Formula Sheet: Wastewater Systems

Wastewater Flow and Loading

Flow Rates

Average Daily Flow: \[Q_{avg} = \frac{\text{Total Volume}}{\text{Time Period}}\]

• Q_avg: Average daily flow (gpd or MGD)
• Typically expressed in gallons per day (gpd), million gallons per day (MGD), or cubic meters per day (m³/d)

Peak Flow Factors: \[Q_{peak} = Q_{avg} \times \text{Peaking Factor}\]

• Q_peak: Peak flow rate (gpd or MGD)
• Peaking Factor: Ratio of peak to average flow, typically 2.0-4.0 for municipal systems
• Peak hourly flow typically 1.5-3.0 times average daily flow
• Minimum flow typically 0.25-0.50 times average daily flow

Per Capita Flow: \[q = \frac{Q}{P}\]

• q: Per capita flow (gpcd - gallons per capita per day)
• Q: Total flow (gpd)
• P: Population served
• Typical domestic wastewater: 80-120 gpcd

Organic Loading

BOD Mass Loading: \[L_{BOD} = Q \times C \times 8.34\]

• L_BOD: BOD mass loading (lb/day)
• Q: Flow rate (MGD)
• C: BOD concentration (mg/L)
• 8.34: Conversion factor (lb-L/MG-mg)

Per Capita BOD Loading: \[BOD_{pc} = \frac{L_{BOD}}{P}\]

• BOD_pc: Per capita BOD loading (lb/capita/day)
• L_BOD: Total BOD loading (lb/day)
• P: Population
• Typical domestic value: 0.17-0.20 lb BOD/capita/day

Organic Loading Rate: \[OLR = \frac{Q \times S_0}{V}\]

• OLR: Organic loading rate (lb BOD/day/1000 ft³ or kg/m³/day)
• Q: Flow rate (MGD or m³/day)
• S₀: Influent BOD concentration (mg/L)
• V: Volume of reactor (ft³ or m³)

Suspended Solids Loading

TSS Mass Loading: \[L_{TSS} = Q \times TSS \times 8.34\]

• L_TSS: Total suspended solids mass loading (lb/day)
• Q: Flow rate (MGD)
• TSS: Total suspended solids concentration (mg/L)
• Typical domestic TSS: 0.20-0.24 lb/capita/day

Preliminary Treatment

Bar Screens and Screening

Head Loss Through Bar Screens: \[h_L = K \times \frac{v^2}{2g}\]

• h_L: Head loss through screen (ft or m)
• K: Head loss coefficient (typically 0.07-0.3 for clean screens)
• v: Approach velocity through openings (ft/s or m/s)
• g: Gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• Typical approach velocity: 1.5-3.0 ft/s (0.45-0.9 m/s)

Screenings Volume: \[V_s = Q \times q_s\]

• V_s: Volume of screenings (ft³/day or m³/day)
• Q: Flow rate (MGD or m³/day)
• q_s: Screenings per unit flow (ft³/MG or m³/1000 m³)
• Typical screenings: 0.5-3.5 ft³/MG (0.004-0.03 m³/1000 m³)

Grit Removal

Stokes' Law (Settling Velocity): \[v_s = \frac{g \times (ρ_p - ρ_w) \times d^2}{18μ}\]

• v_s: Settling velocity (ft/s or m/s)
• g: Gravitational acceleration (32.2 ft/s² or 9.81 m/s²)
• ρ_p: Particle density (lb/ft³ or kg/m³)
• ρ_w: Water density (lb/ft³ or kg/m³)
• d: Particle diameter (ft or m)
• μ: Dynamic viscosity (lb-s/ft² or N-s/m²)
• Applicable to laminar flow (Re <>

Horizontal Flow Grit Chamber Detention Time: \[t = \frac{L}{v_h} = \frac{H}{v_s}\]

• t: Detention time (s)
• L: Length of chamber (ft or m)
• v_h: Horizontal velocity (ft/s or m/s)
• H: Depth of chamber (ft or m)
• v_s: Settling velocity (ft/s or m/s)
• Typical horizontal velocity: 0.8-1.3 ft/s (0.25-0.4 m/s)
• Typical detention time: 45-90 seconds

Grit Chamber Volume: \[V = Q \times t\]

• V: Grit chamber volume (ft³ or m³)
• Q: Flow rate (ft³/s or m³/s)
• t: Detention time (s)

Grit Quantity: \[V_g = Q \times q_g\]

• V_g: Volume of grit (ft³/day or m³/day)
• Q: Flow rate (MGD or m³/day)
• q_g: Grit per unit flow (ft³/MG or m³/1000 m³)
• Typical grit quantity: 0.5-27 ft³/MG (0.004-0.2 m³/1000 m³)

Primary Sedimentation

Settling Tank Hydraulics

Detention Time: \[t = \frac{V}{Q}\]

• t: Detention time (hr or s)
• V: Tank volume (gal or ft³ or m³)
• Q: Flow rate (gph, gpd, or ft³/s, or m³/s)
• Typical primary clarifier detention time: 1.5-2.5 hours

Surface Overflow Rate (Surface Loading Rate): \[SOR = \frac{Q}{A_s}\]

