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WPI Class III Wastewater Formula Sheet

BNR kinetics · SRT/F/M · MBR flux · biosolids · industrial pretreatment · regulatory calculations

🌿

Biological Nutrient Removal (BNR)

Nitrogen Mass Balance (BNR System)

TN_removed = TN_in − TN_effluent

• TN_in = TKN_in + NO₃_in (mg/L)

• TN_effluent = NH₄_eff + NO₃_eff + NO₂_eff + Org-N_eff (mg/L)

• TN_removal efficiency = (TN_in − TN_eff) / TN_in × 100%

EXAMPLE

TN_in = 35 mg/L; TN_eff = 8 mg/L

→ TN removal = (35 − 8) / 35 × 100 = 77%

Nitrification Rate (Monod Kinetics)

μ_AOB = μ_max × [NH₄] / (K_NH₄ + [NH₄]) × [DO] / (K_O + [DO])

• μ_max = maximum specific growth rate of AOB (0.8–1.0 /d at 20°C)

• K_NH₄ = half-saturation constant for ammonia (~0.5–1.0 mg NH₄-N/L)

• K_O = half-saturation constant for DO (~0.3–0.5 mg/L)

• SRT_min for nitrification = 1 / μ_AOB (safety factor ×2–3)

EXAMPLE

μ_max = 0.9/d; [NH₄] = 5 mg/L; K_NH₄ = 0.7 mg/L; [DO] = 2 mg/L; K_O = 0.4 mg/L

→ μ_AOB = 0.9 × (5/5.7) × (2/2.4) = 0.9 × 0.877 × 0.833 = 0.66/d → SRT_min = 1.5 d

Denitrification Rate

SDNR = (NO₃_in − NO₃_out) × Q / (V × MLVSS)

• SDNR = specific denitrification rate (kg NO₃-N/kg MLVSS·d)

• NO₃_in, NO₃_out = nitrate entering and leaving anoxic zone (mg/L)

• Q = flow through anoxic zone (m³/d)

• V = anoxic zone volume (m³); MLVSS = mg/L

• Typical SDNR: 0.03–0.11 kg NO₃-N/kg MLVSS·d

EXAMPLE

NO₃_in = 20 mg/L; NO₃_out = 5 mg/L; Q = 10,000 m³/d; V = 2,000 m³; MLVSS = 2,500 mg/L

→ SDNR = (20−5) × 10,000 / (2,000 × 2,500) × 1,000 = 150,000 / 5,000,000 × 1,000 = 0.03 kg/kg·d

Internal Recycle (IR) Rate for Denitrification

NO₃_eff ≈ TKN_in / (IR/Q + RAS/Q + 1) − (denitrification in anoxic)

• IR = internal recycle flow (m³/d)

• RAS = return activated sludge flow (m³/d)

• Q = influent flow (m³/d)

• Typical IR/Q ratio: 2–4× for 70–80% TN removal

• Higher IR/Q increases pumping energy with diminishing returns

EXAMPLE

TKN_in = 30 mg/L; IR/Q = 3; RAS/Q = 1; assume complete denitrification in anoxic zone

→ NO₃_eff ≈ 30 / (3 + 1 + 1) = 6 mg/L (theoretical minimum with IR = 3Q)

Phosphorus Removal — EBPR

P_removed = P_in − P_eff (mg/L) or P_in × Q × 1/1,000 (kg/d)

• EBPR requires anaerobic zone (P release) followed by aerobic zone (P uptake)

• PAOs store phosphorus as polyphosphate (poly-P) granules

• P content of PAO-rich sludge: 5–15% of VSS (vs. 1.5–2% for normal sludge)

• P removal via WAS: P_WAS = WAS_flow × WAS_TSS × P_fraction / 1,000

EXAMPLE

P_in = 8 mg/L; P_eff = 0.5 mg/L; Q = 10,000 m³/d

→ P removed = (8 − 0.5) × 10,000 / 1,000 = 75 kg/d

Aerobic SRT for BNR

SRT_aerobic = SRT_total × (V_aerobic / V_total)

• SRT_total = overall mean cell residence time (days)

• V_aerobic = volume of aerobic zones (m³)

• V_total = total aeration tank volume (m³)

• Minimum aerobic SRT for nitrification at 15°C: ~6–8 days

• Minimum aerobic SRT for nitrification at 10°C: ~10–15 days

EXAMPLE

SRT_total = 15 d; V_aerobic = 3,000 m³; V_total = 5,000 m³

→ SRT_aerobic = 15 × (3,000/5,000) = 9 days (adequate for nitrification at 15°C)

