Sherwood Number: A Thorough Guide to the Sherwood Number in Mass Transfer

Manager Misc 12. May 2025 | 0

The Sherwood Number is a foundational concept in mass transfer, providing a dimensionless measure of convective transport relative to molecular diffusion at a boundary. In practical terms, it converts complex flow and boundary-layer phenomena into a single, scalable parameter that engineers can use to predict how quickly a species moves from a surface into a moving fluid (or vice versa). The Sherwood Number, typically denoted as Sh, is indispensable in the design of contactors, absorbers, dryers, reactors and countless other units where mass transfer between phases drives performance. This article journeys through the physics, mathematics, correlations, and real-world applications of the Sherwood Number, with practical guidance for students, researchers and practising engineers in the United Kingdom and beyond.

What is the Sherwood Number?

The Sherwood Number is defined as Sh = kL / D_AB, where:

k is the external mass transfer coefficient, with units of metres per second (m/s).
L is a characteristic length that depends on geometry (for a sphere, L is often taken as the diameter; for a plate, it can be the length along the flow; for a cylinder, appropriate choices are used depending on orientation).
D_AB is the diffusion coefficient of species A in the carrier phase B, with units of square metres per second (m^2/s).

In essence, the Sherwood Number compares the rate of mass transfer by convection to the rate that would occur by diffusion alone across a boundary layer. A higher Sherwood Number indicates a more effective convective transport, while a lower value points to diffusion-dominated transfer. When used alongside the Reynolds Number (Re) and the Schmidt Number (Sc), Sh becomes a powerful predictive tool because it encapsulates the influences of fluid flow, geometry and molecular diffusion in a single dimensionless form.

Historical context and the naming of the Sherwood Number

The term Sherwood Number has its roots in classic mass-transfer research from the early to mid-twentieth century. The concept emerged as researchers sought a simple way to compare systems with different fluids, temperatures and geometries. The name honours a British researcher (or group of researchers) whose work popularised the approach to external mass transfer. Since then, a large library of correlations has grown up around the Sherwood Number, enabling engineers to predict mass-transfer coefficients from readily available flow data and fluid properties. The enduring value of the Sherwood Number lies in its generality: it applies from small laboratory cells to large industrial towers, across gases and liquids, whenever diffusion competes with convection at a boundary.

Mathematical foundations: connecting Sh, Re and Sc

The power of the Sherwood Number comes from its connection to other dimensionless groups. The three most important are:

Reynolds Number (Re): Re = ULρ/μ, a measure of the relative importance of inertial to viscous forces in the flowing fluid. It informs whether the flow is laminar or turbulent and influences the thickness of the convective boundary layer.
Schmidt Number (Sc): Sc = μ/(ρD_AB), the ratio of momentum diffusivity to mass diffusivity. A high Sc indicates that mass diffusion is slow compared with momentum diffusion, which tends to thicken the boundary layer and reduce Sh for a given flow.
Sherwood Number (Sh): Sh = kL / D_AB, the dimensionless mass-transfer coefficient that bridges the gap between diffusion-controlled transfer and convection-enhanced transfer.

In practice, engineers express Sh as a function of Re and Sc: Sh = f(Re, Sc). The exact form of f depends on geometry, flow regime and sometimes temperature, viscosity and concentration. A broad set of correlations has been developed from experimental data and validated through numerical simulations, offering a toolbox for rapid design and scale‑up without the need for complex, bespoke experiments each time.

Geometry-based Sherwood Number correlations

Correlations for the Sherwood Number differ with geometry and flow regime. The following are representative, commonly used forms that appear in many mass-transfer textbooks and engineering handbooks. They provide practical starting points for estimation and design.

Spheres and droplets: external mass transfer around a spherical object

The classic Ranz–Marshall correlation is widely applied for spheres in external flow. A broadly accepted form is:

Sh ≈ 2 + 0.6 Re^1/2 Sc^1/3, valid for Re up to about 10^3 and typical liquids and gases. This equation captures the asymptotic approach from diffusion-dominated transfer (Sh ≈ 2 for very small Re) to convection-enhanced transfer as flow strength increases. For many real systems, this simple expression provides a reliable estimate and serves as a baseline against which more complex models are tested.

Flow over a flat plate: boundary layer mass transfer

For laminar flow over a flat plate, the widely used correlation is:

Sh ≈ 0.664 Re^1/2 Sc^1/3, with Re based on the distance from the leading edge, Re_x. This relation is valid for Re_x up to roughly 5 × 10^5 in many liquids. It reflects the growth of the boundary layer thickness along the plate and the resulting reduction in mass-transfer velocity away from the stagnation point.

In turbulent boundary layers or different plate configurations, other correlations become more appropriate, often taking the form Sh = A Re^m Sc^n with adjusted coefficients to reflect the enhanced mixing in the turbulent regime.