• SOR: Surface overflow rate (gpd/ft² or m³/m²/day)
• Q: Flow rate (gpd or m³/day)
• A_s: Surface area (ft² or m²)
• Typical primary clarifier SOR: 600-1200 gpd/ft² (25-50 m/day)
• At peak flow: 1000-1500 gpd/ft² (40-60 m/day)

Weir Overflow Rate: \[WOR = \frac{Q}{L_w}\]

• WOR: Weir overflow rate (gpd/ft or m³/m/day)
• Q: Flow rate (gpd or m³/day)
• L_w: Weir length (ft or m)
• Typical WOR: < 20,000="" gpd/ft="">< 250="">

Solids Loading Rate: \[SLR = \frac{Q \times SS}{A_s}\]

• SLR: Solids loading rate (lb/day/ft² or kg/m²/day)
• Q: Flow rate (MGD or m³/day)
• SS: Suspended solids concentration (mg/L)
• A_s: Surface area (ft² or m²)
• Use 8.34 conversion factor when Q is in MGD and SS in mg/L

Primary Clarifier Performance

Removal Efficiency: \[E = \frac{C_{in} - C_{out}}{C_{in}} \times 100\%\]

• E: Removal efficiency (%)
• C_in: Influent concentration (mg/L)
• C_out: Effluent concentration (mg/L)
• Typical primary clarifier BOD removal: 25-40%
• Typical primary clarifier TSS removal: 50-65%

Sludge Production: \[Q_s = \frac{Q \times (SS_{in} - SS_{out}) \times 8.34}{ρ_s \times (SG_s) \times 62.4}\]

• Q_s: Sludge flow rate (gpd)
• Q: Influent flow rate (MGD)
• SS_in: Influent suspended solids (mg/L)
• SS_out: Effluent suspended solids (mg/L)
• ρ_s: Sludge solids concentration (%)
• SG_s: Specific gravity of sludge solids (typically 1.0-1.05)

Activated Sludge Process

Process Fundamentals

Hydraulic Retention Time (HRT): \[HRT = θ = \frac{V}{Q}\]

• HRT or θ: Hydraulic retention time (hr or days)
• V: Aeration tank volume (gal or ft³ or m³)
• Q: Influent flow rate (gpd, MGD, or ft³/day, or m³/day)
• Typical range for conventional activated sludge: 4-8 hours

Solids Retention Time (SRT) / Mean Cell Residence Time (MCRT): \[SRT = θ_c = \frac{V \times X}{Q_w \times X_r + Q_e \times X_e}\]

• SRT or θ_c: Solids retention time (days)
• V: Aeration tank volume (MG or m³)
• X: MLSS concentration in aeration tank (mg/L)
• Q_w: Waste activated sludge (WAS) flow rate (MGD or m³/day)
• X_r: WAS concentration (mg/L)
• Q_e: Effluent flow rate (MGD or m³/day)
• X_e: Effluent TSS concentration (mg/L)

Simplified SRT (neglecting effluent solids): \[θ_c = \frac{V \times X}{Q_w \times X_r}\]

• Used when effluent suspended solids are negligible compared to wasted sludge

Food-to-Microorganism Ratio (F/M): \[F/M = \frac{Q \times S_0}{V \times X}\]

• F/M: Food-to-microorganism ratio (lb BOD/day/lb MLSS or kg BOD/kg MLSS/day)
• Q: Influent flow rate (MGD or m³/day)
• S₀: Influent BOD concentration (mg/L)
• V: Aeration tank volume (MG or m³)
• X: MLSS concentration (mg/L)
• Use 8.34 conversion factor when Q in MGD and concentrations in mg/L
• Typical F/M for conventional activated sludge: 0.2-0.5 lb BOD/lb MLSS/day

Volumetric Organic Loading: \[L_v = \frac{Q \times S_0}{V}\]

• L_v: Volumetric organic loading (lb BOD/day/1000 ft³ or kg/m³/day)
• Q: Flow rate (MGD or m³/day)
• S₀: Influent BOD (mg/L)
• V: Aeration tank volume (1000 ft³ or m³)

Sludge Production and Wasting

Observed Yield: \[Y_{obs} = \frac{P_x}{Q \times (S_0 - S)}\]

• Y_obs: Observed yield (lb VSS/lb BOD or kg VSS/kg BOD)
• P_x: Net waste activated sludge produced (lb VSS/day or kg VSS/day)
• Q: Flow rate (MGD or m³/day)
• S₀: Influent BOD (mg/L)
• S: Effluent BOD (mg/L)

Relationship Between Observed Yield and SRT: \[Y_{obs} = \frac{Y}{1 + k_d \times θ_c}\]

• Y_obs: Observed yield (lb VSS/lb BOD)
• Y: True yield coefficient (lb VSS/lb BOD), typically 0.4-0.8
• k_d: Endogenous decay coefficient (day^-1), typically 0.025-0.075
• θ_c: Solids retention time (days)

Sludge Wasting Rate: \[Q_w = \frac{V \times X}{θ_c \times X_r}\]

• Q_w: Waste sludge flow rate (MGD or m³/day)
• V: Aeration tank volume (MG or m³)
• X: MLSS concentration (mg/L)
• θ_c: Desired SRT (days)
• X_r: Return or waste sludge concentration (mg/L)

Sludge Production (Mass Balance): \[P_x = Y_{obs} \times Q \times (S_0 - S) \times 8.34\]

