Key formulas for the WPI Class II Wastewater Treatment Operator exam — secondary treatment, nutrient removal, biosolids management, and advanced treatment.

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🦠 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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