Cylinders, ellipsoids and more complex geometries

For cylinders in cross-flow or for other elongated bodies, the exponent values and coefficients in Sh = f(Re, Sc) differ from those for spheres or plates. In practice, engineers rely on correlations tailored to the exact cross-section, aspect ratio, surface roughness and flow orientation. When exact data are not available, a common approach is to select a sphere or plate correlation as a first approximation and then refine it with experimental data or CFD results for the specific geometry.

Practical strategies for estimating the Sherwood Number

In design work, a practical, systematic approach helps ensure reliable predictions while avoiding unnecessary complexity. Here is a concise workflow you can apply in many industrial settings:

Clarify geometry and flow regime: Decide whether you are dealing with a sphere, a plate, a cylinder, or a more intricate body. Determine whether the flow is laminar or turbulent and estimate Re accordingly.
Choose an appropriate correlation: Start with a well-documented Sh = f(Re, Sc) correlation that matches the geometry and Re regime. If no perfect match exists, justify the closest analogue and note the uncertainty.
Compute the diffusion coefficient: Obtain D_AB for the solute in the carrier phase at the operating temperature. This parameter often controls how easily the species diffuses through the fluid.
Calculate the Schmidt number: Sc = μ/(ρD_AB). This quantity captures the relative diffusion rates of momentum and mass and helps indicate whether conduction or convection dominates.
Determine the characteristic length: Select L carefully based on geometry and how the boundary layer develops in your system. Document this choice for transparency and repeatability.
Compute the mass-transfer coefficient and flux: k = Sh·D_AB / L and N_A = k·A·(C_A,bulk − C_A,surface). Ensure all units are consistent and the area A is defined precisely.
Validate and refine: If possible, compare the predicted flux with experimental data or high-fidelity simulations. Use discrepancies to guide refinements in geometry description, correlations or fluid-property inputs.

Worked example: external mass transfer from a small sphere in water

To illustrate how the Sherwood Number is used in practice, consider a small sphere of diameter 5 mm (L = 0.005 m) in water, with a solute diffusing in the surrounding liquid. The diffusion coefficient D_AB is about 1 × 10^-9 m^2/s. The ambient flow gives a characteristic velocity U ≈ 0.05 m/s. Water properties are ρ ≈ 1000 kg/m^3 and μ ≈ 1.0 × 10^-3 Pa·s. The Schmidt number is Sc ≈ μ/(ρD_AB) ≈ 1.0 × 10^-3 / (1000 × 1 × 10^-9) ≈ 1000. The Reynolds number is Re = ULρ/μ ≈ 0.05 × 0.005 × 1000 / (1 × 10^-3) ≈ 250.

Applying the Ranz–Marshall form for a sphere, Sh ≈ 2 + 0.6 Re^1/2 Sc^1/3. This yields Re^1/2 ≈ √250 ≈ 15.81 and Sc^1/3 ≈ 1000^1/3 ≈ 10. Therefore Sh ≈ 2 + 0.6 × 15.81 × 10 ≈ 2 + 94.9 ≈ 97.0.

From here, the external mass-transfer coefficient is k = Sh·D_AB / L ≈ 97 × (1 × 10^-9) / 0.005 ≈ 1.94 × 10^-7 m/s. If the surface area of the sphere is A ≈ 4π(L/2)^2 ≈ 7.85 × 10^-5 m^2, and the driving concentration difference is ΔC, the flux is N_A ≈ k·A·ΔC ≈ 1.94 × 10^-7 × 7.85 × 10^-5 × ΔC ≈ 1.52 × 10^-11 × ΔC (mol/s). This example demonstrates how a relatively modest mass-transfer coefficient can translate into a quantifiable, design-relevant flux when scaled by area and concentration driving force. It also highlights why accurate inputs for D_AB, L and the appropriate correlation are essential for credible predictions.

Applications across industry and science

The Sherwood Number features prominently in many sectors. Here are some representative areas where Sh informs design, optimisation and analysis:

In gas–liquid contactors such as absorbers and scrubbers, Sh connects the gas-side or liquid-side mass-transfer coefficient to concentration differences, allowing the sizing of towers, packed beds and trays to meet separation targets.
During convective drying, Sh helps predict drying rates as air velocity, particle size and temperature rise or fall. Coating systems rely on Sh to gauge how quickly a liquid film dries under flow conditions.
In bioreactors, external mass transfer around gas bubbles, impellers and porous beads controls nutrient supply. The Sherwood Number guides optimisation of agitation, aeration and reactor geometry to avoid diffusion limitations that restrict growth or product formation.
For packed beds and porous catalysts, Sh interacts with internal diffusion (characterised by pore diffusivity) to determine overall reaction rates. Correctly predicting Sh helps ensure effective utilisation of catalyst surfaces.
In electrodeposition, the mass-transfer rate to the electrode surface affects coating quality, uniformity and current efficiency. The Sherwood Number underpins cell design and stirring strategies to maintain smooth operation.

