♻️ WPI Class II Wastewater Formula Sheet
Key formulas for the WPI Class II Wastewater Treatment Operator exam — secondary treatment, nutrient removal, biosolids management, and advanced treatment.
🦠 Secondary Treatment — Activated Sludge
Food-to-Microorganism (F/M) Ratio
F/M = BOD Load (kg/d) ÷ MLVSS (kg)
- BOD Load (kg/d) = Influent BOD (mg/L) × Flow (m³/d) ÷ 1,000
- MLVSS (kg) = MLVSS (mg/L) × Basin Volume (m³) ÷ 1,000
- Typical range: 0.2–0.5 kg BOD/kg MLVSS/d (conventional)
Example:
BOD Load = 250 mg/L × 10,000 m³/d ÷ 1,000 = 2,500 kg/d; MLVSS = 2,500 mg/L × 4,000 m³ ÷ 1,000 = 10,000 kg
→ F/M = 2,500 ÷ 10,000 = 0.25 kg BOD/kg MLVSS/d
Sludge Volume Index (SVI)
SVI (mL/g) = SSV₃₀ (mL/L) ÷ MLSS (g/L)
- SSV₃₀ = Settled sludge volume after 30 min (mL/L)
- MLSS = Mixed liquor suspended solids (g/L)
- Good settling: SVI 80–150 mL/g; Bulking: >150 mL/g
Example:
SSV₃₀ = 250 mL/L; MLSS = 3.0 g/L
→ SVI = 250 ÷ 3.0 = 83 mL/g (good settling)
Sludge Retention Time (SRT / MCRT)
SRT (d) = Total Solids in System (kg) ÷ Solids Wasted per Day (kg/d)
- Total Solids = MLSS (mg/L) × Basin Volume (m³) ÷ 1,000
- Solids Wasted = WAS flow (m³/d) × WAS TSS (mg/L) ÷ 1,000 + Effluent TSS loss
- Typical SRT: 5–15 days (with nitrification: 10–20 days)
Example:
MLSS = 3,000 mg/L × 4,000 m³ ÷ 1,000 = 12,000 kg; WAS = 200 m³/d × 8,000 mg/L ÷ 1,000 = 1,600 kg/d
→ SRT = 12,000 ÷ 1,600 = 7.5 days
Hydraulic Retention Time (HRT)
HRT (h) = Basin Volume (m³) ÷ Flow Rate (m³/h)
- Basin Volume = Length × Width × Depth (m³)
- Flow Rate = Daily flow ÷ 24 (m³/h)
- Typical HRT: 4–8 hours (conventional activated sludge)
Example:
Basin = 4,000 m³; Flow = 10,000 m³/d ÷ 24 = 416.7 m³/h
→ HRT = 4,000 ÷ 416.7 = 9.6 hours
🧪 Nutrient Removal
Nitrogen Loading Rate
TN Load (kg/d) = TN (mg/L) × Flow (m³/d) ÷ 1,000
- TN = Total Nitrogen (ammonia + nitrate + organic N)
- Typical municipal wastewater TN: 30–50 mg/L
- Effluent TN target (nutrient-sensitive): <3 mg/L
Example:
TN = 40 mg/L; Flow = 10,000 m³/d
→ TN Load = 40 × 10,000 ÷ 1,000 = 400 kg/d
Denitrification Rate
NO₃-N Removed (mg/L) = Influent NO₃-N − Effluent NO₃-N
- Carbon required for denitrification ≈ 3–4 g BOD per g NO₃-N removed
- Methanol dose ≈ 2.5–3.0 g methanol per g NO₃-N (post-anoxic)
- Internal recycle ratio (MLE): typically 2–4× influent flow
Example:
Influent NO₃-N = 25 mg/L; Effluent NO₃-N = 3 mg/L
→ NO₃-N Removed = 25 − 3 = 22 mg/L; Carbon needed ≈ 22 × 3.5 = 77 mg/L BOD
Chemical Phosphorus Removal — Alum Dose
Alum Dose (mg/L) = Molar Ratio × P Concentration × (MW Alum ÷ MW P)
- Al₂(SO₄)₃·18H₂O MW = 666 g/mol; P MW = 31 g/mol
- Molar ratio Al:P ≈ 1.5–2.0 for effective removal
- Practical rule: 10–15 mg/L alum per 1 mg/L P to be removed
Example:
P to remove = 4 mg/L; Molar ratio = 1.5
→ Alum Dose ≈ 4 × 1.5 × (666/2 ÷ 31) = 4 × 1.5 × 10.7 ≈ 64 mg/L alum
Phosphorus Mass Balance
P Removed (kg/d) = P In − P Out = (P_in − P_eff) × Q ÷ 1,000
- P_in = Influent total phosphorus (mg/L)
- P_eff = Effluent total phosphorus (mg/L)
- Q = Flow rate (m³/d)
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
♻️ Biosolids Management
Volatile Solids Destruction
