Introduction & Context

Steady-state mass transfer through a thin film describes the movement of a chemical species across a barrier driven by a concentration or partial pressure gradient. In process engineering, this calculation is fundamental to the design of separation processes, packaging technology, and membrane science. It is primarily used to determine the barrier properties of materials, such as the oxygen transmission rate (OTR) in food packaging or the permeability of polymer membranes in gas separation units, and to ensure food contact material compliance. By assuming steady-state conditions, engineers can predict the rate at which a permeate will cross a film of known thickness and surface area, provided the material permeability remains constant under the operating conditions.

Methodology & Formulas

The calculation follows a systematic approach to determine the molar flux and the total molar flow rate across a film. The process begins by standardizing units to the SI system, specifically converting film thickness to meters and partial pressures to Pascals.

The pressure differential across the film is defined as:

$\Delta P = P_1 - P_2$

The molar flux (J), which quantifies the amount of substance passing through a unit area per unit time, is derived from Fick’s Law of diffusion as applied to permeability, and its relationship to the overall mass transfer coefficient can be explored in detail in the discussion of the overall mass transfer coefficient using the two‑film model.

$J = \frac{P_m}{L} \cdot \Delta P$

Finally, the total molar flow rate (ṅ) is calculated by scaling the flux by the total surface area of the film:

$\dot{n} = J \cdot A$

Parameter	Symbol	Constraint/Condition
Film Thickness	L	L > 0 (Must be a positive physical dimension)
Pressure Gradient	ΔP	If ΔP < 0, flux is negative (reverse diffusion)
Permeability	P_m	P_m > 0 (Must be a positive material property)
Flow Regime	Steady-State	Assumes constant concentration profile over time

A system is considered to be in a steady state when the concentration profile within the film does not change with respect to time. You can verify this by monitoring the following indicators:

The molar flux remains constant across all points in the film.
The concentration at the boundaries remains fixed over the observation period.
The accumulation term in the general mass balance equation is equal to zero.

To derive analytical solutions for steady-state mass transfer through a film, process engineers typically rely on these standard assumptions:

The system is isothermal and isobaric.
There are no chemical reactions occurring within the film.
The diffusion coefficient is constant throughout the medium.
The flow is one-dimensional, occurring only in the direction of the concentration gradient.

In many industrial applications, the stagnant film acts as the primary resistance to mass transfer. The impact is characterized by:

The film thickness, which is inversely proportional to the mass transfer coefficient.
The molecular diffusivity of the species, which dictates the rate of transport through the stagnant layer.
The total resistance, which is the sum of the film resistance and any additional resistances present in the bulk phases.

Worked Example: Steady-State Gas Permeation Through a Polymer Film

In a gas separation process, a thin polymer membrane is utilized to recover a specific component from a process stream. We must determine the molar flow rate of the gas permeating through a 50-micron-thick film under a steady-state pressure gradient.

Knowns:

Film thickness (L): 50.000 µm
Upstream pressure (p₁): 0.210 bar
Downstream pressure (p₂): 0.010 bar
Permeability coefficient: 2.000×10^-15 mol/(m·s·Pa)
Surface area (A): 0.500 m²

Calculation Steps:

Convert units to the SI system:
- L = 50.000 µm × 1×10^-6 m/µm = 5.000×10^-5 m
- p₁ = 0.210 bar × 100,000 Pa/bar = 21,000.000 Pa
- p₂ = 0.010 bar × 100,000 Pa/bar = 1,000.000 Pa
- Pressure difference (Δp) = 21,000.000 Pa − 1,000.000 Pa = 20,000.000 Pa
Calculate the molar flux (J) using Fick's Law for permeation:
$J = \frac{Permeability \times \Delta p}{L}$

$J = \frac{2.000e-15 \times 20,000.000}{5.000e-5} = 8.000e-7 \text{ mol/(m2·s)}$
Calculate the total molar flow rate (ṅ) by multiplying the flux by the surface area:
$\dot{n} = J \times A$

$\dot{n} = 8.000e-7 \times 0.500 = 4.000e-7 \text{ mol/s}$

Final Answer:

The steady-state molar flow rate of the gas through the polymer film is 4.000×10^-7 mol/s.

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