• P_x: Sludge produced (lb VSS/day)
• Y_obs: Observed yield (lb VSS/lb BOD)
• Q: Flow rate (MGD)
• S₀: Influent BOD (mg/L)
• S: Effluent BOD (mg/L)

Return Activated Sludge (RAS)

Return Sludge Ratio: \[R = \frac{Q_r}{Q}\]

• R: Return sludge ratio (dimensionless)
• Q_r: Return activated sludge flow rate (MGD or m³/day)
• Q: Influent flow rate (MGD or m³/day)
• Typical range: 0.25-1.0 (25-100%)

Solids Balance for Return Sludge: \[Q_r = \frac{Q \times X}{X_r - X}\]

• Q_r: Return sludge flow rate (MGD or m³/day)
• Q: Influent flow rate (MGD or m³/day)
• X: MLSS concentration (mg/L)
• X_r: Return sludge concentration (mg/L)

Alternative Form: \[R = \frac{X}{X_r - X}\]

Oxygen Requirements

Oxygen Demand for BOD Removal and Endogenous Respiration: \[O_2 = Q \times (S_0 - S) \times 8.34 - 1.42 \times P_x\]

• O₂: Oxygen required (lb/day)
• Q: Flow rate (MGD)
• S₀: Influent BOD (mg/L)
• S: Effluent BOD (mg/L)
• P_x: Net waste activated sludge VSS produced (lb/day)
• 1.42: Factor for oxygen equivalence of cell tissue

Oxygen Requirement with Nitrification: \[O_2 = Q \times (S_0 - S) \times 8.34 - 1.42 \times P_x + 4.57 \times Q \times N_{ox} \times 8.34\]

• N_ox: Nitrogen oxidized (mg/L as N)
• 4.57: Oxygen required per unit nitrogen oxidized (lb O₂/lb N)

Actual Oxygen Transfer Required: \[AOR = \frac{O_2}{t}\]

• AOR: Actual oxygen transfer rate required (lb O₂/hr)
• O₂: Total oxygen required (lb/day)
• t: Time period (24 hr for daily)

Aeration

Oxygen Transfer Rate: \[OTR = SOTR \times \frac{C_{s,T,P} - C_L}{C_{s,20}} \times α \times β \times (1.024)^{T-20}\]

• OTR: Oxygen transfer rate in field conditions (lb O₂/hr)
• SOTR: Standard oxygen transfer rate at clean water conditions (lb O₂/hr)
• C_s,T,P: Oxygen saturation concentration at temperature T and pressure P (mg/L)
• C_L: Dissolved oxygen concentration maintained in basin (mg/L)
• C_s,20: Oxygen saturation at 20°C and 1 atm (mg/L)
• α: Oxygen transfer correction factor for wastewater (typically 0.4-0.8)
• β: Oxygen saturation correction factor for wastewater (typically 0.9-0.98)
• T: Operating temperature (°C)

Standard Aeration Efficiency (SAE): \[SAE = \frac{SOTR}{P}\]

• SAE: Standard aeration efficiency (lb O₂/hp-hr or kg O₂/kW-hr)
• SOTR: Standard oxygen transfer rate (lb O₂/hr or kg O₂/hr)
• P: Power input (hp or kW)

Oxygen Saturation Temperature Correction: \[C_{s,T} = C_{s,20} \times (1.024)^{20-T}\]

• C_s,T: Saturation concentration at temperature T (mg/L)
• C_s,20: Saturation concentration at 20°C (9.09 mg/L at sea level)
• T: Temperature (°C)

Secondary Clarification

Design Parameters

Surface Overflow Rate: \[SOR = \frac{Q + Q_r}{A_s}\]

• SOR: Surface overflow rate (gpd/ft² or m/day)
• Q: Influent flow (gpd or m³/day)
• Q_r: Return sludge flow (gpd or m³/day)
• A_s: Surface area (ft² or m²)
• Typical range: 400-800 gpd/ft² at average flow (16-32 m/day)
• Peak: 1000-1200 gpd/ft² (40-48 m/day)

Solids Loading Rate: \[SLR = \frac{(Q + Q_r) \times MLSS \times 8.34}{A_s}\]

• SLR: Solids loading rate (lb/day/ft² or kg/m²/day)
• Q: Influent flow (MGD or m³/day)
• Q_r: Return sludge flow (MGD or m³/day)
• MLSS: Mixed liquor suspended solids (mg/L)
• A_s: Surface area (ft² or m²)
• Typical range: 20-30 lb/day/ft² at average flow
• Peak: 40-50 lb/day/ft²

Sludge Blanket and Settling

Sludge Volume Index (SVI): \[SVI = \frac{SV_{30} \times 1000}{MLSS}\]

• SVI: Sludge volume index (mL/g)
• SV₃₀: Settled sludge volume after 30 minutes (mL/L)
• MLSS: Mixed liquor suspended solids (mg/L or g/L)
• Good settling sludge: SVI < 100="">
• Bulking sludge: SVI > 150 mL/g

Sludge Density Index (SDI): \[SDI = \frac{100}{SVI}\]

• SDI: Sludge density index (%)
• SVI: Sludge volume index (mL/g)

Trickling Filters

Design Parameters

Hydraulic Loading Rate: \[HLR = \frac{Q}{A}\]

