Reference ID: MET-9623 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Steady-state mass transfer through a stagnant film is a cornerstone concept in membrane separations, barrier packaging, and gas–liquid contacting equipment. The calculation quantifies how fast a species diffuses across a thin layer when the driving force (partial-pressure difference) and the film thickness are known. Engineers use the result to:
Size membrane area for oxygen or CO2 removal in bioreactors.
Predict shelf-life of food packaged under modified atmospheres.
Validate whether a diffusion barrier meets permeation specifications.
Methodology & Formulas
Convert geometric units
The film thickness \(z\) is supplied in micrometres; convert to centimetres for consistency with the diffusion coefficient.
\[ z\ [\text{cm}] = \frac{z\ [\mu\text{m}]}{10\,000} \]
Define permeability
Permeability \(P\) combines solubility and diffusivity into a single transport property.
\[ P = D\ S \]
where
\(D\) = diffusion coefficient, cm2 s−1
\(S\) = solubility coefficient, cm3(STP) cm−3 bar−1
Driving force
The partial-pressure difference across the film is
\[ \Delta p = p_{1} - p_{2} \]
with both pressures in bar.
Steady-state flux (Fick & Henry)
The one-dimensional flux in cm3(STP) per cm2 of film per second is
\[ J = \frac{P\ \Delta p}{z} \]
Convert to engineering units
To report the flux per square metre per day:
\[ J'\ \left[\frac{\text{cm}^{3}(\text{STP})}{\text{m}^{2}\ \text{day}}\right] = J\ \left[\frac{\text{cm}^{3}(\text{STP})}{\text{cm}^{2}\ \text{s}}\right] \times 10\,000\ \left[\frac{\text{cm}^{2}}{\text{m}^{2}}\right] \times 86\,400\ \left[\frac{\text{s}}{\text{day}}\right] \]
Typical transport regimes for gas diffusion through dense films
Regime
Reynolds analogy
Applicability
Molecular diffusion
Knudsen < 0.01
Dense polymer films, < 100 µm
Knudsen diffusion
Knudsen ≥ 0.01
Microporous membranes, < 0.1 µm pores
For diffusion across a stagnant film of thickness δ, k is defined as the ratio of the diffusion coefficient DAB to the film thickness: k = DAB/δ. Units are length/time (e.g., m s-1 or cm s-1). This form assumes dilute conditions and a linear concentration gradient.
The profile is effectively linear when the transferred species is dilute (< 5 mol %) or when the total molar flux is dominated by the non-diffusing component. Switch to a logarithmic (log-mean) driving force when equimolar counter-diffusion is absent and concentrations are high, so that bulk flow terms become significant.
Temperature strongly affects DAB via a T1.5–2 dependence, while pressure inversely affects gas-phase DAB (DAB ∝ 1/P). Correct k using:
k2 = k1 (T2/T1)n (P1/P2) for gases, where n ≈ 1.75–2
Liquid-phase k increases with T and decreases with μ; use Stokes–Einstein or Wilke–Chang corrections
Always re-evaluate δ if hydrodynamics change with P or T.
Worked Example: Steady-State O₂ Transfer Across a 25 µm Polymer Film
A small-scale bioreactor is designed to supply oxygen through a thin, gas-permeable silicone membrane. To verify that the membrane can meet the required O₂ flux, we need to predict the steady-state transfer rate across the film when the upstream partial pressure is 0.21 bar (air) and the downstream partial pressure is 0.01 bar (swept away by the culture medium).
Knowns
Film thickness, z = 25 µm = 0.0025 cm
Upstream O₂ partial pressure, p1 = 0.21 bar
Downstream O₂ partial pressure, p2 = 0.01 bar
Diffusion coefficient of O₂ in silicone, D = 4 × 10−9 cm2 s−1
Solubility coefficient, s = 0.25 cm3 (STP) cm−3 bar−1
