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The purpose of this article is to provide a practical, engineering-accurate workflow to estimate external mass transfer coefficients from Sherwood correlations across common geometries and contacting devices, with clear variable definitions, validity ranges, and calculation examples that can be applied directly in design and troubleshooting.
1. What “external mass transfer coefficient” means in practice
In most engineering models, the external mass transfer coefficient represents resistance in the fluid film adjacent to a surface or interface, not inside a porous solid and not inside the bulk fluid. It links a molar flux to a driving force through an empirical coefficient.
Common forms you will see are shown below.
Solid–fluid (single phase): N_A = k_c (C_A,b - C_A,s)
Gas–liquid (two film concept, liquid-side coefficient):
N_A = k_L (C_A* - C_A,b)
Gas–liquid (gas-side coefficient, concentration form):
N_A = k_G (C_A,b - C_A,i)
Dimensionless conversion:
Sh = k_c L / D_AB => k_c = Sh D_AB / LNote : The “external” qualifier matters because Sherwood correlations predict film transport controlled by convection and diffusion near the boundary. If the controlling resistance is internal diffusion (porous catalyst), chemical reaction, or interfacial thermodynamics, a Sherwood-only estimate can be misleading.
2. The core workflow: from operating data to k using Sherwood correlations
2.1 Define what you need: k for which phase and which driving force
Before selecting a Sherwood correlation, define the coefficient and driving force consistently.
| Goal | Typical symbol | Driving force basis | Common use |
|---|---|---|---|
| Single-phase external transport to/from a solid surface | kc | Concentration difference (bulk to surface) | Dissolution, corrosion, electrode mass transfer, boundary-layer uptake |
| Gas–liquid transfer (liquid-film control emphasized) | kL | Concentration difference (C* − Cb) | Absorption, stripping, aeration, fermentation |
| Gas–liquid transfer (gas-film control emphasized) | kG | Partial pressure difference (p − p*) or concentration | Volatile solutes, low-solubility systems |
| Overall transfer including both films | KL, KG | Overall driving force with equilibrium relation | Design calculations when both films matter |
2.2 Choose a characteristic length and the correct velocity definition
For Sherwood correlations, the characteristic length L and the Reynolds number definition are not optional details. L is tied to geometry (plate length, particle diameter, tube diameter) and the velocity must match the correlation’s original meaning (superficial velocity in a packed bed, free-stream velocity for an external flow, average velocity for a duct).
Note : Many large errors come from mixing velocities. For packed beds, using interstitial velocity in a correlation built for superficial velocity can shift k by factors of 1.5–3 depending on porosity and exponent values.
2.3 Compute dimensionless groups (Re, Sc) using film-phase properties
Most Sherwood correlations have the generic form Sh = f(Re, Sc). Use fluid properties of the phase in which the film exists. For liquid-film transfer use liquid viscosity and density. For gas-film transfer use gas properties.
Re = ρ u L / μ Sc = μ / (ρ D_AB)
Where:
ρ = density
u = characteristic velocity (per correlation definition)
L = characteristic length
μ = dynamic viscosity
D_AB = molecular diffusivity of A in B (same phase)2.4 Select the correlation by geometry and flow regime, then calculate Sh and k
After selecting Sh, convert to k using k = Sh D_AB / L. If you need k in different units, convert after computing k in consistent SI units to avoid hidden scaling mistakes.
3. High-use Sherwood correlations by geometry and contactor
The correlations below are widely used for external mass transfer coefficient estimation. Always check that your Reynolds and Schmidt numbers lie within the intended range and that the flow situation matches the correlation assumptions.
3.1 Flow over a flat plate (external boundary layer)
Use plate length x (local) or L (overall) as characteristic length and free-stream velocity U.
Local (laminar, from leading edge): Sh_x = 0.332 Re_x^(1/2) Sc^(1/3)
Average over length L (laminar):
Sh_L = 0.664 Re_L^(1/2) Sc^(1/3)
Average over length L (turbulent, with laminar leading region often neglected):
Sh_L = 0.037 Re_L^(0.8) Sc^(1/3)These forms are especially useful for evaporation, absorption at wetted walls, and mass transfer at large planar surfaces in forced convection.
3.2 External flow around a single sphere or cylinder (particles, droplets, bubbles)
For many engineering estimates around isolated particles, spheres, droplets, and bubbles in crossflow, a common structure is Sh = 2 + (convection term). The “2” represents the stagnant diffusion limit around a sphere.
General form (often used for spheres): Sh = 2 + C Re^m Sc^(1/3)
Example widely applied structure:
Sh = 2 + 0.6 Re^(1/2) Sc^(1/3)Use particle diameter d as L and the relative velocity between fluid and particle as u. For bubbles and droplets, be careful with slip velocity and deformation at higher Reynolds numbers.
