Environmental Engineering — Formula Sheet

⭐ BOD exerted at time t:

y_t = L₀(1 – e^–k_dt)

L_t = L₀ e^–k_dt (BOD remaining)

y_t + L_t = L₀    (always)

Base-10 form: y_t = L₀(1 – 10^–K₁t); k_d = 2.303 K₁

⭐ BOD temperature correction:

k_d,T = k_d,20 × θ^(T–20)    θ = 1.047

At 25°C: k_d,25 = 1.261 × k_d,20

At 15°C: k_d,15 = 0.794 × k_d,20

Dissolved oxygen:

DO deficit D = DO_sat – DO_actual

DO_sat at 20°C = 9.1 mg/L; at 25°C = 8.26 mg/L; at 30°C = 7.54 mg/L

⭐ Streeter-Phelps DO sag curve:

D_t = [k_dL₀/(k_r–k_d)](e^–k_dt – e^–k_rt) + D₀ e^–k_rt

Critical time: t_c = [1/(k_r–k_d)] ln[(k_r/k_d)(1 – D₀(k_r–k_d)/(k_dL₀))]

Critical deficit: D_c = (k_d/k_r) × L₀ × e^–k_dt_c

Minimum DO = DO_sat – D_c

Mixing equations:

DO_mixed = (Q_r×DO_r + Q_s×DO_s)/(Q_r+Q_s)

L₀_mixed = (Q_r×BOD_r + Q_s×BOD_s)/(Q_r+Q_s)

Hardness as CaCO₃:

Hardness (mg/L CaCO₃) = mg/L ion × (50 / equivalent weight of ion)

Ca²⁺: eq wt = 20; factor = 2.5;   Mg²⁺: eq wt = 12; factor = 4.17

⭐ Sedimentation (CPHEEO):

Overflow rate = Q/A    [m³/m²/day]

Detention time t = V/Q

Particle removed completely if v_s ≥ overflow rate

Partial removal (v_s < overflow rate) = v_s/overflow rate

Water treatment: overflow rate 12,000–24,000 L/m²/day; detention 2–4 hours

⭐ Stokes’ law (settling velocity):

v_s = (ρ_s–ρ_w) g d² / (18μ)

Valid for Re_p = ρ_wv_sd/μ < 1 (Stokes regime)

Filtration rates:

Slow sand filter (SSF): 0.1–0.4 m/h

Rapid sand filter (RSF): 4–6 m/h = 120–150 m³/m²/day

⭐ Chlorination:

Chlorine demand = Cl₂ applied – Residual chlorine

IS 10500: Residual free chlorine = 0.2–1.0 mg/L at consumer tap

Breakpoint: Cl₂/NH₃-N ≈ 10:1 (by mass); beyond breakpoint → free residual

Softening reactions:

Ca(OH)₂ + Ca(HCO₃)₂ → 2CaCO₃↓ + 2H₂O (temporary Ca hardness)

CaSO₄ + Na₂CO₃ → CaCO₃↓ + Na₂SO₄ (permanent Ca hardness)

Lime required: Ca(OH)₂/Ca(HCO₃)₂ = 74/162 (molar ratio 1:1)

⭐ Population forecasting:

Arithmetic: P_n = P₀ + n × ΔP_avg    (n = decades)

Geometric: P_n = P₀ × (1 + r/100)ⁿ    (r = % growth per decade)

Incremental: P_n = P₀ + n×ΔP_avg + n(n+1)/2 × Δ²P_avg

⭐ Per capita demand (IS 1172):

Domestic: 135 LPCD (Class I cities > 1 million)

Rural (Jal Jeevan Mission norm): 55 LPCD FHTC

Sewage = 80% of water supply

⭐ Peak demand factors:

Maximum daily demand (MDD) = 1.8 × ADD

Maximum hourly demand (MHD) = 2.7 × ADD/24

Design: Treatment plant → MDD; Distribution mains → MHD

Fire demand (Kuichling):

Q_f = 3182√P    (L/min; P in thousands)

Freeman: Q_f = 1136.5(P/5 + 10)    (L/min; P in thousands)

Balancing storage ≈ 1/3 to 2/3 of daily demand (depends on pumping hours)

Sewage quantity:

Sewage = 80% × water supply    (Indian practice)

DWF = 100–110 LPCD (for 135 LPCD water supply)

Harmon’s peak factor: P_f = 1 + 14/(4 + √P)    (P in thousands)

Peak sewage = P_f × DWF

⭐ Manning’s equation for sewer flow:

V = (1/n) R^2/3 S^1/2; Q = A × V

Circular sewer full: R = D/4; A = πD²/4

Half-full: R_half = R_full = D/4; V_half = V_full

n = 0.013 (concrete/vitrified clay)

Self-cleansing velocity ≥ 0.6–0.9 m/s

Rational formula (storm runoff):

Q = C × i × A / 360    (Q in m³/s; i in mm/hr; A in ha)

⭐ Treatment efficiencies:

