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.
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):
\[
\frac{1}{L_p^\text{total}} = \frac{1}{L_p^\text{membrane}} + R_\text{fouling} + R_\text{CP}
\]
\[
\frac{1}{L_p^\text{total}} = 0.010 + 0.005 + 0.003 = 0.018\ \text{bar h m}^2\ \text{L}^{-1}
\]
