The PE Environmental exam covers a wide range of wastewater treatment calculations. This comprehensive guide lists the most important equations you'll need, organized by treatment process. Each equation links to our interactive calculator with Excel formula export.
All equations in one place - Interactive calculators with Excel export
Browse Environmental Equations1. Fundamental Wastewater Equations
Detention Time (Hydraulic Retention Time)
t = V / Q
Where: t = detention time (hours), V = tank volume (gal or L), Q = flow rate (gal/hr or L/hr)
Mass Loading (lbs/day)
Mass = 8.34 × Q (MGD) × C (mg/L)
Where: 8.34 = conversion factor for water, Q = flow in million gallons/day, C = concentration in mg/L
PE Exam Tip: Memorize 8.34 lbs/gallon - it's the weight of water and appears constantly in wastewater calculations. Mass (lbs/day) = 8.34 × MGD × mg/L
2. BOD and Kinetics
BOD Exertion (First-Order Kinetics)
BODt = BODu × (1 - e-kt)
Where: BODt = BOD exerted at time t, BODu = ultimate BOD, k = reaction rate constant (day-1), t = time (days)
First-Order Decay in CSTR
Ce = Co / (1 + k × t)
Where: Ce = effluent concentration, Co = influent concentration, k = rate constant, t = detention time
First-Order Decay in PFR
Ce = Co × e-kt
Plug flow reactor - more efficient than CSTR for same volume
3. Clarifier Design
Surface Overflow Rate (SOR)
SOR = Q / A
Where: SOR = gpd/ft² or m³/m²/day, Q = flow rate, A = surface area. Typical: 600-1200 gpd/ft² for secondary clarifiers
Weir Loading Rate
WLR = Q / Lw
Where: WLR = gpd/ft or m³/m/day, Lw = weir length. Typical: 10,000-20,000 gpd/ft
Hydraulic Loading Rate (HLR)
HLR = Q / A
Same as SOR but used for filters. Typical: 2-4 gpm/ft² for gravity filters
4. Activated Sludge
Solids Retention Time (SRT) / Mean Cell Residence Time
SRT = (V × X) / (Qw × Xw + Qe × Xe)
Where: V = aeration volume, X = MLSS, Qw = waste flow, Xw = waste concentration, Qe = effluent flow, Xe = effluent TSS
Food to Microorganism Ratio (F/M)
F/M = (Q × BODo) / (V × MLVSS)
Where: BODo = influent BOD, MLVSS = mixed liquor volatile suspended solids. Typical: 0.2-0.5 for conventional
| Process | F/M (day-1) | SRT (days) | MLSS (mg/L) |
|---|---|---|---|
| Conventional | 0.2-0.5 | 5-15 | 1500-3500 |
| Extended Aeration | 0.05-0.15 | 15-30 | 3000-6000 |
| High Rate | 0.5-1.0 | 2-4 | 1000-2000 |
| Contact Stabilization | 0.2-0.6 | 5-10 | 1000-3000 |
MLSS from MLVSS
MLSS = MLVSS / 0.7 to 0.85
MLVSS is typically 70-85% of MLSS. Use 0.80 if not specified.
5. Disinfection
CT Value (Disinfection)
CT = C × t
Where: C = disinfectant residual (mg/L), t = contact time (minutes). Required CT depends on pathogen and log removal.
Chick-Watson Law
ln(N/No) = -k' × Cn × t
Where: N = surviving organisms, No = initial organisms, k' = rate constant, n = coefficient of dilution
Common Exam Trap: CT calculations often require baffling factors. Actual contact time = theoretical HRT × baffling factor. Unbaffled tanks use 0.1-0.3, well-baffled use 0.5-0.7.
6. Sludge Processing
Sludge Volume Index (SVI)
SVI = (Settled Volume in mL/L × 1000) / MLSS (mg/L)
Good settling: SVI < 100 mL/g. Bulking sludge: SVI > 150 mL/g
Solids Loading Rate
SLR = (Q × MLSS) / A
Where: SLR = lbs/ft²/day, Q = flow + RAS, A = clarifier area. Typical: 20-30 lbs/ft²/day
Quick Reference: Key Numbers to Memorize
| Parameter | Typical Value | Units |
|---|---|---|
| Mass conversion | 8.34 | lbs/gal × mg/L × MGD = lbs/day |
| BOD5/BODu ratio | 0.68 | dimensionless |
| k for BOD (20°C) | 0.1-0.3 | day-1 |
| MLVSS/MLSS ratio | 0.70-0.85 | dimensionless |
| Secondary clarifier SOR | 600-1200 | gpd/ft² |
| Conventional AS F/M | 0.2-0.5 | day-1 |
Practice with interactive calculators - Excel formula export for all equations
Browse All Environmental Equations