🦠

Activated Sludge Process Control

Sludge Retention Time (SRT / MCRT)

SRT = (V × MLSS) / (Q_WAS × SS_WAS + Q_eff × SS_eff)

• V = aeration tank volume (m³)

• MLSS = mixed liquor suspended solids (mg/L)

• Q_WAS = waste activated sludge flow (m³/d)

• SS_WAS = WAS suspended solids concentration (mg/L)

• Q_eff = effluent flow (m³/d); SS_eff = effluent TSS (mg/L)

EXAMPLE

V = 4,000 m³; MLSS = 3,000 mg/L; Q_WAS = 150 m³/d; SS_WAS = 8,000 mg/L; Q_eff = 10,000 m³/d; SS_eff = 10 mg/L

→ SRT = (4,000 × 3,000) / (150 × 8,000 + 10,000 × 10) = 12,000,000 / 1,300,000 = 9.2 days

Food-to-Microorganism (F/M) Ratio

F/M = (Q × BOD_in) / (V × MLVSS) [kg BOD/kg MLVSS·d]

• Q = influent flow (m³/d)

• BOD_in = influent BOD (mg/L)

• V = aeration tank volume (m³)

• MLVSS = volatile suspended solids in aeration tank (mg/L)

• Typical F/M: 0.05–0.15 (extended aeration), 0.2–0.4 (conventional), 0.4–1.5 (high-rate)

EXAMPLE

Q = 10,000 m³/d; BOD_in = 200 mg/L; V = 4,000 m³; MLVSS = 2,200 mg/L

→ F/M = (10,000 × 200) / (4,000 × 2,200) × 1/1,000 = 2,000,000 / 8,800,000 = 0.23 kg/kg·d

Sludge Volume Index (SVI)

SVI (mL/g) = SSV₃₀ (mL/L) / MLSS (g/L)

• SSV₃₀ = settled sludge volume after 30 min in 1-L cylinder (mL/L)

• MLSS = mixed liquor suspended solids (g/L)

• SVI < 80: excellent settling; 80–150: good; 150–250: poor (bulking); >250: severe bulking

• DSVI used when MLSS > 3,500 mg/L (dilute sample 1:2 before test)

EXAMPLE

SSV₃₀ = 220 mL/L; MLSS = 3.2 g/L

→ SVI = 220 / 3.2 = 69 mL/g (excellent settling)

Oxygen Uptake Rate (OUR) and SOUR

SOUR = OUR / MLVSS [mg O₂/g VSS·h]

• OUR = oxygen uptake rate measured in respirometer (mg O₂/L·h)

• MLVSS = volatile suspended solids (g/L)

• SOUR 8–20: active, healthy biomass

• SOUR < 5: inhibited or starved biomass

• SOUR > 30: young sludge, high F/M

EXAMPLE

OUR = 45 mg O₂/L·h; MLVSS = 2.5 g/L

→ SOUR = 45 / 2.5 = 18 mg O₂/g VSS·h (healthy, active biomass)

Oxygen Demand (AOR)

AOR = 1.5 × BOD_removed + 4.57 × NH₄_nitrified − 2.86 × NO₃_denitrified

• AOR = actual oxygen requirement (kg O₂/d)

• BOD_removed = kg BOD/d removed

• NH₄_nitrified = kg NH₄-N/d nitrified (×4.57 for O₂ demand)

• NO₃_denitrified = kg NO₃-N/d denitrified (×2.86 O₂ credit)

• Factor 1.5 accounts for endogenous respiration

EXAMPLE

BOD removed = 1,500 kg/d; NH₄ nitrified = 200 kg/d; NO₃ denitrified = 150 kg/d

→ AOR = 1.5×1,500 + 4.57×200 − 2.86×150 = 2,250 + 914 − 429 = 2,735 kg O₂/d

🔬

Membrane Bioreactor (MBR)

Membrane Flux

J = Q_permeate / A_membrane [L/m²·h = LMH]

• J = permeate flux (LMH)

• Q_permeate = permeate flow rate (L/h)

• A_membrane = total membrane area (m²)

• Typical MBR flux: 15–30 LMH (submerged), 30–60 LMH (pressurized)

