Formula sheet

Membrane Bioreactor Process Control Formula Sheet

MBR formulas for BOD load, F/M, SRT, MLSS, flux, TMP, permeability, oxygen demand, backwash, CIP recovery, fouling rate, limits, and validation.

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

This formula sheet collects first-pass calculations for membrane bioreactor control. Use it to connect biological loading, sludge inventory, membrane flux, pressure, aeration, cleaning and release decisions. Site-specific design must still account for kinetics, membrane supplier limits, redundancy, safety factors, temperature, sensor quality and permit basis.

How to Use This Formula Sheet

Use this sheet as a process-control and release screen for membrane bioreactor operation. Start with the decision: daily process review, high-TMP investigation, ammonia breakthrough response, aeration adjustment, cleaning decision, module derating, commissioning acceptance or permit-risk review. The same formula can mean different things depending on whether the plant is at normal flow, peak wet-weather flow, recovery after cleaning, reduced membrane availability or abnormal biological loading.

State the averaging period, flow basis, active reactor volume, active membrane area, online module count, MLSS basis, temperature, pressure reference, cleaning state and whether permeability is temperature-normalized. MBR calculations are easy to misuse when biological and membrane data come from different operating states.

Use biological formulas to check organic load, F/M, SRT and oxygen demand. Use membrane formulas to check flux, TMP, permeability, fouling rate, backwash loss and CIP recovery. Use the validation checks before changing setpoints or releasing capacity, because a membrane train can pass a flux calculation while biological process control or integrity evidence remains weak.

Organic Load

For wastewater flow in (\text{m}^3/\text{d}) and concentration in (\text{mg/L}):

L=QC10^{-3}

For (Q=12000\ \text{m}^3/\text{d}) and influent BOD (C=220\ \text{mg/L}):

L_{BOD}=12000(220)10^{-3}=2640\ \text{kg/d}

This load drives biomass activity, oxygen demand, sludge production and fouling pressure on the membrane.

Mixed-Liquor Inventory

Convert MLSS from (\text{mg/L}) to (\text{kg/m}^3):

X_{\text{kg/m}^3}=X_{\text{mg/L}}10^{-3}

For (X=8000\ \text{mg/L}):

X=8.0\ \text{kg/m}^3

If biological volume is (V=4500\ \text{m}^3), the mixed-liquor inventory is:

VX=4500(8.0)=36000\ \text{kg}

High inventory can improve retention but also increases viscosity, oxygen-transfer difficulty and membrane fouling risk.

Food-to-Microorganism Ratio

\displaystyle F/M=\frac{L_{BOD}}{VX}

Using the values above:

\displaystyle F/M=\frac{2640}{36000}=0.073\ \text{kg BOD/kg MLSS d}

This is a biological loading screen, not a membrane-capacity proof. Interpret it with SRT, temperature, oxygen, sludge quality and effluent objectives.

Solids Retention Time

\displaystyle SRT=\frac{VX}{Q_wX_w+Q_eX_e}

Let (Q_w=220\ \text{m}^3/\text{d}), (X_w=10\ \text{kg/m}^3), (Q_e=12000\ \text{m}^3/\text{d}) and (X_e=0.005\ \text{kg/m}^3):

\displaystyle SRT=\frac{36000}{220(10)+12000(0.005)}=\frac{36000}{2260}=15.9\ \text{d}

The permeate solids term is small but should not be ignored if membrane integrity is questionable or if solids breakthrough occurs.

Membrane Flux

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

For (Q_p=120\ \text{m}^3/\text{h}) and (A_m=3000\ \text{m}^2):

\displaystyle J=\frac{120}{3000}=0.040\ \text{m/h}=40\ \text{L}/\text{m}^2\text{h}

Flux should be compared with feed solids, MLSS, air scour, backwash interval and TMP trend.

TMP and Permeability

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

If (P_f=160\ \text{kPa}), (P_c=150\ \text{kPa}) and (P_p=15\ \text{kPa}):

\displaystyle TMP=\frac{160+150}{2}-15=140\ \text{kPa}

Permeability is:

\displaystyle K=\frac{J}{TMP}=\frac{40}{140}=0.286\ \text{L}/\text{m}^2\text{h}/\text{kPa}

With viscosity correction (\mu_T/\mu_{20}=1.15):

K_{20}=0.286(1.15)=0.329\ \text{L}/\text{m}^2\text{h}/\text{kPa}

TMP Rise Rate

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

If TMP rises from (140) to (170\ \text{kPa}) in 10 days:

\displaystyle r_{TMP}=\frac{170-140}{10}=3.0\ \text{kPa/d}

Rate-of-rise should be reviewed at comparable flux, temperature, MLSS and air-scour state.

Nitrification Oxygen Demand

For ammonia removed as nitrogen:

L_{NH4,\text{removed}}=Q(C_{in}-C_{out})10^{-3}

If (C_{in}=25\ \text{mg/L as N}), (C_{out}=2\ \text{mg/L as N}) and (Q=12000\ \text{m}^3/\text{d}):

L_{NH4,\text{removed}}=12000(25-2)10^{-3}=276\ \text{kg/d as N}

The nitrification oxygen demand is:

O_{2,\text{nit}}=4.57(276)=1261\ \text{kg O}_2/\text{d}

Add carbon oxidation, endogenous respiration, alpha factor, diffuser condition and control margin separately.

