Formula sheet

Biological Nutrient Removal Formula Sheet

BNR formulas for nutrient loads, nitrification oxygen, alkalinity, denitrification carbon, recycle load, SRT, F/M, oxygen margin 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 used in biological nutrient removal review. It is intended for screening, comparison, commissioning discussions and troubleshooting. Plant-specific design must still account for temperature, kinetics, safety factors, equipment curves, lab methods, permit basis and responsible engineering judgement.

Use and Reporting Basis

State the flow basis, concentration basis and averaging period before using any equation. Use (\text{mg/L as N}) for nitrogen species and (\text{mg/L as P}) for phosphorus species unless a problem states otherwise. For wastewater flows in (\text{m}^3/\text{d}) and concentrations in (\text{mg/L}), multiply by (10^{-3}) to obtain (\text{kg/d}).

Minimum Species Set

A BNR review should normally track:

\{NH_4-N,\ NO_2-N,\ NO_3-N,\ TN,\ TP,\ PO_4-P,\ alkalinity,\ pH,\ DO\}

One species rarely proves performance. Ammonia can fall while nitrate rises, phosphate can release without uptake, and total nitrogen can miss the limit even when nitrification is strong.

Nutrient Load

L = Q C 10^{-3}

where (L) is load in (\text{kg/d}), (Q) is flow in (\text{m}^3/\text{d}) and (C) is concentration in (\text{mg/L}).

Use this for ammonia, nitrate, total nitrogen, orthophosphate, total phosphorus, BOD or COD when the concentration and flow describe the same stream and time period.

For multiple streams:

L_{total}=\sum_i Q_iC_i10^{-3}

Use this form when sidestream return, recycle, bypass or equalization flows materially change the nutrient balance.

Ammonia Removal Load

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

Example: (Q=18000\ \text{m}^3/\text{d}), (C_{in}=28\ \text{mg/L as N}) and (C_{out}=4\ \text{mg/L as N}):

L_{NH4,\text{removed}} = 18000(28-4)10^{-3}=432\ \text{kg/d as N}

This value drives nitrification oxygen demand, alkalinity consumption and expected nitrate formation.

Total Nitrogen Removal

For a simple plant boundary:

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

Removal fraction is:

\displaystyle \eta_{TN}=\frac{C_{TN,in}-C_{TN,out}}{C_{TN,in}}

This screen captures the broader nitrogen outcome. Strong ammonia removal without total-nitrogen removal may indicate nitrification without enough denitrification.

Nitrification Oxygen Demand

O_{2,\text{nit}} \approx 4.57 L_{NH4,\text{removed}}

For the example ammonia load:

O_{2,\text{nit}} = 4.57(432)=1974\ \text{kg O}_2/\text{d}

This is only the nitrification component. Carbon oxidation, endogenous respiration, alpha factor, diffuser condition and control margin must be added separately.

Sidestream-Adjusted Oxygen Demand

If sidestream ammonia is returned to the main process:

O_{2,\text{nit,total}}=4.57(L_{NH4,\text{main}}+L_{NH4,\text{side}})

The sidestream fraction of nitrification oxygen demand is:

\displaystyle f_{O2,side}=\frac{L_{NH4,\text{side}}}{L_{NH4,\text{main}}+L_{NH4,\text{side}}}

This helps explain why a small sidestream flow can consume a large share of aeration capacity.

Nitrification Alkalinity Demand

A_{\text{nit}} \approx 7.14 L_{NH4,\text{removed}}

where (A_{\text{nit}}) is alkalinity demand in (\text{kg/d as CaCO}_3). For (432\ \text{kg/d as N}):

A_{\text{nit}} = 7.14(432)=3084\ \text{kg/d as CaCO}_3

Check residual alkalinity and pH before interpreting poor nitrification as only an aeration problem.

Residual Alkalinity Screen

For an influent alkalinity load (A_{in}):

A_{res}=A_{in}-A_{\text{nit}}+A_{\text{denit,credit}}

A simplified denitrification alkalinity credit may be screened as:

A_{\text{denit,credit}}\approx3.57L_{NOx-N,\text{denit}}

The exact value is process-specific, but the direction matters: nitrification consumes alkalinity, denitrification can recover part of it. Low residual alkalinity can destabilize pH and nitrifier activity.