• Critical flux: flux above which fouling rate increases rapidly

EXAMPLE

Q_permeate = 10,000 L/h; A_membrane = 500 m²

→ J = 10,000 / 500 = 20 LMH (typical submerged MBR flux)

Transmembrane Pressure (TMP)

TMP = P_feed − P_permeate (kPa)

• TMP = transmembrane pressure (kPa)

• P_feed = pressure on feed side (kPa)

• P_permeate = pressure on permeate side (kPa)

• Typical operating TMP: 10–40 kPa (submerged MBR)

• Rising TMP at constant flux indicates membrane fouling

EXAMPLE

P_feed = 30 kPa; P_permeate = 5 kPa

→ TMP = 30 − 5 = 25 kPa (normal operating range)

Specific Membrane Resistance

R = TMP / (μ × J) [m⁻¹]

• R = membrane resistance (m⁻¹)

• TMP = transmembrane pressure (Pa)

• μ = permeate viscosity (Pa·s, ≈ 0.001 Pa·s at 20°C)

• J = flux (m/s, convert from LMH: 1 LMH = 2.78 × 10⁻⁷ m/s)

• Typical clean membrane resistance: 10¹¹–10¹² m⁻¹

EXAMPLE

TMP = 20,000 Pa; μ = 0.001 Pa·s; J = 20 LMH = 5.56 × 10⁻⁶ m/s

→ R = 20,000 / (0.001 × 5.56×10⁻⁶) = 3.6 × 10¹² m⁻¹

MBR Aeration Intensity (SADm)

SADm = Q_air / A_membrane [Nm³/m²·h]

• SADm = specific aeration demand per membrane area

• Q_air = air flow rate for membrane scouring (Nm³/h)

• A_membrane = total membrane area (m²)

• Typical SADm: 0.2–0.5 Nm³/m²·h for submerged MBR

• Higher SADm reduces fouling but increases energy consumption

EXAMPLE

Q_air = 200 Nm³/h; A_membrane = 500 m²

→ SADm = 200 / 500 = 0.4 Nm³/m²·h (typical operating range)

♻️

Advanced Biosolids Management

Volatile Solids Reduction (VSR) — Digester

VSR = (VS_in − VS_out) / VS_in × 100%

• VS_in = volatile solids entering digester (kg/d or %)

• VS_out = volatile solids leaving digester (kg/d or %)

• Typical VSR: 50–60% (mesophilic AD), 55–65% (thermophilic AD)

• Class B biosolids require VSR ≥ 38% or PSRP

• Class A biosolids require additional pathogen reduction (PFRP)

EXAMPLE

VS_in = 1,000 kg/d; VS_out = 420 kg/d

→ VSR = (1,000 − 420) / 1,000 × 100 = 58% (good mesophilic AD performance)

Biogas Production Rate

Q_biogas = VS_destroyed × 0.75–1.12 m³ CH₄/kg VS (+ CO₂)

• Typical biogas yield: 0.75–1.12 m³ CH₄/kg VS destroyed

• Biogas composition: 60–70% CH₄, 30–40% CO₂

• Energy content of CH₄: 35.8 MJ/Nm³

• Combined heat and power (CHP) efficiency: 30–40% electrical, 40–50% thermal

• Net energy recovery: typically 0.5–1.0 kWh/m³ wastewater treated

EXAMPLE

VS destroyed = 500 kg/d; CH₄ yield = 0.9 m³/kg VS

→ CH₄ production = 500 × 0.9 = 450 m³/d; Energy = 450 × 35.8 MJ = 16,110 MJ/d = 4,475 kWh/d

Biosolids Land Application Rate (Agronomic Rate)

Application rate (dry t/ha) = Crop N requirement / Available N in biosolids

• Available N = (Mineral N × 1.0) + (Organic N × mineralization rate)

• Mineralization rate: 20–30% in year 1 for Class B biosolids

• Maximum application rate limited by N, P, or metals (whichever is most restrictive)

• Setback distances: 30 m from watercourses, 100 m from wells

EXAMPLE

Crop N requirement = 150 kg N/ha; Biosolids: 3% TN, 60% available N

→ Rate = 150 / (30,000 mg/kg × 0.60 × 10⁻³) = 150 / 18 = 8.3 dry t/ha

Sludge Dewatering — Cake Solids Content

Solids recovery = (Cake mass × Cake %TS) / (Feed mass × Feed %TS) × 100%

• Cake %TS: belt press 18–25%, centrifuge 20–28%, filter press 30–45%

• Polymer dose: 3–8 g/kg DS (belt press), 5–15 g/kg DS (centrifuge)

