⁺, Fe³⁺) are transported away. The corrosion rate under mass transfer limitation is proportional to k × (C_bulk − C_wall). Using the pipe flow Sherwood number:
k = 0.023 × Re^0.8 × Sc^0.33 × D_AB / D
Higher velocity (higher Re) gives higher Sh and higher k, increasing the corrosion rate for mass-transfer-limited corrosion mechanisms. This explains why FAC is worst at elbows and downstream of flow disturbances where local velocity and turbulence are elevated — the increased local Sherwood number at these locations gives higher local k and therefore higher local corrosion rate.
Bioreactor Oxygen Transfer
Aerobic fermentation bioreactors supply dissolved oxygen by sparging air. The gas-bubble mass transfer coefficient k_L is calculated from the Sh correlation for rising bubbles:
Sh = 2 + 0.31 × Eo^(1/3) × (g × ΔρL / σ)^(1/3) × d_b (Eötvös number-based correlation)
Or from the simpler penetration theory (Higbie model) for a bubble rising at velocity u_b:
k_L = 2 × √(D_O₂ × u_b / (π × d_b))
Both forms give k_L values of 1 to 5 × 10⁻⁴ m/s for typical air bubbles in water, with the specific value depending on bubble size and rise velocity. The total oxygen transfer rate OTR = k_L × a × (C* − C_L), where a is the specific interfacial area (m²/m³), determines whether the bioreactor can supply sufficient oxygen for the microbial culture at its maximum oxygen demand.
Benefits of the Sherwood Number Concept
Dimensionless framework for mass transfer. The Sh = f(Re, Sc) correlation framework allows mass transfer coefficients for any fluid, any geometry, and any operating condition to be predicted from validated dimensionless correlations without requiring empirical measurements for every specific combination of fluid properties and equipment geometry.
Direct analogy to heat transfer. The Sh-Nu equivalence allows the enormous body of heat transfer experimental data and validated correlations to be directly translated to mass transfer predictions through the (Sc/Pr)^(1/3) conversion factor — dramatically reducing the experimental effort needed to characterise mass transfer in new geometries.
Identifies external mass transfer limitations. Computing the Damköhler number Da = k_rxn / k_external from the Sherwood number immediately identifies whether external film diffusion is rate-limiting in a catalytic reactor, directing the engineer toward the appropriate corrective action (higher fluid velocity, smaller particle size) when Da >> 1.
Limitations to Consider
Correlation geometry specificity. The Ranz-Marshall correlation for spheres does not apply to cylinders, flat plates, or packed beds with different pore structures. The Wakao-Funazkri correlation for packed beds applies to close-packed spheres and may not accurately represent catalyst pellets with unusual shapes (rings, trilobes, honeycombs) or beds with wide particle size distributions. Always verify that the chosen Sh correlation was developed for a geometry that matches the application.
Sc^(1/3) exponent has limits. The Sc^(1/3) dependence in most Sh correlations is an empirical approximation valid for 0.6 < Sc < about 10³. For very high Sc systems (Sc > 10⁴, as encountered with large polymer molecules or protein diffusion in water), the exponent may deviate from 1/3 toward lower values, and dedicated high-Sc correlations are needed. For Sc < 0.1 (liquid metals), the mass transfer is conduction-dominated and the standard Sh ∝ Sc^(1/3) correlations underpredict the mass transfer coefficient.
Interphase correlations require care for multicomponent systems. The Sh correlation gives a binary diffusion coefficient D_AB that is appropriate for dilute systems where species A diffuses through a dominant species B. For concentrated multicomponent systems, the effective diffusivity must account for coupling between diffusion fluxes of different species through the Maxwell-Stefan equations, replacing D_AB with a composition-dependent effective diffusivity that changes along the column or reactor.
Sherwood Number FAQ
What is the Sherwood number and how does it relate to the mass transfer coefficient in process engineering?
The Sherwood number Sh = k × L / D_AB is the dimensionless mass transfer coefficient, giving the ratio of convective to diffusive mass transfer across a boundary layer. Process engineering uses it to calculate the convective mass transfer coefficient k = Sh × D_AB / L from empirical correlations Sh = f(Re, Sc), where Re is the flow Reynolds number and Sc = ν/D_AB is the Schmidt number. It is the exact mass transfer analogue of the Nusselt number in heat transfer — every Nu = f(Re, Pr) correlation has a Sh = f(Re, Sc) counterpart, with the Prandtl number replaced by the Schmidt number. At Re = 0 for a sphere (stagnant fluid), Sh = 2 — the fundamental lower limit from pure diffusion to a spherical surface.
How do Sherwood number correlations determine column height and packing selection in absorption design?
In an absorption column or packed column, the Sherwood number for the gas phase gives k_G and the Sherwood number for the liquid phase gives k_L through their respective phase-specific correlations at the design gas and liquid loadings. The film coefficients k_G and k_L combine through the two-film resistance model (1/K_L = 1/k_L + H/k_G, where H is the Henry’s Law constant) to give the overall mass transfer coefficient K_L. Column height Z = N_OL × H_OL, where H_OL = L/(K_L × a × Ω) is the height of a transfer unit — inversely proportional to K_L × a. Higher Sh (from higher gas or liquid velocity, or from packing with higher interfacial area a) gives lower H_OL and shorter column height for the same separation. Packing selection compares j_D factor correlations — which give Sh normalised by Re × Sc^(1/3) — at the design operating point to identify which packing achieves the highest mass transfer efficiency per unit pressure drop.
How does the Sherwood number analogy with the Nusselt number enable estimation of mass transfer from heat transfer data?
The Chilton-Colburn analogy Sh/Nu = (Sc/Pr)^(1/3) allows mass transfer coefficients to be estimated directly from heat transfer measurements at the same flow conditions and geometry, without separate mass transfer experiments. For gas systems where Sc ≈ Pr ≈ 0.7, the ratio (Sc/Pr)^(1/3) ≈ 1 and k ≈ h/(ρ × Cp) — the heat transfer coefficient divided by the volumetric heat capacity gives the mass transfer coefficient numerically. For liquid systems where Sc >> Pr (because D_AB,liquid << α), the Sh significantly exceeds Nu at the same Re but k remains much lower than h/(ρ × Cp) because D_AB,liquid is so small. This analogy underpins psychrometric calculations (drying rate from wet-bulb heat transfer data), corrosion rate prediction from cooling system heat transfer performance, and scaling of mass transfer correlations between solvents with different Sc values when experimental Sh data exist only for one reference fluid.
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