Library / Dimensional analysis and similarity / DIM-15
Dimensional analysis and similarity
Sherwood number
Ratio of convective to diffusive mass transfer: the mass-transfer coefficient times a length, divided by the mass diffusivity.
- Ref
- DIM-15
- Rev
- v1
- Field
- Dimensional analysis and similarity
- Result
- Sh
Sh = kmL/DAB compares convective mass transfer with pure diffusion across the length L, and is the mass-transfer counterpart of the Nusselt number. km is a concentration-based coefficient, defined so that the species flux equals km times the concentration difference. A value of 1 means convection adds nothing to diffusion over the length L; larger values mean the flow thins the diffusion layer. For a sphere in a stagnant fluid the limit is Sh = 2, based on its diameter.
Ratio of convective to diffusive mass transfer: the mass-transfer coefficient times a length, divided by the mass diffusivity. It is the mass-transfer counterpart of the Nusselt number.
Calculator
- Equivalent diffusion-layer thickness, L/Sh0.1 mm
- Mass-transfer coefficient2×10−5 m/s
- Sh with the length doubled and km unchanged200
- Sh for pure diffusion from a sphere (limit)2
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Sh = 100 means mass transfer is 100 times what diffusion alone would give across 10 mm, equivalent to a stagnant diffusion layer 0.1 mm thick.
- In design the calculation usually runs the other way: a correlation Sh = f(Re, Sc) gives Sh, and then km = Sh DAB/L gives the coefficient.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| km Mass-transfer coefficient (concentration based) | 1e-4 to 100 mm/s | 0.5 | 500,000 | 500,000 | +10% | +10 % |
| DAB Binary mass diffusivity | 0.01 to 20,000 10⁻⁹ m²/s | 0.01 | 20,000 | 20,000 | +10% | −9.09 % |
| L Characteristic length | 0.01 to 1,000 mm | 0.1 | 10,000 | 10,000 | +10% | +10 % |
Sweep of Mass-transfer coefficient (concentration based)
| Mass-transfer coefficient (concentration based) (mm/s) | Sherwood number |
|---|---|
| 1e-4 | 0.5 |
| 5.623e-4 | 2.812 |
| 0.003162 | 15.81 |
| 0.01778 | 88.91 |
| 0.1 | 500 |
| 0.5623 | 2,812 |
| 3.162 | 15,810 |
| 17.78 | 88,910 |
| 100 | 500,000 |
Sweep one variable
Variables and how each one contributes
Sherwood number
What the equation gives you. Everything else feeds it.
Mass-transfer coefficient (concentration based)
The concentration-based mass-transfer coefficient, in mm/s, defined so that the flux equals km times the concentration difference. Raises the Sherwood number in direct proportion. Raising it by 10 % (to 0.022) moves the result by +10 %, from 100 to 110.
Characteristic length
The characteristic length, in mm: a diameter, plate length or pipe diameter, matching the correlation used. Raises the Sherwood number in direct proportion. Raising it by 10 % (to 11) moves the result by +10 %, from 100 to 110.
Binary mass diffusivity
The binary mass diffusivity of the species, in 10⁻⁹ m²/s (a solute in water is about 1 to 2). Lowers the Sherwood number in inverse proportion. Raising it by 10 % (to 2.2) moves the result by −9.09 %, from 100 to 90.91.
Worked example with the values above
- Convert to m: km = 0.02 mm/s = 2×10−5 m/s; L = 10 mm = 0.01 m; DAB = 2 × 10⁻⁹ m²/s.
- Numerator: kmL = 2×10−5 × 0.01 = 2×10−7 m²/s.
- Sh = 2×10−7 / 2×10−9 = 100.
Assumptions and limits
- km is a concentration-based coefficient with a defined reference concentration difference.
- L and DAB are those used in the Sherwood correlation being compared with.
Common mistakes
- Using a coefficient defined on a different driving force, such as partial pressure or mole fraction, without converting it.
- Entering km in m/s rather than mm/s: 2 × 10⁻⁵ m/s is 0.02 mm/s.
- Using a different length from the correlation’s.
Statistics and uncertainty
The result is a product of powers, so to first order its relative error is the root sum of squares of each input’s relative error multiplied by its exponent (km × 1, L × 1, DAB × −1). The diffusivity usually carries the largest uncertainty, and the coefficient km from an experiment depends on the driving concentration difference being well defined.
Where it is used
- Converting a measured mass-transfer coefficient into a dimensionless number for a correlation.
- Finding the coefficient and then the flux in evaporation, dissolution, drying and absorption problems.
Maths you need
- Mass-transfer coefficient: flux per unit concentration difference
- Nusselt number: the heat-transfer analogue