• Cake volume = Wet sludge volume × (1 − %TS_feed) / (1 − %TS_cake)

• Volume reduction: 15% TS cake has ~5× less volume than 3% TS thickened sludge

EXAMPLE

Feed: 10 m³/h at 3% TS; Cake: 0.4 m³/h at 22% TS

→ Solids recovery = (0.4 × 22) / (10 × 3) × 100 = 8.8 / 30 × 100 = 29% (low — check polymer dose)

Alkalinity Requirement for Anaerobic Digestion

Alkalinity (mg/L as CaCO₃) should be 2,000–5,000 mg/L for stable AD

• VFA/Alkalinity ratio < 0.3 indicates stable digester

• VFA/Alkalinity ratio > 0.5 indicates potential souring

• Lime addition to raise alkalinity: CaO dose = (target − current alkalinity) × 0.74

• Bicarbonate alkalinity (TA) is the key buffer in AD

EXAMPLE

Current alkalinity = 1,500 mg/L; Target = 3,000 mg/L; Digester volume = 1,000 m³

→ CaO needed = (3,000 − 1,500) × 0.74 × 1,000 m³ × 1,000 L/m³ / 10⁶ = 1,110 kg CaO

🏭

Industrial Pretreatment

Industrial User Permit Limit (IU Limit)

IU_limit = POTW_limit × (Q_POTW / Q_IU) × (1 − removal_efficiency)

• POTW_limit = POTW effluent permit limit for the pollutant (mg/L)

• Q_POTW = POTW design flow (m³/d)

• Q_IU = industrial user flow (m³/d)

• removal_efficiency = POTW removal efficiency for the pollutant (decimal)

• IU limit must also consider pass-through and interference criteria

EXAMPLE

POTW limit = 1 mg/L Pb; Q_POTW = 10,000 m³/d; Q_IU = 500 m³/d; POTW removal = 80%

→ IU_limit = 1 × (10,000/500) × (1/0.20) = 1 × 20 × 5 = 100 mg/L Pb (simplified)

⚠️ Actual IU limits use more complex calculations including headworks analysis and local limits.

Toxic Unit (TU) — Effluent Toxicity

TU = 100 / LC50 (%) or TU = 1 / (EC50 / 100)

• TU = toxic units (dimensionless)

• LC50 = concentration (% effluent) causing 50% mortality in 96-hr test

• EC50 = concentration causing 50% effect in chronic test

• WSER requires LC50 > 100% (i.e., TU < 1) for acute lethality

• TU > 1 means the effluent is acutely lethal at 100% concentration

EXAMPLE

LC50 = 45% effluent (in 96-hr rainbow trout test)

→ TU = 100 / 45 = 2.2 TU (FAILS — effluent is acutely lethal; must investigate cause)

Equalization Basin Volume

V_eq = ΣQ_in × Δt − ΣQ_out × Δt (cumulative flow method)

• V_eq = required equalization volume (m³)

• Q_in = instantaneous inflow rate (m³/h)

• Q_out = constant outflow rate = average daily flow (m³/h)

• Δt = time interval (h)

• V_eq = maximum cumulative difference between inflow and outflow

EXAMPLE

Peak flow = 2× average; Average flow = 500 m³/h; Peak duration = 4 hours

→ V_eq ≈ (2,000 − 500) × 4 = 6,000 m³ (approximate — use mass balance diagram for accuracy)

📋

Regulatory & Environmental Calculations

Effluent Load Calculation

Load (kg/d) = Concentration (mg/L) × Flow (m³/d) / 1,000

• 1 mg/L × 1 m³/d = 1 g/d = 0.001 kg/d

• 1 mg/L × 1,000 m³/d = 1 kg/d

• Load (kg/d) = C (mg/L) × Q (m³/d) × 10⁻³

• Load (lb/d) = C (mg/L) × Q (MGD) × 8.34

EXAMPLE

Effluent TSS = 12 mg/L; Q = 15,000 m³/d

→ TSS load = 12 × 15,000 / 1,000 = 180 kg/d

Dilution Factor (Receiving Water)

DF = (Q_river + Q_effluent) / Q_effluent

• DF = dilution factor (dimensionless)

• Q_river = receiving water flow (m³/s or m³/d)

• Q_effluent = effluent discharge flow (m³/s or m³/d)

• Mixed concentration: C_mix = (C_eff × Q_eff + C_river × Q_river) / (Q_eff + Q_river)