Denitrification Carbon Requirement

L_{COD,\text{req}} \approx R_{COD/N}L_{NOx-N,\text{removed}}

where (R_{COD/N}) is a planning ratio in (\text{kg COD/kg N}). If (L_{NOx-N}=320\ \text{kg/d}) and (R_{COD/N}=4):

L_{COD,\text{req}}=4(320)=1280\ \text{kg COD/d}

Use readily biodegradable COD when possible. Total COD can overstate available carbon.

Anoxic Carbon Capacity

If available readily biodegradable COD is known:

\displaystyle L_{NOx,\text{cap}}=\frac{L_{rbCOD}}{R_{COD/N}}

If (L_{rbCOD}=900\ \text{kg COD/d}) and (R_{COD/N}=4):

\displaystyle L_{NOx,\text{cap}}=\frac{900}{4}=225\ \text{kg N/d}

Compare this capacity with nitrate entering the anoxic zone. A carbon deficit is a process limitation, not a controller tuning issue.

Internal Recycle Nitrate Load

L_{NOx,R}=Q_R C_{NOx,R}10^{-3}

If internal recycle flow is (Q_R=36000\ \text{m}^3/\text{d}) and nitrate is (8\ \text{mg/L as N}):

L_{NOx,R}=36000(8)10^{-3}=288\ \text{kg/d as N}

This load may help denitrification but can disturb EBPR if it enters an anaerobic selector.

Internal Recycle Ratio

Recycle ratio is:

\displaystyle R_i=\frac{Q_R}{Q}

For (Q_R=36000\ \text{m}^3/\text{d}) and (Q=18000\ \text{m}^3/\text{d}):

R_i=2.0

The ratio is useful only with nitrate concentration and routing. A high recycle ratio with low nitrate may be less important than a lower recycle ratio carrying high nitrate into the wrong zone.

Solids Retention Time

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

where (V) is biological volume, (X) is reactor suspended solids, (Q_wX_w) is wasted solids and (Q_eX_e) is effluent solids loss on a consistent mass basis.

With (V=6000\ \text{m}^3), (X=3000\ \text{mg/L}), (Q_w=250\ \text{m}^3/\text{d}), (X_w=9000\ \text{mg/L}), (Q_e=18000\ \text{m}^3/\text{d}) and (X_e=12\ \text{mg/L}):

\displaystyle \text{SRT}=\frac{6000(3000)10^{-3}}{250(9000)10^{-3}+18000(12)10^{-3}}=7.3\ \text{d}

Cold-weather nitrification may require a higher SRT than warm-weather operation.

Guarded SRT

For a required minimum SRT and uncertainty or operating allowance (\Delta SRT):

SRT_{\text{guard}}=SRT_{\text{calc}}-\Delta SRT

Require:

SRT_{\text{guard}}\ge SRT_{\min}

This is useful when wasting measurements, effluent solids losses or inventory estimates are uncertain.

Food-to-Microorganism Ratio

\displaystyle F/M=\frac{QS_0}{VX}

where (S_0) is influent substrate concentration, often BOD or biodegradable COD, and (X) is biomass concentration. Using (Q=18000), (S_0=220), (V=6000) and (X=3000), with consistent concentration units:

\displaystyle F/M=\frac{18000(220)}{6000(3000)}=0.22\ \text{d}^{-1}

Interpret F/M with SRT, temperature, sludge settleability and nutrient objectives.

Zone Hydraulic Retention Time

For a zone volume (V_z):

\displaystyle HRT_z=\frac{V_z}{Q_z}

Use the actual flow through the zone, not only influent flow. Anoxic HRT may include internal recycle, and aerobic HRT may change when sidestream or return flows enter upstream.