• HLR: Hydraulic loading rate (gpd/ft² or m³/m²/day)
• Q: Flow rate including recirculation (gpd or m³/day)
• A: Surface area of filter (ft² or m²)
• Low-rate filters: 25-100 gpd/ft² (1-4 m/day)
• High-rate filters: 200-1000 gpd/ft² (8-40 m/day)

Organic Loading Rate: \[OLR = \frac{Q \times BOD \times 8.34}{V}\]

• OLR: Organic loading rate (lb BOD/day/1000 ft³ or kg/m³/day)
• Q: Flow rate (MGD or m³/day)
• BOD: Applied BOD concentration (mg/L)
• V: Filter volume (1000 ft³ or m³)
• Low-rate filters: 5-25 lb BOD/day/1000 ft³
• High-rate filters: 25-100 lb BOD/day/1000 ft³

Recirculation Ratio: \[r = \frac{Q_r}{Q}\]

• r: Recirculation ratio
• Q_r: Recirculation flow (gpd or m³/day)
• Q: Influent flow (gpd or m³/day)

Performance Models

NRC (National Research Council) Equation for Single-Stage Filters: \[E = \frac{100}{1 + 0.0561 \times \sqrt{\frac{W}{V \times F}}}\]

• E: BOD removal efficiency at 20°C (%)
• W: BOD loading (lb/day)
• V: Filter volume (1000 ft³)
• F: Recirculation factor

Recirculation Factor: \[F = \frac{1 + r}{(1 + 0.1r)^2}\]

• F: Recirculation factor
• r: Recirculation ratio

NRC Equation for Two-Stage Filters in Series: \[E = \frac{100}{1 + 0.0561 \times \sqrt{\frac{W}{V \times (1 + r)}}}\]

• For second stage, replace W with load to second stage

Velz/Germain Equation: \[\frac{S_e}{S_0} = e^{-k \times \frac{D}{Q^n}}\]

• S_e: Effluent BOD concentration (mg/L)
• S₀: Influent BOD concentration (mg/L)
• k: Treatability coefficient (varies with media and temperature)
• D: Depth of filter (ft or m)
• Q: Hydraulic loading rate (gpd/ft² or m/day)
• n: Constant (typically 0.5)

Rotating Biological Contactors (RBC)

Design Parameters

Hydraulic Loading Rate: \[HLR = \frac{Q}{A}\]

• HLR: Hydraulic loading rate (gpd/ft² or m³/m²/day)
• Q: Flow rate (gpd or m³/day)
• A: Total media surface area (ft² or m²)
• Typical: 1-3 gpd/ft² (0.04-0.12 m³/m²/day)

Organic Loading Rate: \[OLR = \frac{Q \times BOD \times 8.34}{A}\]

• OLR: Organic loading rate (lb BOD/day/1000 ft² or g/m²/day)
• Q: Flow rate (MGD or m³/day)
• BOD: Applied BOD (mg/L)
• A: Media surface area (1000 ft² or m²)
• Typical: 1.5-4.0 lb/day/1000 ft² for first stage

Stabilization Ponds

Design Equations

Detention Time: \[t = \frac{V}{Q}\]

• t: Detention time (days)
• V: Pond volume (ft³ or m³)
• Q: Flow rate (ft³/day or m³/day)
• Typical detention time: 30-180 days

Organic Loading Rate: \[L = \frac{Q \times BOD \times 8.34}{A}\]

• L: Organic loading (lb BOD/acre/day or kg/ha/day)
• Q: Flow rate (MGD or m³/day)
• BOD: Influent BOD (mg/L)
• A: Surface area (acres or ha)
• Typical aerobic pond: < 100="">
• Facultative pond: 15-40 lb/acre/day

BOD Removal (First-Order Kinetics): \[S_e = \frac{S_0}{1 + k \times t}\]

• S_e: Effluent BOD (mg/L)
• S₀: Influent BOD (mg/L)
• k: BOD removal rate constant (day^-1)
• t: Detention time (days)

Temperature Correction for Rate Constant: \[k_T = k_{20} \times θ^{(T-20)}\]

• k_T: Rate constant at temperature T (day^-1)
• k₂₀: Rate constant at 20°C (day^-1)
• θ: Temperature coefficient (typically 1.06-1.09)
• T: Operating temperature (°C)

Nutrient Removal

Nitrogen Removal

Nitrification Oxygen Requirement: \[O_2 = 4.57 \times N_{ox}\]

• O₂: Oxygen required (lb O₂/lb N or kg O₂/kg N)
• N_ox: Nitrogen oxidized (lb N or kg N)
• 4.57: Stoichiometric coefficient for complete nitrification

Nitrification with Mass Balance: \[O_2 = 4.57 \times Q \times (NH_3 - NH_{3,eff}) \times 8.34\]

• O₂: Daily oxygen required (lb/day)
• Q: Flow rate (MGD)
• NH₃: Influent ammonia-nitrogen (mg/L as N)
• NH_3,eff: Effluent ammonia-nitrogen (mg/L as N)

Alkalinity Consumed During Nitrification: \[Alk = 7.14 \times N_{ox}\]

• Alk: Alkalinity consumed (lb as CaCO₃/lb N or mg/L as CaCO₃)
• N_ox: Nitrogen oxidized (lb N or mg/L as N)
• 7.14: Stoichiometric coefficient

