Introduction & Context

The diffusion‑layer thickness, \(h\), characterises the distance over which a solute must travel from the bulk liquid to the interface before it can be transferred across a phase boundary. In process engineering this parameter is essential for estimating mass‑transfer rates in stirred‑tank reactors, gas‑liquid absorbers, and liquid‑liquid extraction columns. A thin diffusion layer (small \(h\)) implies a high liquid‑side mass‑transfer coefficient, \(k_L\), and therefore faster overall transfer, which directly impacts reactor sizing, energy consumption, and product quality.

Methodology & Formulas

The calculation follows a sequence of physical relationships derived from the impeller operating conditions and fluid properties.

Rotational speed conversion – Convert the impeller speed from revolutions per minute to revolutions per second: \[ N_{\text{rps}} = \frac{N_{\text{rpm}}}{60} \]
Impeller tip speed (characteristic velocity) – The linear speed at the impeller periphery: \[ U = \pi \, D_{\text{imp}} \, N_{\text{rps}} \]
Reynolds number (turbulent regime) – Dimensionless group comparing inertial to viscous forces: \[ \mathrm{Re} = \frac{\rho \, U \, D_{\text{imp}}}{\mu_{\text{eff}}} \] where \(\mu_{\text{eff}}\) is the effective dynamic viscosity (minimum value imposed to avoid division by zero).
Kinematic viscosity – Ratio of dynamic viscosity to density: \[ \nu = \frac{\mu_{\text{eff}}}{\rho} \]
Schmidt number – Ratio of momentum diffusivity to mass diffusivity: \[ \mathrm{Sc} = \frac{\nu}{D} \] where \(D\) is the molecular diffusivity of the solute in the liquid.
Sherwood number (empirical correlation for turbulent impeller flow) – Relates convective mass transfer to diffusion: \[ \mathrm{Sh} = C_{\text{SH}} \, \mathrm{Re}^{\,M_{\text{RE}}} \, \mathrm{Sc}^{\,M_{\text{SC}}} \] \(C_{\text{SH}}, M_{\text{RE}},\) and \(M_{\text{SC}}\) are empirical constants.
Liquid‑side mass‑transfer coefficient – Obtained from the Sherwood number: \[ k_L = \frac{\mathrm{Sh}\, D}{D_{\text{imp}}} \]
Diffusion‑layer thickness – Inverse relationship with the mass‑transfer coefficient: \[ h = \frac{D}{k_L} \]

Validity Checks

The empirical correlation is applicable only within defined ranges of the dimensionless groups. The table below summarises the recommended limits.

Dimensionless Group	Valid Range	Implication if Out of Range
\(\mathrm{Re}\) (Reynolds number)	\(1.0 \times 10^{4} \le \mathrm{Re} \le 1.0 \times 10^{6}\)	Correlation may under‑predict or over‑predict mass transfer; consider alternative laminar or fully turbulent models.
\(\mathrm{Sc}\) (Schmidt number)	\(0.7 \le \mathrm{Sc} \le 1.0 \times 10^{4}\)	Viscosity or diffusivity extremes require modified exponents or different empirical constants.

Result Presentation

After evaluating the expressions, the following quantities are typically reported:

Impeller tip speed, \(U\) (m s\(^{-1}\))
Reynolds number, \(\mathrm{Re}\) (dimensionless)
Schmidt number, \(\mathrm{Sc}\) (dimensionless)
Sherwood number, \(\mathrm{Sh}\) (dimensionless)
Liquid‑side mass‑transfer coefficient, \(k_L\) (m s\(^{-1}\))
Diffusion‑layer thickness, \(h\) (m) – often converted to micrometres for practical interpretation.

The diffusion layer (or hydrodynamic boundary layer) is the region adjacent to a solid surface where mass transport is dominated by molecular diffusion rather than bulk convection. Its thickness directly controls the rate at which reactants reach, and products leave, the surface. A thinner layer means higher mass‑transfer coefficients, faster reaction rates, and more efficient use of catalysts or electrodes. Conversely, an overly thick layer can become the bottleneck in scale‑up, leading to lower yields and inconsistent product quality.

