Reference ID: MET-2603 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Fick’s first law for steady-state diffusion quantifies how quickly a species moves through a stagnant medium when concentration gradients are constant with time. In process engineering the result is used to size membranes, predict solvent losses, estimate drying times, design catalytic wash-coats, and rate barrier films. Because the law links concentration driving force to molar flux, it is the mass-transfer analogue of Ohm’s law and underlies the design of any unit operation where diffusion is the rate-limiting step.
Methodology & Formulas
Convert temperature to absolute scale
\[T(\text{K})=T(^\circ\text{C})+273.15\]
Relate partial pressure to molar concentration (ideal-gas approximation for dilute vapour)
\[C_i=\frac{p_i}{R\,T}\]
where
\(C_i\) = molar concentration of diffusing species, kmol m−3
\(p_i\) = partial pressure of species, kPa
\(R\) = 8.314 m3·kPa·kmol−1·K−1
Fick’s first law for one-dimensional steady-state diffusion
\[\dot{N}=\frac{D\,A\,(C_1-C_2)}{z}\]
where
\(\dot{N}\) = molar diffusion rate, kmol s−1
\(D\) = diffusion coefficient, m2 s−1
\(A\) = cross-sectional area normal to flux, m2
\(z\) = diffusion path length, m
\(C_1-C_2\) = concentration difference across path, kmol m−3
Convert molar rate to mass rate
\[\dot{m}=\dot{N}\,M\]
where
\(M\) = molar mass of species, kg kmol−1
\(\dot{m}\) = mass diffusion rate, kg s−1
Convert to convenient units
\[\dot{m}~[\text{mg h}^{-1}]=\dot{m}~[\text{kg s}^{-1}]\times 10^6\times 3600\]
Regime check for ideal-gas assumption
Condition
Criterion
Ideal gas valid
\(p_i\ll p_{\text{total}}\) and \(T\) well above critical temperature
Steady state
\(\partial C/\partial t=0\) (concentration at each point constant with time)
One-dimensional
Area \(\perp\) flux ≫ edge effects
Fick’s First Law states that the molar flux J (mol m⁻² s⁻¹) is proportional to the negative concentration gradient: J = –D dC/dx.
At steady state the concentration profile is linear, so for a flat membrane or catalyst pellet of thickness L the integrated form becomes:
J = D (C₁ – C₂) / L
C₁ and C₂ are the upstream and downstream concentrations (mol m⁻³)
D is the effective diffusivity (m² s⁻¹) in the porous medium or membrane
Process engineers use this to size membranes, predict loss of hydrogen through vessel walls, or calculate flux through catalyst wafers without solving transient equations.
Replace the bulk diffusivity DAB with an effective diffusivity:
Deff = (ε / τ) DAB
ε = pellet porosity (void fraction)
τ = tortuosity factor (typically 2–6 for catalyst supports)
If Knudsen diffusion dominates (pores < 100 nm), calculate DKn and combine using the Bosanquet relation: 1/Deff = 1/DKn + 1/DAB. Always validate with a simple Wicke–Kallenbach experiment if data are scarce.
For dilute species (yi < 0.1) Fick’s Law with an effective Di,eff is usually adequate and keeps hand calculations simple.
When mole fractions are large and components interact (e.g., H₂, H₂O, CO, CO₂ in steam reforming), the Maxwell-Stefan equations give more accurate fluxes.
Use Fick: quick sizing, preliminary vessel rating, or when only one species is limiting
Use Maxwell-Stefan: detailed reactor modeling, catalyst pellet gradients, or when equimolar counter-diffusion occurs
Most steady-state process simulators accept both; choose based on required accuracy and available binary diffusion coefficients.
Assume local equilibrium at the interface:
Csurface = Cbulk_gas / H for sparingly soluble gases (H = Henry constant)
For fast external mass transfer, set Csurface = Cbulk
If external resistance matters, couple Fick’s Law with a film model: J = kg(Cbulk – Csurface)
Match the flux expressions at the interface to eliminate the unknown surface concentration and solve for the overall rate.
Worked Example – Estimating Water Loss from a Storage Tank Vent
A small vent on the roof of a demineralised-water storage tank is fitted with a 2 mm thick microporous membrane that limits liquid carry-over while allowing vapour to escape. Under steady daytime conditions the partial pressure of water vapour inside the tank head-space is 3.17 kPa, whereas the outside air is sufficiently dry that the partial pressure can be taken as zero. Estimate the steady-state loss of water vapour through the vent if the membrane has an effective diffusion area of 0.01 m² and the diffusion coefficient for water vapour in air at 25 °C is 2.6 × 10⁻⁵ m² s⁻¹.
Knowns
Membrane thickness, \( z = 2 \) mm = 0.002 m
Diffusion coefficient, \( D = 2.6 \times 10^{-5} \) m² s⁻¹
Effective area, \( A = 0.01 \) m²
Temperature, \( T = 25 \) °C = 298.15 K
Inside partial pressure, \( p_1 = 3.17 \) kPa
Outside partial pressure, \( p_2 = 0 \) kPa
Universal gas constant, \( R = 8.314 \) kPa m³ kmol⁻¹ K⁻¹
Molar mass of water, \( M_{\text{H}_2\text{O}} = 18.015 \) kg kmol⁻¹
Step-by-step calculation
Convert partial pressures to concentrations using the ideal-gas law:
\[
C = \frac{p}{RT}
\]
\[
C_1 = \frac{3.17}{8.314 \times 298.15} = 0.00128 \ \text{kmol m}^{-3}
\]
\[
C_2 = 0 \ \text{kmol m}^{-3}
\]
Apply Fick’s first law for steady-state diffusion:
\[
N = \frac{DA}{z}(C_1 - C_2)
\]
\[
N = \frac{(2.6 \times 10^{-5})(0.01)}{0.002}(0.00128 - 0)
\]
\[
N = 1.66 \times 10^{-7} \ \text{kmol s}^{-1}
\]