Denitrification Alkalinity Production: \[Alk = 3.57 \times N_{NO3}\]

• Alk: Alkalinity produced (lb as CaCO₃/lb N or mg/L as CaCO₃)
• N_NO3: Nitrate-nitrogen reduced (lb N or mg/L as N)
• 3.57: Stoichiometric coefficient

Denitrification Carbon Requirement: \[C_{req} = 2.86 \times N_{NO3}\]

• C_req: BOD required (lb BOD/lb NO₃-N or mg/L BOD/mg/L N)
• N_NO3: Nitrate-nitrogen to be reduced (lb N or mg/L as N)
• 2.86: Stoichiometric coefficient

Minimum SRT for Nitrification: \[θ_{c,min} = \frac{1}{μ_m \times (1 - f_d) - k_d}\]

• θ_c,min: Minimum SRT for nitrification (days)
• μ_m: Maximum specific growth rate of nitrifiers (day^-1)
• f_d: Fraction of biomass remaining as debris
• k_d: Endogenous decay rate (day^-1)

Temperature Effect on Nitrification Rate: \[μ_T = μ_{20} \times θ^{(T-20)}\]

• μ_T: Growth rate at temperature T (day^-1)
• μ₂₀: Growth rate at 20°C (day^-1)
• θ: Temperature coefficient (typically 1.07-1.10 for nitrifiers)
• T: Operating temperature (°C)

Phosphorus Removal

Chemical Phosphorus Precipitation - Aluminum Salts: \[Al^{3+} + PO_4^{3-} → AlPO_4↓\]

• Stoichiometric Al:P molar ratio = 1:1
• Typical dose: 1.5-2.5 moles Al per mole P for effective removal

Alum Dose for Phosphorus Removal: \[\text{Alum dose (mg/L)} = \frac{P \times MW_{alum}}{MW_P} \times \text{molar ratio}\]

• P: Phosphorus to be removed (mg/L)
• MW_alum: Molecular weight of alum (Al₂(SO₄)₃·14H₂O) = 594
• MW_P: Molecular weight of phosphorus = 31
• Practical alum dose: 9.6 mg alum per mg P (for 1:1 ratio)

Chemical Phosphorus Precipitation - Iron Salts: \[Fe^{3+} + PO_4^{3-} → FePO_4↓\]

• Stoichiometric Fe:P molar ratio = 1:1
• Typical dose: 1.5-3.0 moles Fe per mole P for effective removal

Ferric Chloride Dose: \[\text{FeCl}_3 \text{ dose (mg/L)} = \frac{P \times MW_{FeCl_3}}{MW_P} \times \text{molar ratio}\]

• MW_FeCl3: Molecular weight of ferric chloride = 162
• Practical FeCl₃ dose: 5.2 mg FeCl₃ per mg P (for 1:1 ratio)

Biological Phosphorus Removal - Luxury Uptake:

• Enhanced phosphorus removal through anaerobic/aerobic cycling
• Typical phosphorus content: 1.5-2.0% of biomass (normal)
• Luxury uptake: 5-7% of biomass

Disinfection

Chlorination

Chlorine Dose: \[D = C_r + C_d\]

• D: Chlorine dose (mg/L)
• C_r: Chlorine residual (mg/L)
• C_d: Chlorine demand (mg/L)

Chlorine Feed Rate: \[F = Q \times D \times 8.34\]

• F: Chlorine feed rate (lb/day)
• Q: Flow rate (MGD)
• D: Chlorine dose (mg/L)

Contact Time (CT Value): \[CT = C \times t\]

• CT: Contact time parameter (mg-min/L)
• C: Disinfectant residual concentration (mg/L)
• t: Contact time (min)
• CT values depend on organism, temperature, and pH

Contact Time at t₁₀: \[t_{10} = \frac{V}{Q} \times \text{baffling factor}\]

• t₁₀: Time for 10% of flow to pass through (min)
• V: Basin volume (gal or m³)
• Q: Flow rate (gpm or m³/min)
• Baffling factor typically 0.1-0.7 depending on basin configuration

UV Disinfection

UV Dose: \[D_{UV} = I \times t\]

• D_UV: UV dose (mW-s/cm² or mJ/cm²)
• I: UV intensity (mW/cm²)
• t: Exposure time (s)
• Typical dose for secondary effluent: 30-100 mJ/cm²

Sludge Processing

Sludge Quantities

Sludge Volume: \[V_s = \frac{M}{ρ \times P_s \times 62.4}\]

• V_s: Sludge volume (ft³ or gal)
• M: Mass of dry solids (lb)
• ρ: Specific gravity of sludge (typically 1.0-1.05)
• P_s: Solids content (decimal, e.g., 0.02 for 2%)
• 62.4: Density of water (lb/ft³)

Conversion for Sludge Volume: \[V_{gal} = \frac{M \times 120}{ρ \times P_s \times 100}\]

• V_gal: Sludge volume (gal)
• M: Mass of dry solids (lb)
• ρ: Specific gravity
• P_s: Percent solids (%)

Mass-Volume-Concentration Relationship: \[M = V \times C \times ρ \times 8.34\]