Air-Scour Intensity

A simple membrane air-scour intensity screen is:

\displaystyle I_{air}=\frac{Q_{air}}{A_m}

For (Q_{air}=1800\ \text{Nm}^3/\text{h}) and (A_m=3000\ \text{m}^2):

\displaystyle I_{air}=\frac{1800}{3000}=0.60\ \text{Nm}^3/\text{m}^2\text{h}

This is not an oxygen-transfer calculation. It is a membrane-scour screen that should be checked against blower capacity, diffuser condition, membrane supplier limits and energy use.

Backwash and Net Production

If backwash interval is 30 minutes:

\displaystyle N_{bw}=\frac{24(60)}{30}=48

For (V_{event}=2.0\ \text{m}^3):

V_{bw}=48(2.0)=96\ \text{m}^3/\text{d}

Gross production at (120\ \text{m}^3/\text{h}) is:

V_p=120(24)=2880\ \text{m}^3/\text{d}

If CIP downtime equivalent is (25.7\ \text{m}^3/\text{d}):

Q_{net}=2880-96-25.7=2758\ \text{m}^3/\text{d}

CIP Recovery

\displaystyle \eta_K=\frac{K_{post}-K_{pre}}{K_{clean}-K_{pre}}

For (K_{pre}=0.31), (K_{post}=0.62) and (K_{clean}=0.80):

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

or 63.3 percent. This is partial recovery and should be paired with fouling-source review.

Basis and Validity Limits

These equations are screening tools. A credible MBR calculation set states whether the same operating window was used for MLSS, SRT, flux, TMP, temperature, air scour and cleaning state. It also states what action follows the result: continue, derate, backwash review, clean-in-place, integrity test, wasting adjustment, aeration correction or upstream screening repair.

The load, F/M and SRT formulas assume representative flow, concentration and sludge inventory data. They can mislead during storm dilution, recycle disturbances, sidestream loading, poor sample handling, unusual soluble/particulate COD split, solids breakthrough or rapid wasting changes. Use plant data windows that match the control question.

The flux, TMP and permeability formulas assume known active membrane area, correct pressure reference, comparable temperature and stable operating mode. They are not valid comparisons when module count, air scour, relaxation interval, backwash sequence, viscosity, temperature correction, sensor zero, header losses or cleaning state changed between points.

CIP recovery and fouling-rate checks are trend indicators, not root-cause proof. Fouling can be driven by biological stress, high MLSS viscosity, low dissolved oxygen, filamentous sludge, poor screening, fats/oils/grease, polymer carryover, coagulant dose, aeration maldistribution, damaged diffusers, membrane ageing or operator mode changes.

Common Formula Mistakes

The most common mistake is mixing biological and membrane operating windows. An F/M value from a daily composite, an MLSS value from a grab sample, a TMP value during peak flow and a permeability value after cleaning do not describe one plant state unless the time basis is reconciled.

Do not treat gross permeate flow as net production without subtracting backwash, relaxation, cleaning, offline trains and derated modules. A train can meet instantaneous flux while failing daily production or redundancy requirements.

Do not compare permeability trends without temperature, viscosity and cleaning-state context. A permeability drop may indicate fouling, but it may also reflect colder water, higher MLSS viscosity, pressure-sensor drift, air in lines, header loss, valve position, wrong active-area assumption or changed filtration cycle.

Do not use oxygen demand as an aeration proof by itself. Nitrification and carbon oxidation demand must be checked against oxygen transfer, alpha factor, diffuser condition, blower capacity, DO control tuning, basin mixing and ammonia limits. Passing a theoretical oxygen calculation does not prove stable ammonia removal.

Validation Evidence Package

Before using an MBR calculation for release, derating, cleaning, setpoint change or compliance review, assemble the evidence that connects the formulas to the real plant state. Include influent and effluent sampling basis, flow totalization period, MLSS and MLVSS results, active biological volume, online membrane area, train availability, TMP sensor status, temperature correction, air-scour state, backwash/relaxation sequence and current cleaning state.

For membrane decisions, include flux trend, TMP trend, permeability trend, fouling-rate trigger, integrity-test status, turbidity or particle-count evidence where relevant, CIP recipe, chemical contact time, recovery percentage and post-cleaning stabilization period. For biological decisions, include SRT calculation basis, wasting rate, ammonia and nitrate trend, DO trend, alkalinity and pH, oxygen-transfer margin, sludge settleability or viscosity evidence and upstream disturbance notes.

The release record should state the action taken and the retest trigger: continue operation, restrict flux, bring standby area online, clean-in-place, perform chemically enhanced backwash, repair screening, adjust wasting, retune DO control, investigate biological upset or escalate to integrity testing. A defensible MBR calculation ends with an operating decision, not only a number.

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