VS Destruction (%) = (VS_in − VS_out) ÷ VS_in × 100
- VS_in = Volatile solids entering digester (kg/d)
- VS_out = Volatile solids leaving digester (kg/d)
- Target: 40–60% for mesophilic anaerobic digestion
Example:
VS_in = 1,000 kg/d; VS_out = 450 kg/d
→ VS Destruction = (1,000 − 450) ÷ 1,000 × 100 = 55%
Biogas Production Estimate
Biogas (m³/d) ≈ VS Destroyed (kg/d) × 0.75–1.0 m³/kg
- Typical yield: 0.75–1.0 m³ biogas per kg VS destroyed
- Methane content: 60–70% of biogas
- Energy value: ~22 MJ/m³ methane (6.1 kWh/m³)
Example:
VS Destroyed = 550 kg/d; yield = 0.85 m³/kg
→ Biogas = 550 × 0.85 = 467.5 m³/d; Methane ≈ 467.5 × 0.65 = 304 m³/d
Sludge Thickening — Solids Loading Rate
SLR (kg/m²/d) = Sludge Flow (m³/d) × TSS (mg/L) ÷ 1,000 ÷ Thickener Area (m²)
- Gravity thickener SLR: 25–100 kg/m²/d (primary sludge)
- DAF thickener SLR: 50–150 kg/m²/d (WAS)
- Hydraulic loading: 1–4 m³/m²/h (gravity thickener)
Example:
WAS flow = 500 m³/d; TSS = 8,000 mg/L; Area = 50 m²
→ SLR = 500 × 8,000 ÷ 1,000 ÷ 50 = 80 kg/m²/d
Biosolids Cake Solids Content
Solids Content (%) = Dry Solids (kg) ÷ Total Wet Cake (kg) × 100
- Belt filter press: 15–25% solids
- Centrifuge: 18–28% solids
- Screw press: 18–25% solids
- Heat drying: 85–95% solids
Example:
Wet cake = 10,000 kg; Dry solids = 2,200 kg
→ Solids Content = 2,200 ÷ 10,000 × 100 = 22%
🔬 Advanced Treatment
Membrane Flux
Flux (L/m²/h) = Permeate Flow (L/h) ÷ Membrane Area (m²)
- Typical MBR flux: 10–25 L/m²/h (sustainable)
- Peak flux: up to 40 L/m²/h (short-term)
- Transmembrane pressure (TMP): 0.1–0.5 bar (normal operation)
Example:
Permeate flow = 500 m³/d = 20,833 L/h; Membrane area = 1,000 m²
→ Flux = 20,833 ÷ 1,000 = 20.8 L/m²/h
UV Disinfection Dose
UV Dose (mJ/cm²) = UV Intensity (mW/cm²) × Contact Time (s)
- Typical wastewater UV dose: 40–100 mJ/cm²
- Class A reuse: ≥100 mJ/cm²
- Dose depends on UV transmittance (UVT) of effluent
Example:
UV Intensity = 10 mW/cm²; Contact time = 8 s
→ UV Dose = 10 × 8 = 80 mJ/cm²
Ozone Dose
O₃ Dose (mg/L) = O₃ Applied (g/h) ÷ Flow (m³/h) × 1,000
- Typical wastewater ozone dose: 5–15 mg/L
- Ozone transfer efficiency: 80–95% (fine bubble diffusers)
- CT for disinfection: varies by target organism
Example:
O₃ Applied = 5,000 g/h; Flow = 500 m³/h
→ O₃ Dose = 5,000 ÷ 500 × 1,000 = 10 mg/L
Chlorination — Chlorine Demand
Chlorine Demand (mg/L) = Chlorine Applied − Chlorine Residual
- Chlorine Applied = dose added to effluent (mg/L)
- Chlorine Residual = remaining after contact time (mg/L)
- Typical wastewater effluent demand: 5–20 mg/L
Example:
Applied = 15 mg/L; Residual = 2 mg/L
→ Chlorine Demand = 15 − 2 = 13 mg/L
⚙️ Process Control Calculations
Return Activated Sludge (RAS) Rate
RAS Rate (%) = RAS Flow ÷ Influent Flow × 100
- Typical RAS rate: 50–100% of influent flow
- RAS concentration can be estimated from clarifier underflow
- Higher RAS needed when MLSS target is high or SVI is elevated
Example:
RAS Flow = 6,000 m³/d; Influent Flow = 10,000 m³/d
→ RAS Rate = 6,000 ÷ 10,000 × 100 = 60%
Waste Activated Sludge (WAS) Rate
WAS (m³/d) = (MLSS × V_basin) ÷ (SRT × TSS_WAS) − (Q_eff × TSS_eff ÷ TSS_WAS)
- V_basin = Aeration basin volume (m³)