• M: Mass of solids (lb)
• V: Volume (MG)
• C: Concentration (mg/L or %)
• ρ: Specific gravity

Thickening

Gravity Thickener Solids Loading: \[SLR = \frac{Q \times C \times 8.34}{A}\]

• SLR: Solids loading rate (lb/day/ft² or kg/m²/day)
• Q: Flow rate (MGD or m³/day)
• C: Solids concentration (mg/L or %)
• A: Thickener surface area (ft² or m²)
• Typical loading for WAS: 4-8 lb/day/ft²
• Typical loading for primary sludge: 20-40 lb/day/ft²

Solids Mass Balance for Thickening: \[Q_{in} \times C_{in} = Q_{out} \times C_{out}\]

• Q_in: Influent flow (gpd or m³/day)
• C_in: Influent solids concentration (%)
• Q_out: Effluent (thickened) flow (gpd or m³/day)
• C_out: Effluent solids concentration (%)

Anaerobic Digestion

Volatile Solids Loading Rate: \[L_{VS} = \frac{Q \times VS}{V}\]

• L_VS: Volatile solids loading (lb VS/day/ft³ or kg VS/m³/day)
• Q: Sludge flow rate (MGD or m³/day)
• VS: Volatile solids concentration (mg/L)
• V: Digester volume (ft³ or m³)
• Typical range: 0.03-0.20 lb VS/day/ft³ (0.5-3.2 kg/m³/day)

Digester Detention Time: \[t = \frac{V}{Q}\]

• t: Detention time (days)
• V: Digester volume (gal, ft³, or m³)
• Q: Sludge feed rate (gpd, ft³/day, or m³/day)
• Typical range for mesophilic: 15-20 days
• Typical range for thermophilic: 10-15 days

Volatile Solids Reduction: \[VSR = \frac{VS_{in} - VS_{out}}{VS_{in}} \times 100\%\]

• VSR: Volatile solids reduction (%)
• VS_in: Influent volatile solids (lb/day or kg/day)
• VS_out: Effluent volatile solids (lb/day or kg/day)
• Typical VSR: 40-60%

Gas Production: \[G = VS_{destroyed} \times Y_g\]

• G: Gas production (ft³/day or m³/day)
• VS_destroyed: Volatile solids destroyed (lb/day or kg/day)
• Y_g: Gas yield (ft³/lb VS or m³/kg VS)
• Typical yield: 12-18 ft³/lb VS destroyed (0.75-1.12 m³/kg VS)
• Methane content: 60-70% by volume

Heat Required for Digester: \[Q_h = m \times C_p \times ΔT + Q_{loss}\]

• Q_h: Heat required (Btu/hr or kJ/hr)
• m: Mass flow rate of sludge (lb/hr or kg/hr)
• C_p: Specific heat of sludge (Btu/lb-°F or kJ/kg-°C), approximately 1.0
• ΔT: Temperature rise required (°F or °C)
• Q_loss: Heat loss through walls, roof, floor (Btu/hr or kJ/hr)

Aerobic Digestion

Digestion Time: \[t = \frac{V}{Q}\]

• t: Digestion time (days)
• V: Digester volume (gal or m³)
• Q: Sludge feed rate (gpd or m³/day)
• Typical range: 15-25 days at 20°C

Oxygen Requirements for Aerobic Digestion: \[O_2 = 2.3 \times VS_{destroyed}\]

• O₂: Oxygen required (lb O₂/lb VS or kg O₂/kg VS)
• VS_destroyed: Volatile solids destroyed (lb or kg)
• 2.3: Stoichiometric coefficient

Dewatering

Solids Recovery: \[R = \frac{C_{cake} \times (C_{feed} - C_{filtrate})}{C_{feed} \times (C_{cake} - C_{filtrate})} \times 100\%\]

• R: Solids recovery (%)
• C_cake: Cake solids concentration (%)
• C_feed: Feed solids concentration (%)
• C_filtrate: Filtrate solids concentration (%)

Belt Filter Press Loading: \[L = \frac{Q \times C \times 8.34}{W}\]

• L: Hydraulic loading (lb dry solids/hr/ft belt width)
• Q: Feed flow rate (MGD)
• C: Feed solids concentration (mg/L or %)
• W: Belt width (ft)
• Typical loading: 400-800 lb/hr/ft

Centrifuge Loading: \[L = \frac{Q \times C \times 8.34 \times 60}{V_b}\]

• L: Hydraulic loading (lb dry solids/hr/gal bowl volume)
• Q: Feed flow rate (gpm)
• C: Feed solids concentration (%)
• V_b: Bowl volume (gal)

Polymer Dose: \[D_p = \frac{P \times Q \times 8.34}{C_p}\]

• D_p: Polymer dose (lb/day)
• P: Polymer concentration in feed (mg/L or ppm)
• Q: Sludge flow rate (MGD)
• C_p: Polymer solution concentration (decimal)

Effluent Disposal and Reuse

Land Application

Hydraulic Loading Rate: \[HLR = \frac{Q}{A}\]

• HLR: Hydraulic loading rate (in/week, gpd/acre, or m/day)
• Q: Flow rate (gpd or m³/day)
• A: Land application area (acres or m²)

Nitrogen Loading Rate: \[NLR = \frac{Q \times N \times 8.34}{A}\]

