Exercise set

Membrane Separation Flux, Rejection, and Fouling Exercises

Solved membrane-separation exercises for mass balance, rejection, flux, area, recovery, TMP, permeability, fouling, cleaning and reject release gates.

Branch: Chemical Engineering
Content: Exercise set
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

These exercises focus on membrane separation as a chemical-process unit operation. They cover feed and permeate balances, rejection, retentate concentration, flux, area, recovery, concentration factor, transmembrane pressure, normalized permeability, fouling rate, cleaning recovery, reject load, staged recovery and release gates.

Assume simplified screening calculations unless an exercise states otherwise. Real membrane process design must check membrane chemistry, solvent compatibility, pressure rating, osmotic pressure, concentration polarization, fouling, scaling, cleaning limits, temperature, viscosity, pretreatment and reject handling.

How to use these exercises

Work the page as a membrane-release sequence. Exercises 1 to 5 establish whether the feed, permeate and retentate material balance is credible. Exercises 6 to 10 connect the balance to flux, membrane area, recovery, TMP and permeability. Exercises 11 to 15 test whether fouling, cleaning, staging, recycle and reject handling still make the separation operable. Exercises 16 to 18 convert the technical results into an evidence gate.

Keep units explicit. Membrane calculations often mix cubic meters per hour, liters per hour, square meters, bar, concentration and mass load. A release decision should show both concentration performance and total reject load, because a good permeate can still create an unmanageable retentate.

Release Evidence Notes

Membrane evidence should state feed flow, permeate flow, retentate flow, solute concentration basis, membrane area, pressure, temperature, recovery, rejection and cleaning state.

Flux evidence is incomplete without fouling state. A clean-water flux cannot release a production membrane if real feed has colloids, oil, solids, polymer, scaling ions or biological growth.

Reject evidence should include mass load, concentration, storage, recycle, purge and disposal route. High recovery can create an unreleasable retentate if reject handling is not controlled.

The package should connect process performance to operating controls: pretreatment status, pressure limits, temperature, cleaning trigger, cleaning recovery, integrity test, recycle purge and environmental or product-quality limits. The release basis is weak if rejection, flux and recovery are proved in different feed conditions or at different stages of membrane fouling.

Engineering Boundary Notes

These calculations do not replace vendor design, pilot testing, material compatibility review, pressure safety review, cleaning validation or environmental discharge approval. They are screening exercises for membrane separation release.

Real membrane design must account for osmotic pressure, viscosity, concentration polarization, compaction, module hydraulics, spacer fouling, chemical compatibility, biological growth, scaling saturation, cleaning chemistry and pressure-vessel limits. Use the exercises to locate the controlling constraint, then confirm it with pilot data, vendor limits, pressure protection and representative feed testing before changing operating conditions.

Common Release Mistakes

using clean-water flux for fouling feed;
reporting permeate quality without retentate mass load;
increasing recovery without checking concentration polarization and reject disposal;
interpreting TMP alone without normalized permeability;
releasing after cleaning without proving recovered flux and integrity;
optimizing for high permeate recovery while creating an overloaded recycle or purge stream;
comparing flux runs at different temperature or viscosity without normalization;
treating one short stable run as proof that fouling rate is acceptable;
accepting permeate concentration while ignoring pressure, cleaning frequency and membrane aging.

Scenario Map

Scenario	Exercises	Primary check	Engineering decision
Mass balance and rejection	1, 2, 3, 4, 5	permeate, retentate, solute, rejection and concentration	Decide whether separation performance closes.
Flux and pressure	6, 7, 8, 9, 10	membrane area, flux, recovery, TMP and permeability	Decide whether area and pressure are acceptable.
Fouling and cleaning	11, 12, 13, 14, 15	decline, cleaning recovery, staged recovery, reject and recycle	Decide whether operation can continue.
Release gate	16, 17, 18	evidence, RPN and all-of membrane release	Decide whether the membrane package can close.

Exercise 1: Membrane Solute Feed Load

A membrane unit treats $10\ \text{m}^3/\text{h}$ feed at $2.0\ \text{kg/m}^3$ solute. Compute solute feed load.

Solution

M_F=10(2.0)=20\ \text{kg/h}

Engineering Comment

All membrane recovery and rejection claims should start from a feed mass load.

Plausibility Check

Ten cubic meters per hour at two kilograms per cubic meter gives twenty kilograms per hour.

Exercise 2: Permeate Solute Load

Permeate flow is $7\ \text{m}^3/\text{h}$ with $0.4\ \text{kg/m}^3$ solute. Compute permeate solute load.

Solution

M_P=7(0.4)=2.8\ \text{kg/h}

Engineering Comment

Permeate quality should be checked as both concentration and mass load if discharge or product recovery matters.

