What Is Mass Transfer?
Mass transfer refers to the net movement of a chemical species from one location to another — driven by a concentration gradient, a pressure difference, or an external force. It is a cornerstone of chemical engineering and is central to processes such as distillation, absorption, drying, membrane separation, and biological transport.
Just as Fourier's Law governs heat conduction, Fick's Laws form the mathematical foundation of diffusive mass transfer.
Fick's First Law: Steady-State Diffusion
Adolf Fick proposed his first law in 1855. For a binary system in one dimension, it states:
J = −D · (dC/dx)
Where:
- J = molar flux of species A (mol/m²·s)
- D = binary diffusion coefficient (m²/s)
- dC/dx = concentration gradient (mol/m⁴)
The negative sign confirms that diffusion always occurs from regions of high concentration to low concentration. Fick's First Law applies strictly under steady-state conditions — when the concentration profile does not change with time.
Fick's Second Law: Transient Diffusion
When concentration changes with time — such as during the early stages of drying, gas absorption, or carburising a steel surface — Fick's Second Law applies:
∂C/∂t = D · ∂²C/∂x²
This is a partial differential equation. Solutions depend on the boundary conditions and geometry of the problem. The error function (erf) solution is commonly used for semi-infinite solid geometries, which is directly applicable to surface hardening of metals and gas absorption into liquids.
The Diffusion Coefficient
The diffusion coefficient D is a measure of how easily a species moves through a medium. It depends on:
- Temperature: D increases with temperature (gases follow a T^(3/2) dependence; liquids follow an Arrhenius relationship).
- Pressure: For gas-phase diffusion, D is inversely proportional to total pressure.
- Molecular size: Larger molecules diffuse more slowly (described by the Stokes-Einstein equation in liquids).
- Medium: Diffusion in gases is orders of magnitude faster than in liquids, which is itself faster than in solids.
| System | Typical D (m²/s) |
|---|---|
| Gas–gas (e.g. CO₂ in air) | ~1 × 10⁻⁵ |
| Liquid–liquid (e.g. salt in water) | ~1 × 10⁻⁹ |
| Solid–solid (e.g. carbon in iron) | ~1 × 10⁻¹¹ to 10⁻¹⁴ |
Analogy Between Heat and Mass Transfer
The mathematical structure of Fick's laws closely mirrors Fourier's law of heat conduction. This analogy is powerful — solutions developed for one transport problem can often be adapted directly for the other.
| Quantity | Heat Transfer | Mass Transfer |
|---|---|---|
| Driving force | Temperature gradient (dT/dx) | Concentration gradient (dC/dx) |
| Flux | Heat flux q (W/m²) | Molar flux J (mol/m²·s) |
| Transport property | Thermal conductivity k | Diffusivity D |
| Governing equation | Fourier's Law | Fick's Law |
| Dimensionless group | Nusselt number (Nu) | Sherwood number (Sh) |
Practical Engineering Applications
- Case hardening of steel: Carbon diffuses into the steel surface under high-temperature conditions. Fick's second law predicts the carbon concentration profile and hardened depth as a function of time.
- Membrane separation: Gas permeability through polymer membranes is characterised by solution-diffusion models built on Fick's laws.
- Drying of porous materials: Moisture migration from the interior to the surface of foods, ceramics, and timber is governed by diffusion.
- Drug delivery: Controlled-release medical patches rely on Fickian diffusion to deliver active compounds at a predictable rate through the skin.
Limitations of Fick's Laws
Fick's laws assume that diffusion is the only mechanism of mass transfer and that the diffusion coefficient is constant. In reality:
- Convective (bulk flow) contributions must be added in many industrial systems.
- D can vary significantly with concentration, particularly in liquid and solid systems.
- Multicomponent systems require more advanced frameworks such as the Maxwell-Stefan equations.
Despite these limitations, Fick's laws remain the starting point for virtually all mass transfer analysis in engineering education and practice.