Reference ID: MET-566E | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In membrane-based separations, the sieving coefficient and rejection quantify how effectively a membrane retains a target solute. These dimensionless figures are pivotal for:
Membrane selection and validation
Process scale-up and troubleshooting
Quality control in pharmaceutical, food, and water-treatment plants
They are routinely reported for proteins, polymers, and colloids during ultrafiltration, nanofiltration, or reverse-osmosis operations.
Methodology & Formulas
Measure the steady-state solute concentrations on the retentate (feed side) and permeate (filtrate side).
Compute the sieving coefficient:
\[ S = \frac{C_{\text{perm}}}{C_{\text{retn}}} \]
where \(C_{\text{perm}}\) and \(C_{\text{retn}}\) are mass concentrations in identical units.
Convert to percentage rejection:
\[ R = (1 - S)\times 100\% \]
Because membranes ideally pass solvent while fully rejecting solute, the theoretical ranges are:
Parameter
Lower Limit
Upper Limit
Physical Meaning
\(S\)
0
1
0 = complete rejection, 1 = complete transmission
\(R\)
0%
100%
0% = no rejection, 100% = perfect rejection
Values outside these bounds indicate measurement error, membrane defect, or concentration-polarisation anomalies.
The sieving coefficient is simply the ratio of the solute concentration in the permeate to that in the feed at steady state:
S = Cpermeate / Cfeed
Collect samples only after the system has run for at least three residence times to avoid startup artifacts.
Use the same analytical method for both streams to cancel out calibration bias.
If the membrane is asymmetric, measure the feed concentration at the membrane wall (Cw) rather than the bulk feed; otherwise S can exceed 1 and appear physically meaningless.
Rejection is preferred when you need a quick performance snapshot or when comparing different membranes under identical pressure and flow conditions.
R = 1 – S, so a high-sieving solute gives a low rejection number that is easy to interpret.
Use rejection in plant dashboards because operators intuitively understand “95% rejection” more than “S = 0.05”.
Switch to sieving coefficient when modeling transport equations, because S is linearly proportional to the intrinsic membrane property and avoids the asymptote at R = 100%.
Higher pressure increases solvent flux faster than solute passage, so the concentration-polarization layer thickens and the wall concentration (Cw) rises.
The apparent rejection Robs = 1 – Cpermeate/Cbulk falls because Cbulk is fixed while Cpermeate increases slightly.
Correct for polarization by measuring the true rejection Rtrue = 1 – Cpermeate/Cw using a sample port right at the membrane surface.
If Rtrue still drops, check for membrane compaction or defects that raise the intrinsic sieving coefficient.
Only semi-quantitatively; solute–solute and solute–membrane interactions shift the effective size and charge.
For neutral solutes of similar chemistry, the geometric mean Smix ≈ √(S1 × S2) gives a first estimate.
When one solute is charged, ionic strength and Donnan effects can raise or lower the sieving of the neutral species by up to 30%.
Always validate with a pilot trial at the target pH and ionic strength; use a design-of-experiments approach to keep lab workload manageable.
Worked Example – Sieving Coefficient and Rejection in a Whey UF Loop
A small dairy plant is running a batch ultrafiltration (UF) unit to concentrate whey protein. After the system reaches steady state, samples are taken from the retentate and permeate lines. The task is to quantify how well the membrane rejects lactose.
Retentate lactose concentration, \(C_\text{retn}\): 8.0 g L⁻¹
Permeate lactose concentration, \(C_\text{perm}\): 0.4 g L⁻¹
Calculate the sieving coefficient \(S\):
\[ S = \frac{C_\text{perm}}{C_\text{retn}} = \frac{0.4}{8.0} = 0.05 \]
Convert the sieving coefficient to percentage rejection \(R\):
\[ R = (1 - S) \times 100\% = (1 - 0.05) \times 100\% = 95.0\% \]
Final Answer: The membrane exhibits a sieving coefficient of 0.05 and rejects 95.0% of the lactose.
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