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Sherwood Number Calculator

Sherwood number equals mass transfer coefficient times length divided by diffusion coefficient

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How It Works

The Sherwood number is the mass-transfer analogue of the Nusselt number. It compares the total (convective) mass transfer at a surface to what pure molecular diffusion alone would provide. A Sh of 1 means there is no convective enhancement; in turbulent flows Sh can reach hundreds or thousands, indicating that convective mixing dominates. Chemical engineers use Sherwood correlations -- functions of Reynolds and Schmidt numbers -- to predict mass-transfer coefficients for reactor design, gas absorption columns, and separation membranes without running full experiments.

Example Problem

A 0.05 m sphere dissolves in water. The mass-transfer coefficient is measured at k = 3.6 × 10⁻⁵ m/s and the diffusion coefficient of the solute in water is 1.2 × 10⁻⁹ m²/s. What is the Sherwood number?

  1. Identify the known values: mass-transfer coefficient k = 3.6 × 10⁻⁵ m/s, characteristic length L = 0.05 m (sphere diameter), and diffusion coefficient D = 1.2 × 10⁻⁹ m²/s.
  2. Write the Sherwood number formula: Sh = kL / D.
  3. Multiply k by L: kL = 3.6 × 10⁻⁵ × 0.05 = 1.8 × 10⁻⁶.
  4. Divide by D: Sh = 1.8 × 10⁻⁶ / 1.2 × 10⁻⁹ = 1,500.
  5. Interpret the result: Sh = 1,500 means convective mass transfer is 1,500 times more effective than diffusion alone, typical for a sphere in a moderate liquid flow.
  6. Verify with a correlation check: for Re ~ 10⁴ and Sc ~ 800, the Ranz-Marshall correlation (Sh = 2 + 0.6 Re½ Sc⅓) gives a value in the same order of magnitude, confirming the measurement.

Sh = 1,500 means convective mass transfer is 1,500 times more effective than diffusion alone, typical for a sphere in a moderate liquid flow.

When to Use Each Variable

  • Solve for Sherwood Number — when you know the mass-transfer coefficient, characteristic length, and diffusion coefficient, e.g., evaluating convective enhancement in a packed column.
  • Solve for Mass Transfer Coefficient — when you know Sh, length, and diffusivity, e.g., predicting how fast a solute dissolves from a sphere into a flowing liquid.
  • Solve for Characteristic Length — when you know the other variables, e.g., sizing catalyst pellets to achieve a target mass-transfer rate.
  • Solve for Diffusion Coefficient — when you know Sh, length, and mass-transfer coefficient, e.g., determining diffusivity from experimental mass-transfer data.

Key Concepts

The Sherwood number is the dimensionless ratio of total (convective + diffusive) mass transfer to diffusive-only mass transfer at a surface. A Sh of 1 means no convective enhancement — only molecular diffusion operates. In turbulent flows, Sh can reach hundreds or thousands, indicating that fluid motion dramatically accelerates mass transport. Sherwood-number correlations (functions of Re and Sc) allow engineers to predict mass-transfer rates without full experimentation.

Applications

  • Gas absorption tower design: predicting mass-transfer coefficients for sizing packed and tray columns
  • Catalytic reactor engineering: estimating external mass-transfer resistance around catalyst particles
  • Dissolution studies: modeling drug dissolution rates in pharmaceutical development
  • Corrosion engineering: predicting oxygen transfer rates to metal surfaces in flowing water

Common Mistakes

  • Confusing the mass-transfer coefficient k with thermal conductivity k — they share the same symbol but have completely different units and meanings
  • Using the wrong characteristic length — for a sphere it is the diameter, for a flat plate it is the plate length in the flow direction
  • Applying laminar correlations to turbulent flows — Sh correlations are regime-specific and give large errors if misapplied
  • Forgetting the minimum Sh = 2 for a sphere — any result below 2 for a sphere indicates a calculation error

Frequently Asked Questions

What does the Sherwood number tell you about mass transfer?

