Reference ID: MET-14C6 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
Membrane processes (micro‑, ultra‑, nano‑filtration and reverse osmosis) are sized by their permeance \(L_p\), the proportionality constant between trans‑membrane pressure (TMP) and permeate flux. In practice the membrane is never “clean”; fouling and concentration‑polarisation add extra hydraulic resistances that act in series with the intrinsic membrane resistance. The resistance‑in‑series model collapses these effects into a single equivalent permeance \(L_{p,\text{tot}}\) that can be used for scale‑up, energy calculations and scheduling of cleaning cycles, and its accuracy can be further refined by considering the findings of an asymmetric membrane structure analysis as well as the concentration polarization flux limit (film theory).
Methodology & Formulas
Convert individual resistances to a common basis
The inverse of permeance is resistance per unit area. For the membrane itself:
\[
R_{\text{mem}} = \frac{1}{L_p}
\]
with \(L_p\) given in the same units as the fouling and concentration-polarisation resistances.
Sum resistances in series
\[
R_{\text{tot}} = R_{\text{mem}} + R_{\text{fouling}} + R_{\text{CP}}
\]
where:
\(R_{\text{fouling}}\) accounts for cake, biofilm or scaling layers;
\(R_{\text{CP}}\) represents the additional resistance caused by the elevated solute concentration at the wall (concentration-polarisation).
Recover the total permeance
\[
L_{p,\text{tot}} = \frac{1}{R_{\text{tot}}}
\]
Calculate the permeate flux
\[
J = L_{p,\text{tot}} \cdot \Delta P
\]
with \(\Delta P\) the applied TMP.
Validity regime for the linear resistance model
Parameter
Lower limit
Upper limit
Remark
Flux \(J\)
—
120 L m−2 h−1
Linear relation assumed; cake compressibility and non-linear CP become significant above this value.
Pressure \(\Delta P\)
—
3 bar
Same as above; compressible cakes invalidate linear additivity.
Fouling resistance \(R_{\text{fouling}}\)
—
0.02 bar h L−1
Exceeding this limit implies cake compressibility; pressure-independent resistance no longer holds.
Reynolds number \(Re\)
500
10,000
Correlation used for \(R_{\text{CP}}\) is valid only in this cross-flow turbulent regime.
Treat each layer as an electrical resistor in series: add the individual resistances.
\(R_{\text{total}} = R_1 + R_2 + \dots + R_n\)
Units must be consistent (e.g., all in m²·bar·h·L⁻¹ or all in m⁻¹ if normalized to area).
Include support layers, active skin, and any fouling or gel layers you have quantified.
The highest single resistance governs flux almost entirely.
A 0.5 bar skin on top of a 0.05 bar support means 91 % of the driving force is lost across the skin.
Designers often relax support porosity to cut cost once the skin resistance is fixed.
Yes, if you know the clean-membrane resistance at the same temperature.
Pressure does not enter the sum directly; it only sets the driving force once \(R_{\text{total}}\) is known.
Worked Example: Estimating Permeate Flux Through a Fouled Membrane
A small ultrafiltration skid is processing 25 °C de-ionised water at 1 bar.
The clean membrane has a water permeability of 100 L m−2 h−1 bar−1.
After several hours a thin fouling layer plus a concentration-polarisation gel add their own hydraulic resistances.
We need the resulting permeate flux.
Knowns
Clean-membrane permeability, \(L_p^\text{membrane}\) = 100 L m−2 h−1 bar−1
Fouling resistance, \(R_\text{fouling}\) = 0.005 bar h m2 L−1
Concentration-polarisation resistance, \(R_\text{CP}\) = 0.003 bar h m2 L−1
Applied pressure, ΔP = 1 bar
Step-by-step calculation
Convert the clean-membrane permeability to its inverse (resistance):
\[
\frac{1}{L_p^\text{membrane}} = \frac{1}{100} = 0.010\ \text{bar h m}^2\ \text{L}^{-1}
\]
Add resistances in series (fouling and CP are additional resistances):
\[
\frac{1}{L_p^\text{total}} = \frac{1}{L_p^\text{membrane}} + R_\text{fouling} + R_\text{CP}
\]
\[
\frac{1}{L_p^\text{total}} = 0.010 + 0.005 + 0.003 = 0.018\ \text{bar h m}^2\ \text{L}^{-1}
\]