Plausibility Check

The permeate concentration is low, so permeate solute load is much lower than feed load.

Exercise 3: Retentate Concentration

Feed solute load is $20\ \text{kg/h}$ , permeate solute load is $2.8\ \text{kg/h}$ , feed flow is $10\ \text{m}^3/\text{h}$ and permeate flow is $7\ \text{m}^3/\text{h}$ . Compute retentate concentration.

Solution

Retentate solute:

M_R=20-2.8=17.2\ \text{kg/h}

Retentate flow:

Q_R=10-7=3\ \text{m}^3/\text{h}

Concentration:

C_R=\dfrac{17.2}{3}=5.73\ \text{kg/m}^3

Engineering Comment

Retentate concentration controls viscosity, scaling, recycle and disposal risk.

Plausibility Check

Most solute remains in a smaller flow, so retentate concentration rises above feed concentration.

Exercise 4: Solute Rejection

Feed concentration is $2.0\ \text{kg/m}^3$ and permeate concentration is $0.4\ \text{kg/m}^3$ . Compute rejection.

Solution

R=1-\dfrac{0.4}{2.0}=0.80=80\%

Engineering Comment

Rejection should be measured at the same temperature, pressure and recovery condition as the release claim.

Plausibility Check

The permeate has one fifth of the feed concentration, so rejection is eighty percent.

Exercise 5: Concentration Factor

Retentate concentration is $5.73\ \text{kg/m}^3$ and feed concentration is $2.0\ \text{kg/m}^3$ . Compute concentration factor.

Solution

CF=\dfrac{5.73}{2.0}=2.87

Engineering Comment

Concentration factor should be checked against viscosity, osmotic pressure and fouling limits.

Plausibility Check

The retentate is almost three times the feed concentration.

Exercise 6: Membrane Area From Flux

Required permeate flow is $6.0\ \text{m}^3/\text{h}$ . Sustainable flux is $24\ \text{L/(m}^2\text{ h)}$ . Compute membrane area.

Solution

Convert permeate flow:

Q_P=6000\ \text{L/h}

Area:

A=\dfrac{6000}{24}=250\ \text{m}^2

Engineering Comment

Sustainable flux should come from fouling feed or pilot data, not clean-water tests.

Plausibility Check

Thousands of liters per hour at a few dozen liters per square meter-hour requires hundreds of square meters.

Exercise 7: Actual Flux

Membrane area is $280\ \text{m}^2$ and permeate flow is $6.4\ \text{m}^3/\text{h}$ . Compute flux.

Solution

J=\dfrac{6400}{280}=22.9\ \text{L/(m}^2\text{ h)}

Engineering Comment

Flux should be normalized before comparing across temperature or viscosity changes.

Plausibility Check

The flow and area are close to the sizing basis, so flux near $24$ is plausible.

Exercise 8: Recovery

Feed flow is $10\ \text{m}^3/\text{h}$ and permeate flow is $7\ \text{m}^3/\text{h}$ . Compute recovery.

Solution

Y=\dfrac{7}{10}=0.70=70\%

Engineering Comment

Higher recovery reduces liquid waste volume but increases retentate concentration and fouling risk.

Plausibility Check

Seven out of ten units of feed become permeate.

Exercise 9: Transmembrane Pressure

Feed pressure is $5.2\ \text{bar}$ , concentrate pressure is $4.6\ \text{bar}$ and permeate pressure is $0.8\ \text{bar}$ . Estimate TMP as average feed/concentrate pressure minus permeate pressure.

Solution

TMP=\dfrac{5.2+4.6}{2}-0.8=4.1\ \text{bar}

Engineering Comment

TMP should be trended with flux; rising TMP at constant flux indicates fouling or scaling.

Plausibility Check

Average feed-side pressure is just below $5$ bar, and subtracting permeate pressure leaves about $4$ bar.

Exercise 10: Normalized Permeability

Flux is $22.9\ \text{L/(m}^2\text{ h)}$ and TMP is $4.1\ \text{bar}$ . Compute permeability.

Solution

K=\dfrac{22.9}{4.1}=5.59\ \text{L/(m}^2\text{ h bar)}

Engineering Comment

Permeability is more diagnostic than flux alone when pressure changes during operation.

Plausibility Check

Dividing a flux in the twenties by about four bar gives a value near six.

Exercise 11: Flux Decline

Initial filtration flux is $80\ \text{L/(m}^2\text{ h)}$ . After a run, flux is $52\ \text{L/(m}^2\text{ h)}$ . Compute decline.

Solution

\text{decline}=\dfrac{80-52}{80}=0.35=35\%

Engineering Comment

A large decline should trigger cleaning review, pretreatment review or lower operating flux.