- SRT = Target sludge retention time (days)
- TSS_WAS = TSS in waste sludge stream (mg/L)
- Simplified: WAS ≈ Total Solids ÷ (SRT × WAS TSS concentration)
Example:
Total solids = 12,000 kg; SRT = 10 d; WAS TSS = 8,000 mg/L = 8 kg/m³
→ WAS = 12,000 ÷ (10 × 8) = 150 m³/d
Surface Overflow Rate (SOR) — Secondary Clarifier
SOR (m³/m²/d) = Flow (m³/d) ÷ Surface Area (m²)
- Typical SOR: 16–32 m³/m²/d (average flow)
- Peak SOR: up to 48 m³/m²/d
- Lower SOR = better settling performance
Example:
Flow = 10,000 m³/d; Clarifier area = 500 m²
→ SOR = 10,000 ÷ 500 = 20 m³/m²/d
Oxygen Uptake Rate (OUR) / Specific OUR (SOUR)
SOUR (mg O₂/g VSS/h) = OUR (mg/L/h) ÷ MLVSS (g/L)
- OUR = rate of DO decrease in a sealed sample (mg/L/h)
- Typical SOUR: 8–20 mg O₂/g VSS/h (healthy sludge)
- Low SOUR (<5) may indicate old sludge or inhibition
Example:
OUR = 40 mg/L/h; MLVSS = 2.5 g/L
→ SOUR = 40 ÷ 2.5 = 16 mg O₂/g VSS/h
🛡️ Safety & Regulatory
Effluent Compliance — BOD Removal Efficiency
Removal (%) = (Influent BOD − Effluent BOD) ÷ Influent BOD × 100
- Typical secondary treatment target: ≥85% BOD removal
- Effluent BOD limit (Ontario ECA): 25 mg/L (30-day avg)
- Effluent TSS limit: 25 mg/L (30-day avg)
Example:
Influent BOD = 200 mg/L; Effluent BOD = 12 mg/L
→ Removal = (200 − 12) ÷ 200 × 100 = 94%
Chlorine Contact Time (CT) for Disinfection
CT (mg·min/L) = Residual Cl₂ (mg/L) × Contact Time (min)
- Wastewater effluent CT for 2-log E. coli reduction: ~30–50 mg·min/L
- Contact time typically 15–30 minutes
- Effective CT = T₁₀ (time for 10% of water to pass through)
Example:
Residual = 2 mg/L; Contact time = 20 min
→ CT = 2 × 20 = 40 mg·min/L
Biosolids Land Application Rate
Application Rate (t/ha) = Crop N Requirement (kg/ha) ÷ Available N in Biosolids (kg/t)
- Available N = (Ammonia-N × 0.5) + (Organic N × mineralization rate)
- Typical Class B biosolids available N: 10–20 kg/t dry weight
- Application rate limited by N, P, or metals — use most restrictive
Example:
Crop N need = 150 kg/ha; Available N = 15 kg/t dry biosolids
→ Application Rate = 150 ÷ 15 = 10 t/ha dry biosolids
Dilution Factor for Effluent Toxicity
Dilution Factor = Receiving Water Flow ÷ (Receiving Water Flow + Effluent Flow)
- Higher dilution = lower toxicity risk to receiving water
- Whole effluent toxicity (WET) testing uses serial dilutions
- LC50 = concentration lethal to 50% of test organisms
Example:
River flow = 10 m³/s; Effluent = 0.1 m³/s
→ Dilution Factor = 10 ÷ (10 + 0.1) ≈ 99:1 dilution
📋 Quick Reference — Typical Operating Ranges
MLSS (conventional AS)2,000–4,000 mg/L
MLSS (MBR)8,000–15,000 mg/L
SVI (good settling)80–150 mL/g
F/M ratio (conventional)0.2–0.5 kg BOD/kg MLVSS/d
SRT (with nitrification)10–20 days
HRT (conventional AS)4–8 hours
DO target (aeration basin)2–4 mg/L
RAS rate50–100% of influent flow
Secondary clarifier SOR16–32 m³/m²/d
VS destruction (anaerobic digestion)40–60%
Biogas methane content60–70%
Belt filter press cake solids15–25%
Effluent BOD limit (Ontario)25 mg/L (30-day avg)
Effluent TSS limit (Ontario)25 mg/L (30-day avg)
UV dose (wastewater)40–100 mJ/cm²
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