Primary: BOD 40–60%; SS 50–70%

Secondary (ASP): BOD 85–95% total from raw sewage

Overall: E_total = 1 – (1–E₁)(1–E₂)

BOD removal efficiency: E = (BOD_in–BOD_out)/BOD_in

⭐ Primary sedimentation (CPHEEO):

Overflow rate: 24,000–32,000 L/m²/day

Detention time: 1.5–2.5 hours

Secondary clarifier (ASP): 16,000–24,000 L/m²/day; 2–4 hours

Oxidation pond:

Complete mix: Ce = Ci/(1 + k_rt)

Plug flow: Ce = Ci × e^–k_rt

BOD surface loading: 100–350 kg/ha/day (tropical India)

HRT: 10–30 days (facultative pond)

Sludge digestion:

Biogas: 0.03–0.05 m³/capita/day; 65–70% CH₄

Digestion period: 25–30 days (mesophilic, 30–38°C)

Digester capacity: 70–100 L/capita

Sludge drying bed: 0.1–0.2 m² per capita

⭐ F/M ratio (Food-to-Microorganism):

F/M = Q × S₀ / (V × X) = S₀ / (X × θ)    [kg BOD/kg MLSS/day]

Conventional ASP: F/M = 0.2–0.5; MLSS = 1500–3000 mg/L; HRT = 4–8 hr

Extended aeration: F/M = 0.05–0.15; MLSS = 3000–6000 mg/L; HRT = 18–36 hr

⭐ Sludge Volume Index (SVI):

SVI = settled volume (mL/L) × 1000 / MLSS (mg/L)    [mL/g]

Good settling: SVI = 50–150 mL/g; Bulking: SVI > 150 mL/g

RAS concentration: X_r ≈ 10⁶/SVI (mg/L)

Return ratio: R = X/(X_r–X)

⭐ Sludge age (SRT):

θ_c = V × X / (Q_w × X_w + Q_e × X_e)    [days]

Conventional ASP: θ_c = 5–15 days

Nitrification requires θ_c > 10 days (at 20°C)

Oxygen requirement:

O₂ = a’ × BOD_removed + b’ × MLVSS × V

a’ ≈ 0.5–0.65 kg O₂/kg BOD; b’ ≈ 0.05–0.10 kg O₂/kg MLVSS/day

⭐ NRC formula (trickling filter):

E = 1/[1 + 0.4432√(W/(VF))]    (single stage)

F = (1+R)/(1+R/10)²; W = BOD applied (kg/day); V = media volume (m³)

Two-stage: E₂ = 1/[1 + {0.4432/(1–E₁)} × √(W₂/(V₂F₂))]

Overall: E_total = 1 – (1–E₁)(1–E₂)

Generation rates:

Class I cities (metro): 0.45–0.60 kg/capita/day

Small towns: 0.20–0.35 kg/capita/day

India total: ~160,000 TPD

⭐ Compaction ratio and volume:

Compaction ratio (CR) = ρ_compacted/ρ_loose

Volume reduction % = (1 – 1/CR) × 100%

Daily volume = daily waste / compacted density

Daily total landfill volume = waste volume × (1 + cover fraction)

Landfill life = total volume / daily total volume

Composting C:N ratio:

Optimal C:N = 25–35:1

Blend C:N = Total C / Total N (weighted; not simple average)

⭐ Landfill gas (first-order model):

Q(t) = M × k × L₀ × e^–kt

L₀ = ultimate CH₄ potential (m³/tonne); k = decay rate (0.02–0.10 yr⁻¹)

LFG: 50–60% CH₄; 40–50% CO₂

⭐ Leachate (simplified):

Q_leachate ≈ infiltration fraction × precipitation × area

BOD of leachate: 1,000–10,000 mg/L (young landfill)

⭐ Gaussian plume — ground-level centreline concentration:

C(x,0,0) = [Q/(π σ_y σ_z u)] × exp[–H²/(2σ_z²)]

Full Gaussian (any y, z):

C(x,y,z) = [Q/(2π σ_y σ_z u)] × exp[–y²/(2σ_y²)] × {exp[–(z–H)²/(2σ_z²)] + exp[–(z+H)²/(2σ_z²)]}

C_max ∝ Q; C_max ∝ 1/u; C_max ∝ 1/H²

⭐ Effective stack height:

H = h + Δh (physical height + plume rise)

Holland’s plume rise: Δh = (v_sd/u)[1.5 + 2.68×10⁻³ P(T_s–T_a)/T_s × d]

CPCB minimum stack height: H = 14Q^0.3 (H in m; Q = SO₂ in kg/hr)

⭐ Particulate control efficiency:

η = (C_in–C_out)/C_in; P = 1–η (penetration)

Series devices: P_total = P₁ × P₂ × …; η_total = 1–P_total

ESP: 95–99.5%; Baghouse: 99–99.9%; Cyclone: 50–90%

FGD (wet limestone):

CaCO₃ + SO₂ + ½O₂ + 2H₂O → CaSO₄·2H₂O (gypsum) + CO₂

SO₂ removal: 90–98%