Plausibility Check

The flux lost is $28$ out of $80$ , a little over one third.

Exercise 12: Cleaning Recovery

Fouled permeability is $5.59\ \text{L/(m}^2\text{ h bar)}$ . After cleaning, permeability is $7.8\ \text{L/(m}^2\text{ h bar)}$ . Baseline clean permeability is $8.6\ \text{L/(m}^2\text{ h bar)}$ . Compute cleaning recovery relative to baseline.

Solution

R_c=\dfrac{7.8}{8.6}=0.907=90.7\%

Engineering Comment

Incomplete recovery can indicate irreversible fouling, wrong cleaning chemistry or membrane aging.

Plausibility Check

The cleaned value is close to the baseline value, so recovery just above ninety percent is plausible.

Exercise 13: Retentate Reject Load

Retentate solute load is $17.2\ \text{kg/h}$ . If $15\%$ is recycled and the rest is purged, compute purge solute load.

Solution

M_P=17.2(1-0.15)=14.6\ \text{kg/h}

Engineering Comment

Reject purge must have treatment, disposal or recovery capacity before high recovery is accepted.

Plausibility Check

Most of the retentate is purged, so purge load is close to total retentate load.

Exercise 14: Staged Recovery

Stage 1 recovery is $70\%$ . Stage 2 recovers $50\%$ of the Stage 1 retentate. Compute overall recovery.

Solution

Unrecovered after Stage 1:

f_1=1-0.70=0.30

Unrecovered after Stage 2:

f=0.30(1-0.50)=0.15

Overall recovery:

Y=1-0.15=85\%

Engineering Comment

Staging improves recovery but increases concentration in later stages and can increase fouling.

Plausibility Check

Recovering half of the remaining thirty percent adds fifteen percentage points.

Exercise 15: Recycle Impurity Screen

Membrane recycle flow is $1.2\ \text{m}^3/\text{h}$ at impurity concentration $4.0\ \text{kg/m}^3$ . Compute impurity recycle load.

Solution

M=1.2(4.0)=4.8\ \text{kg/h}

Engineering Comment

Recycle load should be compared with purge capacity to prevent impurity buildup.

Plausibility Check

One to two cubic meters per hour at several kilograms per cubic meter gives a few kilograms per hour.

Exercise 16: Membrane Evidence Completion

The release package requires feed flow, feed assay, permeate flow, permeate assay, retentate flow, TMP, temperature, flux trend, cleaning state, integrity status, reject route and recycle purge. Ten of twelve records are complete. Compute completion.

Solution

C=\dfrac{10}{12}=83.3\%

Engineering Comment

Missing reject route or integrity status should hold release even if flux and quality pass.

Plausibility Check

Ten of twelve is five sixths, or about $83\%$ .

Exercise 17: Fouling RPN

A rapid-fouling failure mode has severity $7$ , occurrence $5$ and detection $4$ . Compute RPN.

Solution

RPN=7(5)(4)=140

Engineering Comment

This score supports stronger pretreatment, monitoring or cleaning trigger controls.

Plausibility Check

Moderate-high ratings multiply to a three-digit value.

Exercise 18: Membrane Release Gate

A release gate requires rejection at least $75\%$ , recovery at least $70\%$ , cleaning recovery above $90\%$ , reject route assigned, and evidence completion above $90\%$ . Current values are rejection $80\%$ , recovery $70\%$ , cleaning recovery $90.7\%$ , reject route assigned and evidence completion $83.3\%$ . Decide release status.

Solution

Rejection, recovery, cleaning recovery and reject route pass. Evidence completion fails:

83.3\%<90\%

Release status:

\text{hold}

Engineering Comment

The membrane looks technically capable, but release should wait until missing records are closed.

Plausibility Check

An all-of gate fails when a mandatory evidence threshold fails.

Validation Package Checklist

Feed, permeate, retentate, recycle and purge flows close a mass balance.
Rejection, recovery, flux, TMP and permeability are evaluated together.
Fouling and cleaning evidence comes from production-representative feed conditions.
Retentate reject and recycle impurity routes are assigned before release.
Pretreatment, pressure limits, temperature, pH and cleaning chemistry match the release case.
Integrity status and post-cleaning recovery are documented before performance claims are accepted.
Environmental, product-quality and waste-handling constraints are checked on mass load as well as concentration.
Operating procedures define when to lower flux, clean, divert, recycle, purge or hold release.

The final acceptance question is whether the same feed case closes quality, capacity, fouling control and reject management at the same time. If any one of those four dimensions is borrowed from a better operating case, the membrane package is not mature release evidence.

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