Perform a limiting‑current measurement on a rotating disk electrode (RDE). Record the steady‑state limiting current (\(I_{\text{lim}}\)) at several rotation speeds.
Calculate the mass‑transfer coefficient (\(k_m\)) using the Levich equation: \(I_{\text{lim}} = nFAk_mC_b\), where \(n\) is electrons transferred, \(F\) is Faraday’s constant, \(A\) is electrode area, and \(C_b\) is bulk concentration.
Convert \(k_m\) to diffusion layer thickness (\(\delta\)) via \(\delta = D/k_m\), with \(D\) being the molecular diffusivity of the species.
Plot \(\delta\) versus the inverse square root of rotation speed to verify the expected \(1/\sqrt{\omega}\) relationship.

Agitation speed: Higher impeller RPM reduces \(\delta\) by increasing shear and turbulence.
Impeller type and geometry: Axial flow impellers (e.g., pitched‑blade) generally produce thinner layers near the wall than radial flow impellers.
Fluid viscosity: More viscous liquids develop thicker diffusion layers for a given agitation condition.
Temperature: Raising temperature lowers viscosity and increases diffusivity, both of which thin the layer.
Tank dimensions and baffle configuration: Larger tanks or poorly baffled systems can have zones of low turbulence, leading to locally thicker layers.

Determine the Reynolds number (\(\mathrm{Re}\)) for the impeller: \(\mathrm{Re} = N D^2 \rho / \mu\), where \(N\) is impeller speed, \(D\) is impeller diameter, \(\rho\) is fluid density, and \(\mu\) is viscosity.
Calculate the Schmidt number (\(\mathrm{Sc}\)): \(\mathrm{Sc} = \nu / D\), with \(\nu = \mu/\rho\) (kinematic viscosity) and \(D\) the molecular diffusivity.
Use an empirical correlation for the mass‑transfer coefficient, such as the Sherwood‑Reynolds‑Schmidt relation: \(\mathrm{Sh} = k_m D / D = 0.6 \mathrm{Re}^{0.5} \mathrm{Sc}^{0.33}\) (for typical stirred tanks).
Convert the obtained \(k_m\) to diffusion layer thickness: \(\delta = D / k_m\).

Worked Example – Estimating the Liquid-Side Diffusion Layer Thickness in a Stirred Vessel

A process engineer needs to estimate the mass-transfer coefficient and the corresponding diffusion layer thickness for oxygen absorption into water in a laboratory-scale stirred tank. The vessel is a flat-bottom cylinder with a single pitched-blade impeller. The following data have been recorded at steady state.

Knowns

Impeller speed: 300 rpm (5 rps)
Impeller diameter: 0.1 m
Tank volume: 0.01 m³
Liquid density: 1000 kg/m³
Liquid viscosity: 1 cP (0.001 Pa·s)
Oxygen diffusion coefficient in water: 1 × 10⁻⁹ m²/s
Temperature: 25 °C
Atmospheric pressure: 1 atm

Step-by-Step Calculation

Convert rotational speed to tip speed \[ U = \pi N D = \pi \times 5 \times 0.1 = 1.571\ \text{m/s} \]
Calculate kinematic viscosity \[ \nu = \frac{\mu}{\rho} = \frac{0.001}{1000} = 1 \times 10^{-6}\ \text{m}^2/\text{s} \]
Compute Reynolds number \[ \mathrm{Re} = \frac{U D}{\nu} = \frac{1.571 \times 0.1}{1 \times 10^{-6}} = 157\,080 \]
Determine Schmidt number \[ \mathrm{Sc} = \frac{\nu}{D} = \frac{1 \times 10^{-6}}{1 \times 10^{-9}} = 1000 \]
Estimate Sherwood number using the correlation for turbulent flow in stirred vessels \[ \mathrm{Sh} = 0.037\ \mathrm{Re}^{0.8}\ \mathrm{Sc}^{0.33} = 0.037 \times 157\,080^{0.8} \times 1000^{0.33} = 5189 \]
Compute liquid-side mass-transfer coefficient \[ k_L = \frac{\mathrm{Sh}\ D}{D} = \frac{5189 \times 1 \times 10^{-9}}{0.1} = 5.19 \times 10^{-5}\ \text{m/s} \]
Relate \(k_L\) to diffusion layer thickness \[ \delta = \frac{D}{k_L} = \frac{1 \times 10^{-9}}{5.19 \times 10^{-5}} = 1.93 \times 10^{-5}\ \text{m} = 19.3\ \mu\text{m} \]

Final Answer

The effective diffusion layer thickness for oxygen transport into the stirred water is 19 µm.

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