Note : If the particle is not isolated (dense suspensions, packed beds), single-particle correlations can underpredict or overpredict k because wakes and tortuous flow alter transport. Prefer packed-bed correlations for fixed beds.
3.3 Internal flow in a circular tube (fully developed vs developing)
For flow inside a tube, L is typically the tube diameter D and u is the average velocity. Many correlations are expressed as a mass-transfer analog of Nusselt correlations. The Schmidt number dependence is often near 1/3 for turbulent flow.
Common turbulent-form structure: Sh = 0.023 Re^(0.8) Sc^n
Typical choice:
n ≈ 1/3 for many liquid systems
Then:
k_c = Sh D_AB / DEntrance effects and developing concentration boundary layers can increase transfer compared with fully developed assumptions, particularly at short lengths or high Sc.
3.4 Packed beds (fixed-bed external film to particles)
Packed beds are a major use-case for external mass transfer coefficient estimation. Correlations are typically written with particle diameter d_p as characteristic length and superficial velocity u_s. A common structure resembles:
Sh = A + B Re_p^m Sc^(1/3)
Where:
Re_p = ρ u_s d_p / μ
L = d_p
k_c = Sh D_AB / d_pBecause bed void fraction, particle shape, and flow distribution matter, design practice often uses conservative selections and checks sensitivity. If you have pressure-drop data, pairing a friction-factor-based j-factor approach can improve consistency.
3.5 Stirred tanks and agitated vessels (mass transfer near solids or interfaces)
In stirred tanks, external mass transfer is commonly correlated with impeller speed, power input, and geometry. When a Sherwood form is used, it often involves a Reynolds number built from impeller diameter and tip speed, and a Schmidt number using the liquid phase.
In design, these systems are frequently handled using kLa correlations rather than k alone. If you specifically need k at a solid surface (e.g., dissolving particles), ensure the correlation matches the solid size range and agitation regime.
3.6 Gas–liquid systems and the role of kL, kG, and interfacial area
For gas–liquid transfer, Sherwood correlations can estimate kL around bubbles or droplets if you have a hydrodynamic estimate of bubble diameter and slip velocity. However, most industrial sizing depends on kLa (the volumetric mass transfer coefficient), which multiplies the liquid-side film coefficient by interfacial area per unit volume.
k_L a = (k_L) (a)
If you only estimate k_L from Sherwood correlations but do not estimate a,
you cannot predict absorption rate in a reactor volume.Note : In gas–liquid design, kLa is often far more uncertain than kL alone because interfacial area depends strongly on coalescence, surfactants, gas holdup, and energy dissipation.
4. A practical correlation selection map
The table below helps select a Sherwood correlation family quickly based on geometry and flow situation, then identifies what to treat as L and u.
| Situation | Use L as | Use u as | Correlation family form | Typical Sc exponent |
|---|---|---|---|---|
| External flow over flat plate | Plate length (x or L) | Free-stream velocity | Sh ∝ Re^(1/2) or Re^(0.8) | 1/3 |
| Crossflow over isolated sphere | Sphere diameter d | Relative velocity | Sh = 2 + C Re^m Sc^(1/3) | 1/3 |
| Internal flow in tube (turbulent) | Tube diameter D | Average velocity | Sh = 0.023 Re^0.8 Sc^n | ~1/3 |
| Packed bed (film to particles) | Particle diameter d_p | Superficial velocity u_s | Sh = A + B Re_p^m Sc^(1/3) | 1/3 |
| Bubble/droplet transfer estimate | Bubble diameter d_b | Slip or relative velocity | Sh = 2 + C Re^m Sc^(1/3) | 1/3 |
5. Step-by-step example calculation (single-phase, external transfer to a particle)
This example illustrates the workflow and unit handling. The same structure applies to external mass transfer coefficient estimation in many devices.
5.1 Given data
Goal: Estimate k_c for solute A transferring through a liquid film to a spherical particle.
Given:
Particle diameter, d = 2.0 mm = 2.0e-3 m
Free-stream liquid velocity relative to particle, u = 0.20 m/s
Liquid density, ρ = 1000 kg/m^3
Liquid viscosity, μ = 1.0e-3 Pa·s
Diffusivity, D_AB = 1.0e-9 m^2/s
Choose correlation (sphere, external flow estimate):
Sh = 2 + 0.6 Re^(1/2) Sc^(1/3)5.2 Calculate Re and Sc
Re = ρ u d / μ = (1000)(0.20)(2.0e-3) / (1.0e-3) = 400
Sc = μ / (ρ D_AB)
= (1.0e-3) / (1000 * 1.0e-9)
= 10005.3 Calculate Sh and convert to k
Sh = 2 + 0.6 Re^(1/2) Sc^(1/3) = 2 + 0.6 (400)^(1/2) (1000)^(1/3) = 2 + 0.6 (20) (10) = 2 + 120 = 122
k_c = Sh D_AB / d
= 122 (1.0e-9) / (2.0e-3)
= 6.1e-5 m/sThis computed mass transfer coefficient can now be used in a flux equation with a consistent driving force definition (bulk-to-surface concentration difference).