• NLR: Nitrogen loading rate (lb N/acre/yr or kg N/ha/yr)
• Q: Annual flow (MG/yr or m³/yr)
• N: Nitrogen concentration (mg/L)
• A: Application area (acres or ha)

Constructed Wetlands

BOD Removal Rate: \[k_a = -\frac{\ln(C_e/C_0)}{t}\]

• k_a: Areal removal rate constant (day^-1)
• C_e: Effluent BOD concentration (mg/L)
• C₀: Influent BOD concentration (mg/L)
• t: Hydraulic residence time (days)

Wetland Area Required: \[A = \frac{Q \times \ln(C_0/C_e)}{k_T \times d \times n}\]

• A: Wetland area (m² or acres)
• Q: Flow rate (m³/day or acre-ft/day)
• C₀: Influent concentration (mg/L)
• C_e: Effluent concentration (mg/L)
• k_T: Temperature-dependent rate constant (day^-1)
• d: Depth of wetland (m or ft)
• n: Porosity (typically 0.65-0.75)

Pumping and Collection Systems

Pump Sizing

Pump Power: \[P = \frac{Q \times H \times ρ}{33000 \times η}\]

• P: Pump power (hp)
• Q: Flow rate (gpm)
• H: Total dynamic head (ft)
• ρ: Specific weight of fluid (lb/ft³), water ≈ 62.4
• η: Pump efficiency (decimal)
• 33000: Conversion factor (ft-lb/min/hp)

Simplified Pump Power (Water): \[P = \frac{Q \times H}{3960 \times η}\]

• P: Pump power (hp)
• Q: Flow rate (gpm)
• H: Total dynamic head (ft)
• η: Pump efficiency (decimal)

Wire-to-Water Efficiency: \[η_{w-w} = η_{pump} \times η_{motor}\]

• η_w-w: Wire-to-water efficiency
• η_pump: Pump efficiency
• η_motor: Motor efficiency

Gravity Flow in Sewers

Manning's Equation for Open Channel Flow: \[Q = \frac{1.486}{n} \times A \times R^{2/3} \times S^{1/2}\]

• Q: Flow rate (ft³/s)
• n: Manning's roughness coefficient
• A: Cross-sectional area of flow (ft²)
• R: Hydraulic radius (ft) = A/P
• P: Wetted perimeter (ft)
• S: Slope of energy grade line (ft/ft)
• 1.486: Unit conversion constant (US Customary)
• For SI units, replace 1.486 with 1.0

Velocity in Pipe: \[v = \frac{Q}{A}\]

• v: Velocity (ft/s or m/s)
• Q: Flow rate (ft³/s or m³/s)
• A: Cross-sectional area (ft² or m²)
• Minimum self-cleansing velocity: 2.0 ft/s (0.6 m/s)

Hydraulic Radius for Circular Pipe: \[R = \frac{d}{4}\]

• R: Hydraulic radius for full pipe flow (ft or m)
• d: Pipe diameter (ft or m)

Hydraulic Radius for Partially Full Circular Pipe: \[R = \frac{A}{P}\]

• A: Area of flow
• P: Wetted perimeter
• Requires geometric calculations based on depth-to-diameter ratio

Wet Well Sizing

Pump Cycle Time: \[t_{cycle} = \frac{V}{Q_{in} - Q_{pump}}\]

• t_cycle: Pump cycle time (min)
• V: Working volume between on/off levels (gal or ft³)
• Q_in: Inflow rate (gpm or ft³/min)
• Q_pump: Pump discharge rate (gpm or ft³/min)

Wet Well Volume: \[V = \frac{Q_{pump} \times t_{cycle}}{4}\]

• V: Working volume (gal)
• Q_pump: Pump capacity (gpm)
• t_cycle: Desired cycle time (min)
• Typical minimum cycle time: 10 minutes

Process Control and Monitoring

Mass Balance

General Mass Balance: \[Accumulation = Input - Output + Generation - Consumption\]

• For steady-state: Accumulation = 0
• For conservative substances: Generation = Consumption = 0

Reactor Mass Balance (Steady State): \[Q \times C_{in} = Q \times C_{out} + V \times r\]

• Q: Flow rate (MGD or m³/day)
• C_in: Influent concentration (mg/L)
• C_out: Effluent concentration (mg/L)
• V: Reactor volume (MG or m³)
• r: Reaction rate (mg/L/day)

Quality Control

Percent Removal: \[R = \frac{C_{in} - C_{out}}{C_{in}} \times 100\%\]

• R: Percent removal (%)
• C_in: Influent concentration
• C_out: Effluent concentration

Unit Process Efficiency: \[E = \frac{Load_{in} - Load_{out}}{Load_{in}} \times 100\%\]

• E: Efficiency (%)
• Load_in: Mass loading into unit (lb/day or kg/day)
• Load_out: Mass loading out of unit (lb/day or kg/day)

Kinetics and Biological Processes

Monod Kinetics

Monod Equation: \[μ = \frac{μ_{max} \times S}{K_s + S}\]

• μ: Specific growth rate (day^-1)
• μ_max: Maximum specific growth rate (day^-1)
• S: Substrate concentration (mg/L)
• K_s: Half-saturation constant (mg/L)

Substrate Utilization Rate: \[r_{su} = -\frac{k \times X \times S}{K_s + S}\]

