Formula sheet

Membrane Filtration and Fouling Control Formula Sheet

Membrane filtration formulas for flux, area, TMP, permeability, recovery, fouling, cleaning, integrity and release checks.

Branch: Environmental Engineering
Content: Formula sheet
Updated: Jul 03, 2026
Revision: v1.1.0 · reviewed

This formula sheet collects first-pass calculations for membrane filtration and fouling control in water and wastewater treatment. Use it for operating review, troubleshooting, capacity screening and validation planning. It does not replace vendor limits, pilot testing, membrane-specific flux guidance, chemical compatibility checks or permit acceptance criteria.

How to Use This Formula Sheet

Use this sheet as a membrane train release screen. Start with the decision: capacity acceptance, high-TMP investigation, cleaning trigger, module derating, backwash review, integrity-test response, peak-flow readiness or post-maintenance release. Then keep the same train boundary, active membrane area, pressure reference, water quality, temperature, cleaning state and operating mode through every calculation.

State whether flux is reported as (\text{L}/\text{m}^2\text{h}), (\text{m}^3/\text{m}^2\text{d}) or another site convention. State whether pressure is gauge pressure, absolute pressure or a vendor-specific TMP basis. Permeability should be reported with temperature, cleaning state, online module count and whether the value is gross, net or normalized. A membrane train can pass a flux calculation and still fail release if backwash loss, recovery, pressure margin, integrity evidence or fouling rate is not acceptable.

Gross Permeate Flux

\displaystyle J=\frac{Q_p}{A_m}

where (J) is flux, (Q_p) is permeate flow and (A_m) is active membrane area.

If (Q_p=150\ \text{m}^3/\text{h}) and (A_m=3000\ \text{m}^2):

\displaystyle J=\frac{150}{3000}=0.050\ \text{m/h}=50\ \text{L}/\text{m}^2\text{h}

Flux should be compared with feed quality, recovery, cleaning interval and TMP.

Required Membrane Area

\displaystyle A_m=\frac{Q_{\text{req}}}{J_{\text{allow}}}

For a required peak flow of (3600\ \text{m}^3/\text{d}=150\ \text{m}^3/\text{h}) and allowable flux of (45\ \text{L}/\text{m}^2\text{h}=0.045\ \text{m/h}):

\displaystyle A_m=\frac{150}{0.045}=3333\ \text{m}^2

This area is a screening value. Redundancy, backwash downtime and module availability must still be included.

Effective Online Area

If only part of the installed membrane area is available:

A_{\text{eff}}=A_{\text{installed}}f_{\text{online}}f_{\text{available}}

For (A_{\text{installed}}=3600\ \text{m}^2), online fraction (0.92) and availability (0.95):

A_{\text{eff}}=3600(0.92)(0.95)=3146\ \text{m}^2

Use effective area for release decisions. Installed area can overstate capacity when modules are isolated, offline for integrity testing or unavailable after chemical cleaning.

Released Net Capacity

A practical release capacity should combine allowable flux, effective area and operating loss:

Q_{\text{rel}}=J_{\text{allow}}A_{\text{eff}}(1-f_{bw}-f_{dt})

where (f_{bw}) is backwash loss fraction and (f_{dt}) is downtime or cleaning loss fraction for the review period. If (J_{\text{allow}}=45\ \text{L}/\text{m}^2\text{h}=0.045\ \text{m/h}), (A_{\text{eff}}=3146\ \text{m}^2), (f_{bw}=0.05) and (f_{dt}=0.03):

Q_{\text{rel}}=0.045(3146)(1-0.05-0.03)=130\ \text{m}^3/\text{h}

This is closer to the capacity that operations can promise than a gross clean-water flux calculation. It should still be checked against feed quality, pressure margin and integrity status.

Transmembrane Pressure

\displaystyle TMP=\frac{P_f+P_c}{2}-P_p

where (P_f) is feed pressure, (P_c) is concentrate-side pressure and (P_p) is permeate pressure on the same basis.

With (P_f=190\ \text{kPa}), (P_c=170\ \text{kPa}) and (P_p=20\ \text{kPa}):

\displaystyle TMP=\frac{190+170}{2}-20=160\ \text{kPa}

Do not compare TMP values unless the pressure reference and flow state are consistent.