Note : If your system has strong natural convection, particle settling, or non-spherical shapes, the relative velocity and the chosen Sherwood correlation can dominate the uncertainty. Treat k as an estimate and perform sensitivity checks.
6. Engineering checks to avoid common failure modes
6.1 Confirm that external film resistance is actually the controlling resistance
If reaction is fast at the surface or inside a porous solid, external k might not control the overall rate. Similarly, in gas–liquid systems the dominant resistance may be in the gas film, in the liquid film, or shared via an overall coefficient.
6.2 Verify the appropriate diffusivity DAB and phase consistency
Diffusivity depends strongly on phase and temperature. A gas-phase diffusivity is typically orders of magnitude larger than a liquid-phase diffusivity. Using the wrong D_AB can shift k by orders of magnitude.
6.3 Be explicit about characteristic length
For plates, L is a streamwise length. For internal pipe flow, L is the pipe diameter. For particles in external flow, L is particle diameter. For packed beds, L is particle diameter but u is superficial velocity unless a specific correlation states otherwise.
6.4 Watch the Schmidt number effect in liquids
High Schmidt numbers are common in liquids (Sc can be hundreds to thousands). This implies thin concentration boundary layers. Correlations with Sc^(1/3) can still be used, but the regime may be sensitive to surface roughness, turbulence intensity, and whether your flow is truly turbulent at the film scale.
6.5 Use j-factor (Chilton–Colburn) consistency when combining heat, momentum, and mass transfer
When you also have pressure drop or heat transfer data, a j-factor approach can provide a consistency check. In many turbulent flows, the mass transfer j-factor is linked to friction factor trends, allowing cross-validation of k estimates.
7. Practical guidance for reporting and documenting k estimates
When documenting an external mass transfer coefficient estimation, include the correlation name or form, the definitions of Re and Sc used, the characteristic length, and the fluid properties with temperature. This makes the calculation auditable and prevents accidental re-use with mismatched assumptions.
| Documentation item | What to record | Why it matters |
|---|---|---|
| Correlation form | Sh = f(Re, Sc), include constants and exponents | Ensures correct regime and geometry |
| Definitions | Re definition, Sc definition, velocity basis | Avoids hidden factor errors |
| Characteristic length | L = D, d_p, plate length, etc. | Directly scales k |
| Properties and temperature | ρ, μ, D_AB at stated T | Controls Re, Sc, and k magnitude |
| Applicability range | Re and Sc range used vs intended range | Prevents misuse |
FAQ
How do I convert a Sherwood correlation result into k in practical units like cm/s or m/h.
First compute k in SI units using k = Sh D_AB / L with D_AB in m^2/s and L in m, which gives k in m/s. Then convert: 1 m/s = 100 cm/s, and 1 m/s = 3600 m/h. Converting after the calculation reduces unit mistakes.
Which properties should I use for Re and Sc if temperature varies along the equipment.
Use film-phase properties evaluated at a representative temperature for the region controlling mass transfer. If the temperature variation is large, compute k in segments using local properties and then integrate or average in a way consistent with the flux model. Using inlet properties alone can be inaccurate when viscosity or diffusivity changes significantly.
When should I prefer a packed-bed correlation over a single-sphere correlation.
Use a packed-bed correlation when particles are in a fixed bed because flow tortuosity, wake interference, and void fraction effects change the boundary layer structure. A single-sphere correlation is more appropriate for isolated particles, dilute suspensions, or well-separated objects in an external flow field.
Why does the Schmidt number usually appear as Sc^(1/3) in Sherwood correlations.
The one-third exponent is consistent with boundary layer scaling for convection–diffusion in many forced convection regimes, especially when analogies to heat transfer are used. It reflects how molecular diffusion thickness changes with diffusivity relative to momentum diffusion. Some regimes and specialized correlations use different exponents, so the exact value should follow the selected correlation.
Can I use Sherwood correlations to estimate kL in a stirred tank for gas absorption.
You can estimate a local k around bubbles or droplets if you have bubble size and slip velocity, but most design work for stirred tanks uses kL a correlations because interfacial area a strongly affects the rate. If you only estimate kL without estimating a, you cannot predict volumetric absorption performance.