• r_su: Substrate utilization rate (mg/L/day)
• k: Maximum substrate utilization rate (mg substrate/mg biomass/day)
• X: Biomass concentration (mg/L)
• S: Substrate concentration (mg/L)
• K_s: Half-saturation constant (mg/L)

Effluent Substrate Concentration: \[S = \frac{K_s \times (1 + k_d \times θ_c)}{θ_c \times (μ_{max} - k_d) - 1}\]

• S: Effluent substrate concentration (mg/L)
• K_s: Half-saturation constant (mg/L)
• k_d: Endogenous decay coefficient (day^-1)
• θ_c: Solids retention time (days)
• μ_max: Maximum specific growth rate (day^-1)

BOD Kinetics

First-Order BOD Removal: \[L_t = L_0 \times e^{-kt}\]

• L_t: BOD remaining at time t (mg/L)
• L₀: Initial BOD (mg/L)
• k: BOD decay rate constant (day^-1, base e)
• t: Time (days)

BOD Exerted: \[BOD_t = L_0 \times (1 - e^{-kt})\]

• BOD_t: BOD exerted at time t (mg/L)
• L₀: Ultimate BOD (mg/L)
• k: BOD decay rate constant (day^-1)
• t: Time (days)

Relationship Between Rate Constants: \[k_{base 10} = 2.303 \times k_{base e}\]

• k_{base 10}: Rate constant to base 10
• k_{base e}: Rate constant to base e

Ultimate BOD: \[BOD_u = \frac{BOD_t}{1 - e^{-kt}}\]

• BOD_u: Ultimate BOD (mg/L)
• BOD_t: BOD at time t (mg/L)
• k: Rate constant (day^-1)
• t: Time (days)

Temperature Correction for k: \[k_T = k_{20} \times θ^{(T-20)}\]

• k_T: Rate constant at temperature T (day^-1)
• k₂₀: Rate constant at 20°C (day^-1)
• θ: Temperature coefficient (typically 1.047 for BOD)
• T: Temperature (°C)

Advanced Treatment Processes

Filtration

Filtration Rate: \[FR = \frac{Q}{A}\]

• FR: Filtration rate (gpm/ft² or m³/m²/hr)
• Q: Flow rate (gpm or m³/hr)
• A: Filter surface area (ft² or m²)
• Typical range: 2-5 gpm/ft² (5-12 m/hr)

Head Loss (Carman-Kozeny): \[h_L = \frac{k \times μ \times v \times L \times (1-ε)^2}{ρ \times g \times d^2 \times ε^3}\]

• h_L: Head loss (ft or m)
• k: Kozeny constant (dimensionless)
• μ: Dynamic viscosity (lb-s/ft² or N-s/m²)
• v: Approach velocity (ft/s or m/s)
• L: Depth of filter bed (ft or m)
• ε: Porosity (decimal)
• ρ: Fluid density (lb/ft³ or kg/m³)
• g: Gravitational acceleration (ft/s² or m/s²)
• d: Particle diameter (ft or m)

Backwash Rate: \[Q_{bw} = A \times v_{bw}\]

• Q_bw: Backwash flow rate (gpm or m³/hr)
• A: Filter surface area (ft² or m²)
• v_bw: Backwash velocity (gpm/ft² or m/hr)
• Typical backwash rate: 15-25 gpm/ft² (37-61 m/hr)

Adsorption (Activated Carbon)

Freundlich Isotherm: \[q_e = K \times C_e^{1/n}\]

• q_e: Mass of adsorbate per unit mass of adsorbent at equilibrium (mg/g)
• K: Freundlich capacity factor
• C_e: Equilibrium concentration of adsorbate (mg/L)
• n: Freundlich intensity parameter

Langmuir Isotherm: \[q_e = \frac{q_m \times K_L \times C_e}{1 + K_L \times C_e}\]

• q_e: Amount adsorbed at equilibrium (mg/g)
• q_m: Maximum adsorption capacity (mg/g)
• K_L: Langmuir constant (L/mg)
• C_e: Equilibrium concentration (mg/L)

Empty Bed Contact Time (EBCT): \[EBCT = \frac{V}{Q}\]

• EBCT: Empty bed contact time (min)
• V: Volume of carbon bed (ft³ or m³)
• Q: Flow rate (ft³/min or m³/min)
• Typical range: 7.5-30 minutes

Membrane Processes

Flux Rate: \[J = \frac{Q}{A}\]

• J: Flux rate (gfd - gallons per square foot per day, or L/m²/hr)
• Q: Permeate flow rate (gpd or L/hr)
• A: Membrane surface area (ft² or m²)

Recovery Ratio: \[R = \frac{Q_p}{Q_f} \times 100\%\]

• R: Recovery ratio (%)
• Q_p: Permeate flow rate (gpd or m³/day)
• Q_f: Feed flow rate (gpd or m³/day)

Rejection: \[Rej = \frac{C_f - C_p}{C_f} \times 100\%\]

• Rej: Rejection (%)
• C_f: Feed concentration (mg/L)
• C_p: Permeate concentration (mg/L)

Concentration Factor: \[CF = \frac{C_c}{C_f}\]

• CF: Concentration factor
• C_c: Concentrate concentration (mg/L)
• C_f: Feed concentration (mg/L)