Permeability

\displaystyle K=\frac{J}{TMP}

Using (J=50\ \text{L}/\text{m}^2\text{h}) and (TMP=160\ \text{kPa}):

\displaystyle K=\frac{50}{160}=0.313\ \text{L}/\text{m}^2\text{h}/\text{kPa}

Permeability is often the most useful operating indicator because it normalizes flow by pressure.

Temperature Normalized Permeability

\displaystyle K_{20}=K_T\frac{\mu_T}{\mu_{20}}

where (\mu_T) is water viscosity at operating temperature and (\mu_{20}) is viscosity at (20^\circ\text{C}). If (K_T=0.313) and (\mu_T/\mu_{20}=1.20):

K_{20}=0.313(1.20)=0.376\ \text{L}/\text{m}^2\text{h}/\text{kPa}

Temperature normalization is a correction, not a substitute for checking feed quality and cleaning state.

Recovery

\displaystyle R=\frac{Q_p}{Q_f}

If feed flow is (190\ \text{m}^3/\text{h}) and permeate flow is (150\ \text{m}^3/\text{h}):

\displaystyle R=\frac{150}{190}=0.789

or 78.9 percent. Higher recovery can increase concentration, scaling and fouling risk.

Concentration Factor

For a simplified conservative solute or colloid screen:

\displaystyle CF=\frac{1}{1-R}

At (R=0.789):

\displaystyle CF=\frac{1}{1-0.789}=4.74

This does not account for precipitation, biodegradation, adsorption or rejection differences. It is a first warning that the concentrate side may see much higher foulant exposure than the feed.

Net Production

\displaystyle Q_{\text{net}}=\frac{V_p-V_{bw}-V_{CIP}}{T}

where (V_p) is produced permeate, (V_{bw}) is backwash water used, (V_{CIP}) is cleaning or downtime-equivalent loss and (T) is the review period.

If a train produces (3600\ \text{m}^3) in a day, uses (180\ \text{m}^3) for backwash and loses (120\ \text{m}^3) equivalent to cleaning downtime:

\displaystyle Q_{\text{net}}=\frac{3600-180-120}{1}=3300\ \text{m}^3/\text{d}

Net production is the capacity relevant to demand planning.

Backwash Loss Fraction

Backwash and chemically enhanced backwash reduce net production:

\displaystyle f_{bw}=\frac{V_{bw}}{V_p}

If (V_{bw}=180\ \text{m}^3) and gross permeate is (3600\ \text{m}^3):

\displaystyle f_{bw}=\frac{180}{3600}=0.050=5.0\%

Backwash loss should be reviewed with turbidity, recovery, chemical use and filtrate quality. A lower backwash frequency can improve net production while accelerating irreversible fouling.

Capacity at TMP Limit

J_{\max}=K_{\text{current}}TMP_{\max}

Q_{\max}=J_{\max}A_m

If current permeability is (0.313\ \text{L}/\text{m}^2\text{h}/\text{kPa}), the TMP limit is (220\ \text{kPa}) and area is (3000\ \text{m}^2):

J_{\max}=0.313(220)=68.9\ \text{L}/\text{m}^2\text{h}

Q_{\max}=0.0689(3000)=207\ \text{m}^3/\text{h}

This capacity is only credible if fouling does not continue during the reviewed period.

TMP Rise Rate

\displaystyle r_{TMP}=\frac{TMP_2-TMP_1}{\Delta t}

If TMP rises from (120) to (160\ \text{kPa}) in 10 days at the same flux:

r_{TMP}=4.0\ \text{kPa/d}

Rate of rise is often a better trigger than a single TMP alarm because it captures accelerating fouling.

Time to TMP Limit

For a roughly linear TMP rise at fixed flux:

\displaystyle t_{\text{limit}}=\frac{TMP_{\max}-TMP_{\text{current}}}{r_{TMP}}

If (TMP_{\max}=220\ \text{kPa}), current TMP is (160\ \text{kPa}) and (r_{TMP}=4.0\ \text{kPa/d}):

\displaystyle t_{\text{limit}}=\frac{220-160}{4.0}=15\ \text{d}

This is a planning estimate, not a guarantee. Accelerating fouling, temperature change, feed-quality deterioration and incomplete backwash can shorten the available run time.