Oxygen Transfer Margin

\displaystyle M_{OTR}=\frac{OTR_{\text{available}}-O_{2,\text{required}}}{O_{2,\text{required}}}

If available oxygen transfer is (2400\ \text{kg O}_2/\text{d}) and nitrification demand alone is (1974\ \text{kg O}_2/\text{d}):

\displaystyle M_{OTR}=\frac{2400-1974}{1974}=0.216

The 21.6 percent margin may be inadequate if carbon oxidation, peak load, diffuser fouling or low alpha factor are not included.

DO Carryover Screen

Dissolved oxygen carried into an anoxic or anaerobic zone can consume carbon or disturb EBPR:

L_{DO,carry}=Q_{carry}C_{DO}10^{-3}

If (Q_{carry}=36000\ \text{m}^3/\text{d}) and (C_{DO}=1.2\ \text{mg/L}):

L_{DO,carry}=43.2\ \text{kg O}_2/\text{d}

This number should be compared with available carbon and the intended zone function.

EBPR Carbon Selection Screen

\displaystyle R_{VFA/P}=\frac{L_{VFA-COD}}{L_{P,in}}

This ratio screens whether readily available carbon is plausible for anaerobic PAO selection. It does not prove EBPR health. Check nitrate intrusion, dissolved oxygen carryover, phosphate release, uptake, wasting and final total phosphorus.

Chemical Phosphorus Backstop

If biological phosphorus removal cannot meet the final target, a chemical phosphorus-removal backstop can be screened as:

D_{chem}=R_{metal/P}L_{P,\text{to remove}}

where (R_{metal/P}) is a site-specific dose ratio. This is only a planning screen; actual chemical dose depends on species, alkalinity, solids production, mixing and permit basis.

Sidestream Nutrient Fraction

\displaystyle f_s=\frac{L_s}{L_{\text{main}}+L_s}

If sidestream ammonia return is (140\ \text{kg/d as N}) and mainstream ammonia load is (432\ \text{kg/d as N}):

\displaystyle f_s=\frac{140}{432+140}=0.245

A 24.5 percent sidestream contribution is large enough to affect aeration, alkalinity and effluent ammonia trends.

Nutrient Release Gate

A compact release gate is:

G = N \land P \land O \land A \land S \land V

where (N) is nitrogen performance, (P) is phosphorus performance, (O) is oxygen-transfer margin, (A) is alkalinity/pH support, (S) is solids/SRT control and (V) is valid monitoring evidence. If any element is false, the result should be treated as conditional or held for review.

Compliance Margin

\displaystyle M_c=\frac{C_{\text{limit}}-C_{\text{observed}}}{C_{\text{limit}}}

Use concentration margin for concentration limits and load margin for load limits. A positive margin does not prove stability if the monitoring period misses peak flow, cold weather, sidestream return or maintenance conditions.

Data Validity

\displaystyle A=\frac{n_{\text{valid}}}{n_{\text{required}}}

where (A) is data availability. Use this with analyzer uptime, lab quality checks, calibration records and missing-period context. Validity percentage alone does not prove representative monitoring.

Validity Limits

These equations are screening tools. They assume consistent units, representative sampling, stable flow basis and appropriate averaging periods. They do not replace kinetic modeling, design standards, site-specific safety factors, permit interpretation, lab QA/QC or commissioning evidence.

Review Checklist

Before accepting a BNR calculation, check:

nitrogen and phosphorus are reported on correct N or P basis;
flows and concentrations describe the same stream and period;
sidestream loads are included where relevant;
nitrification oxygen and alkalinity demand are both checked;
denitrification carbon uses available COD, not only total COD;
SRT includes wasting and effluent solids losses;
recycle nitrate load is interpreted with zone routing;
oxygen-transfer margin includes realistic alpha and fouling assumptions;
compliance margin matches the permit averaging period;
monitoring validity supports the operating decision.

Common Formula Mistakes

Common mistakes include mixing (\text{mg/L as NH}_4) with (\text{mg/L as N}), applying the (4.57) oxygen factor to total ammonia concentration without basis conversion, using total COD as denitrification carbon, ignoring alkalinity, calculating SRT from incomplete solids losses, treating internal recycle as only a flow ratio, and interpreting positive compliance margin as proof of resilient BNR operation.

REF

Disciplines