EXAMPLE

Q_river (7Q10) = 1.5 m³/s; Q_eff = 0.15 m³/s; C_eff = 10 mg/L; C_river = 0 mg/L

→ DF = (1.5 + 0.15) / 0.15 = 11×; C_mix = 10 × 0.15 / 1.65 = 0.91 mg/L

Greenhouse Gas Emissions — N₂O

N₂O emissions (kg N₂O/yr) = TN_load (kg N/yr) × EF_N₂O

• EF_N₂O = emission factor for N₂O from treatment (0.005 kg N₂O-N/kg N for BNR)

• GWP of N₂O = 298 (100-year global warming potential vs CO₂)

• CO₂e = N₂O emissions × 298

• IPCC default EF for direct N₂O from WWTP: 0.0032 kg N₂O-N/kg N influent

EXAMPLE

TN load = 200,000 kg N/yr; EF = 0.005 kg N₂O-N/kg N

→ N₂O = 200,000 × 0.005 × 44/28 = 1,571 kg N₂O/yr; CO₂e = 1,571 × 298 = 468,000 kg CO₂e/yr

Biosolids Metal Loading Rate

Annual loading (kg/ha·yr) = Application rate (dry t/ha) × Metal concentration (mg/kg) / 1,000

• Cumulative loading limits (CCME): Pb 150 kg/ha, Cd 4 kg/ha, Cu 150 kg/ha, Zn 300 kg/ha

• Annual loading limits: Pb 15 kg/ha·yr, Cd 0.5 kg/ha·yr

• Metal concentration in biosolids (mg/kg dry weight)

• Application rate (dry tonnes/ha)

EXAMPLE

Biosolids Zn = 1,200 mg/kg; Application rate = 5 dry t/ha

→ Annual Zn loading = 5 × 1,200 / 1,000 = 6 kg Zn/ha·yr (< 30 kg/ha·yr annual limit)

⚙️

Advanced Process Control

Aeration Energy — Standard Oxygen Transfer Rate (SOTR)

SOTR = AOR / (α × β × θ^(T−20) × (C_s20 − C_L) / C_s20)

• AOR = actual oxygen requirement (kg O₂/d)

• α = process water correction factor (0.4–0.8 for activated sludge)

• β = salinity-surface tension correction (0.95–0.99)

• θ = temperature correction factor (1.024)

• C_s20 = DO saturation at 20°C (9.09 mg/L); C_L = operating DO (mg/L)

EXAMPLE

AOR = 2,000 kg/d; α = 0.6; β = 0.97; T = 15°C; C_L = 2 mg/L; C_s20 = 9.09 mg/L

→ SOTR = 2,000 / (0.6 × 0.97 × 1.024^(15−20) × (9.09−2)/9.09) = 2,000 / (0.6 × 0.97 × 0.887 × 0.780) = 2,000 / 0.402 = 4,975 kg O₂/d

Blower Power Requirement

P = (Q_air × ρ_air × R × T₁) / (η × 29) × [(P₂/P₁)^(0.283) − 1]

• P = shaft power (kW)

• Q_air = air flow rate (m³/s)

• ρ_air = air density (1.2 kg/m³ at 20°C)

• R = gas constant (8,314 J/kmol·K)

• T₁ = inlet air temperature (K)

• η = blower efficiency (0.70–0.85)

• P₂/P₁ = discharge/inlet pressure ratio

EXAMPLE

Q_air = 1 m³/s; T₁ = 293 K; P₁ = 101.3 kPa; P₂ = 115 kPa; η = 0.75

→ P ≈ Q × ΔP / η = 1 × (115,000 − 101,300) / 0.75 ≈ 18.3 kW (simplified isothermal approximation)

Hydraulic Loading Rate (HLR) — Secondary Clarifier

HLR (m/h) = Q_influent / A_clarifier

• HLR = hydraulic loading rate (m/h or m³/m²·h)

• Q_influent = influent flow to clarifier (m³/h)

• A_clarifier = clarifier surface area (m²)

• Typical HLR: 0.5–1.5 m/h (average), 1.5–2.5 m/h (peak)

• Solids loading rate (SLR) = (Q + Q_RAS) × MLSS / A_clarifier

EXAMPLE

Q = 10,000 m³/d; A = 800 m²

→ HLR = (10,000/24) / 800 = 416.7 / 800 = 0.52 m/h (within normal range)

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