The Sherwood number quantifies how much convection enhances mass transfer beyond pure molecular diffusion. A Sh of 1 means only diffusion is at work. Higher values (Sh = 100–1,000+) mean convective motion is moving species far faster than diffusion alone could. Engineers use Sh to compare mass-transfer performance across different geometries and flow conditions.

How is the Sherwood number analogous to the Nusselt number?

Both have the same mathematical structure: Nu = hL/k for heat transfer and Sh = kₘL/D for mass transfer. The Chilton-Colburn analogy relates them directly, so heat-transfer correlations can estimate mass-transfer rates and vice versa. Replacing thermal conductivity with diffusivity and the heat-transfer coefficient with the mass-transfer coefficient converts one to the other.

What is the Sherwood number used for in chemical engineering?

It predicts the mass-transfer coefficient from known flow conditions and fluid properties. Engineers use Sh correlations (e.g., Sh = 2 + 0.6 Re½ Sc⅓) to design gas-liquid contactors, packed columns, and catalytic reactors without requiring expensive pilot-plant tests.

What is the minimum Sherwood number for a sphere?

For a sphere in a stagnant fluid, the exact solution gives Sh = 2. This represents pure diffusion from the sphere surface into an infinite medium. Any fluid motion increases Sh above this minimum value.

How do Reynolds and Schmidt numbers affect the Sherwood number?

Higher Reynolds numbers (faster flow or larger geometry) increase Sh because turbulence enhances mixing. Higher Schmidt numbers (thicker concentration boundary layer) also increase Sh. The classic Ranz-Marshall correlation Sh = 2 + 0.6 Re½ Sc⅓ shows both dependencies explicitly.

What characteristic length should I use for the Sherwood number?

Use the diameter for spheres and cylinders, the plate length in the flow direction for flat plates, and the packing diameter for packed beds. Using the wrong length is the most common source of Sh calculation errors.

Can the Sherwood number be less than 1?

In practice, Sh is always at least 1 (pure diffusion from a flat surface) and at least 2 for a sphere. A value below these minima indicates a calculation error, not a physical result. Convection always enhances or maintains mass transfer relative to diffusion alone.

Sherwood Number Formula

The Sherwood number is defined as:

Sh = kL / D

Where:

  • Sh — Sherwood number (dimensionless)
  • k — mass transfer coefficient (m/s)
  • L — characteristic length (m)
  • D — diffusion coefficient (m²/s)

The Sherwood number quantifies how much convection enhances mass transfer beyond pure molecular diffusion. For a sphere in stagnant fluid, the minimum Sh is 2. In turbulent flows, Sh can reach thousands.

Worked Examples

Chemical Engineering

What is the Sherwood number for a gas absorption tower packing?

A packed column uses 0.025 m Raschig rings. The mass transfer coefficient for CO&sub2; absorption is measured at 0.005 m/s, and the diffusion coefficient of CO&sub2; in water is 1.9 × 10¹²³ m²/s.

  • Sh = kL / D = 0.005 × 0.025 / 1.9 × 10¹²³
  • Sh = 1.25 × 10¹²¹ / 1.9 × 10¹²³
  • Sh ≈ 65.8

Sh = 65.8 indicates convection enhances mass transfer about 66 times over pure diffusion, typical for packed column internals.

Environmental Engineering

How fast does a pollutant disperse from a spherical source?

A 0.1 m sphere releases pollutant into flowing water. The measured Sh = 500 and D = 1 × 10¹²³ m²/s. What is the mass transfer coefficient?

  • k = Sh × D / L
  • k = 500 × 1 × 10¹²³ / 0.1
  • k = 0.005 m/s

Knowing k allows environmental engineers to predict how quickly pollutant concentration drops downstream.

Biomedical Engineering

What diffusion coefficient explains a drug tablet's dissolution rate?

A 0.01 m drug tablet dissolves with k = 2 × 10¹²¹ m/s and the measured Sh = 25. What is the effective diffusion coefficient?

  • D = kL / Sh
  • D = 2 × 10¹²¹ × 0.01 / 25
  • D = 8 × 10¹²³ m²/s

This diffusivity value helps pharmaceutical engineers predict dissolution rates at different flow conditions.

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