Cleaning Recovery

\displaystyle \eta_{K}=\frac{K_{\text{after}}-K_{\text{before}}}{K_{\text{clean}}-K_{\text{before}}}

If clean permeability is (0.80), pre-clean permeability is (0.31) and post-clean permeability is (0.62):

\displaystyle \eta_K=\frac{0.62-0.31}{0.80-0.31}=0.633

Only 63.3 percent of recoverable permeability was restored. That may indicate irreversible fouling, weak cleaning chemistry or upstream foulant change.

CIP Return-to-Service Gate

A simple return-to-service gate can compare recovered permeability with the accepted clean baseline:

\displaystyle G_{CIP}=\frac{K_{\text{after}}}{K_{\text{clean}}}

If (K_{\text{after}}=0.62) and (K_{\text{clean}}=0.80):

\displaystyle G_{CIP}=\frac{0.62}{0.80}=77.5\%

If the site requires at least 85 percent recovery before returning a train to normal duty, the train should remain derated, recleaned or investigated.

Pressure Margin

\displaystyle M_{TMP}=\frac{TMP_{\max}-TMP_{\text{operating}}}{TMP_{\max}}

With (TMP_{\max}=220\ \text{kPa}) and operating TMP (160\ \text{kPa}):

\displaystyle M_{TMP}=\frac{220-160}{220}=27.3\%

Margin should be interpreted with flux, temperature, feed condition and TMP rise rate.

Integrity-Test Log Removal Screen

For a conservative concentration-based integrity screen:

\displaystyle LRV=\log_{10}\left(\frac{C_f}{C_p}\right)

where (C_f) is feed challenge concentration and (C_p) is permeate-side concentration on the same basis. If (C_f=1200\ \text{counts/mL}) and (C_p=3\ \text{counts/mL}):

\displaystyle LRV=\log_{10}\left(\frac{1200}{3}\right)=2.60

Integrity testing is method-specific. Use the approved direct or indirect integrity test, acceptance limit, sensitivity and response procedure before releasing filtered water quality.

Pumping Power Screen

\displaystyle P=\frac{Q\Delta p}{\eta}

Use SI units: (Q) in (\text{m}^3/\text{s}), (\Delta p) in pascals and (P) in watts. For (Q=150\ \text{m}^3/\text{h}=0.0417\ \text{m}^3/\text{s}), (\Delta p=160000\ \text{Pa}) and efficiency (\eta=0.70):

\displaystyle P=\frac{0.0417(160000)}{0.70}=9.5\ \text{kW}

This is a hydraulic screen, not the total train energy.

Validity Limits

These equations are useful only when flow, area, pressure, temperature, cleaning state and operating mode are consistent. They do not by themselves identify the fouling mechanism. Interpretation should include feed turbidity, TSS, colloids, temperature, chemistry, pretreatment condition, backwash history, integrity testing and cleaning records.

Do not use this sheet to override membrane supplier pressure limits, chemical exposure limits, maximum flux rules, recovery limits, warranty requirements, sanitary barriers or permit acceptance criteria. The calculations identify whether the numbers are internally coherent; the release decision still needs representative water quality, validated instruments, alarm settings, maintenance state and operator procedures.

Review Checklist

Before accepting a membrane calculation, check:

flux and area are on the same train boundary;
TMP pressure points and basis are stated;
permeability is normalized when temperature varies;
recovery and concentration factor are reviewed together;
net production includes backwash and cleaning losses;
TMP rise rate is compared at similar flux;
cleaning recovery is judged by permeability, not pressure alone;
pressure margin is tied to feed quality and operating time;
pump power uses SI units and realistic efficiency;
validation evidence includes water quality and membrane integrity;
active online area excludes isolated, failed or test-held modules;
backwash and CIP losses are included before claiming production capacity;
integrity-test acceptance criteria are method-specific and current;
release status states normal duty, derated duty, cleaning hold, integrity hold or investigation.

Common Formula Mistakes

Common mistakes include comparing TMP at different fluxes, using pump discharge pressure as module TMP, reporting flux without membrane area, ignoring backwash downtime in capacity, treating one successful cleaning as permanent recovery, omitting temperature normalization, and assuming clear permeate proves membrane capacity.

Other mistakes are using installed area instead of online area, calling a clean-water test representative of production water, accepting high recovery while concentrate quality is uncontrolled, treating pressure margin as independent of fouling rate, and returning a membrane train to service after CIP without a defined permeability or integrity gate.

REF

Disciplines