Numerical modelling and simulation: the role of CFD

Computational fluid dynamics (CFD) has become a standard tool to study mass transfer in complex geometries. In CFD, the species transport equation is solved concurrently with the Navier–Stokes equations, allowing researchers to observe concentration fields and local mass-transfer coefficients. The Sherwood Number emerges naturally from simulations as Sh = kL/D when a boundary-condition-based approach is used (for example, prescribing a concentration boundary at a surface or using a finite-difference approximation for flux). CFD is particularly valuable for multiphase flows, irregular particle clusters, and transient events where empirical correlations fall short. For practitioners, CFD results are often used to validate correlations, probe regimes outside existing datasets, or optimise reactor internals and packing arrangements with a view to enhancing overall mass transfer.

Common mistakes and how to avoid them

Even with straightforward definitions, misapplication of the Sherwood Number is common. Here are practical tips to avoid typical errors:

Misidentifying L: The characteristic length must reflect the boundary layer geometry. Using a geometric length that is too large or too small will distort k and, consequently, the predicted flux.
Applying the wrong regime: Ensure the Re range and flow regime match the correlation. A laminar-boundary-layer correlation can be inappropriately applied to turbulent flow, leading to underestimation or overestimation of mass transfer.
Ignoring variable properties: Temperature, pressure and composition can alter μ, ρ and D_AB. If these properties vary significantly across the operating range, consider a local or temperature-corrected Sh.
Neglecting multiphase nuances: In gas–liquid–solid systems, interfacial phenomena and phase holdups can influence effective k. Simple single-stage correlations may not capture these effects, necessitating more advanced models or pilot tests.
Presenting Sh without context: Reporting Sh without accompanying Re and Sc information makes it hard to gauge applicability. Always present Sh with the governing Re and Sc for clarity and comparability.

Relating Sherwood Number to heat and mass transfer analogies

The Sherwood Number shares a strong kinship with the Nusselt Number (Nu), which quantifies convective heat transfer. The mathematical form of both Sh and Nu is similar: each is a ratio of convective to diffusive transport in their respective domains. This parallel means that techniques, correlations and experimental methods developed for thermal problems can often be transposed to mass transfer with appropriate substitutions (D for thermal diffusivity α, etc.). The analogy is particularly helpful during scale-up, where confirming your mass-transfer correlations with well-tested heat-transfer correlations can provide an intuitive cross-check. In many teaching laboratories, the Sherwood Number serves as a concrete example of how dimensionless analysis unifies transport phenomena across transport disciplines.

Practical tips for students and new engineers

Start with a clear problem statement: Identify geometry, flow regime, phases involved and the solute to be transported. The accuracy of Sh hinges on these choices.
Choose a reliable correlation: Use well-cited correlations appropriate for your geometry and Re range. If in doubt, consult multiple sources and compare results.
Document assumptions: Record your L choice, the basis for k, the D_AB used, and the property data. Transparency makes peer review and future work much easier.
Use Sh alongside Re and Sc: Always report Re and Sc with Sh. This makes your results portable and easier to compare with literature and data.
Iterate with data: If possible, calibrate your correlation against experimental measurements or high-fidelity simulations. A small amount of data can significantly improve predictive credibility.

Frequently asked questions about the Sherwood Number

What is the Sherwood Number used for?: It quantifies external mass transfer relative to diffusion, enabling prediction of mass transfer rates in a wide range of systems, from beads in suspension to industrial contactors.
Can the Sherwood Number be zero or negative?: In physical systems, Sh is positive and greater than or equal to 2 in diffusion-limited cases for many simple correlations. Negative values are non-physical.
Is the Sherwood Number the same for gas–liquid and liquid–gas systems?: Sh depends on the geometry, flow, and the diffusion coefficient, which differ between gas and liquid phases. The same concepts apply, but correlations must be chosen for the correct phase pair and regime.
How is the Sherwood Number measured?: Experimentally, Sh is inferred from measured mass flux and known diffusion coefficients, or from direct measurement of boundary-layer concentrations. In practice, researchers often determine k from the flux and then compute Sh using Sh = kL/D_AB.

Final reflections: why the Sherwood Number matters

In mass transfer practice, the Sherwood Number acts as a bridge between the microscopic physics of diffusion and the macroscopic engineering realities of design and scale-up. It translates intricate flow patterns, boundary-layer phenomena and diffusion pathways into a compact, dimensionless gauge that can be used to compare, predict and optimise. Across industries—whether refining captive energy usage in dryers, enhancing efficiency in contactors, or accelerating bioprocess throughput—the Sherwood Number remains a pragmatic, widely applicable tool. By combining a sound understanding of geometry, flow regime and material properties with robust correlations and, where possible, experimental calibration, engineers can harness the Sherwood Number to deliver safer, more efficient and more cost-effective processes.

Sherwood Number: A Thorough Guide to the Sherwood Number in Mass Transfer