Streeter-Phelps & Water-Quality Modeling
Key Takeaways
- Streeter-Phelps models dissolved oxygen sag downstream of an organic discharge with deoxygenation and reaeration.
- Ultimate BOD L0 is the total oxygen demand after infinite time; k_d is the deoxygenation rate constant.
- Reaeration rate k_a depends on velocity, depth, and turbulence — faster in riffles than pools.
- Critical deficit occurs at distance Dc where DO is minimum; must meet fisheries standards.
- Mixing at outfalls dilutes pollutant concentrations before biochemical reactions proceed downstream.
Quick Answer: Organic discharge → microbes consume DO (k_d) while atmosphere replenishes it (k_a). The Streeter-Phelps equation predicts the DO sag — find minimum DO and distance to critical point.
Water quality modeling links wastewater treatment performance to receiving stream health. Streeter-Phelps is the classic river DO model on the FE exam.
BOD Kinetics Review
First-order BOD exertion:
[ \frac{dL}{dt} = -k_d L \quad \Rightarrow \quad BOD_t = L_0 (1 - e^{-k_d t}) ]
| Term | Meaning |
|---|---|
| L₀ | Ultimate BOD (mg/L) |
| k_d | Base e deoxygenation rate (day⁻¹ typical) |
| BOD₅ | BOD at 5 days — standard lab test |
Temperature correction: (k_{d,T} = k_{d,20} , \theta^{(T-20)}) with θ ≈ 1.047 (use problem value).
Dissolved Oxygen Balance
Deoxygenation consumes DO proportional to remaining BOD. Reaeration transfers O₂ across air-water interface when DO < saturation C_s.
DO deficit D = C_s − DO.
Streeter-Phelps Equation (Conceptual Form)
Handbook provides the full equation for DO or deficit as function of travel time t (or distance x if velocity u known):
[ D = \frac{k_d L_0}{k_a - k_d}\left(e^{-k_d t} - e^{-k_a t}\right) + D_0 e^{-k_a t} ]
Requires k_a ≠ k_d. Initial deficit D₀ from mixing of effluent and river.
Mixing at the Outfall
[ C_m = \frac{Q_w C_w + Q_r C_r}{Q_w + Q_r} ]
Same for BOD and DO after temperature adjustment if needed. Complete mixing assumed at outfall cross-section.
Worked example: River Q_r = 50 m³/s, DO_r = 8.0 mg/L; effluent Q_w = 2 m³/s, DO_w = 2.0 mg/L.
[ DO_m = \frac{50 \times 8 + 2 \times 2}{52} = \frac{404}{52} = 7.77 \text{ mg/L} ]
If effluent BOD is high, deficit rises even if DO mixing looks acceptable.
Critical Point
Minimum DO (maximum deficit) occurs where:
[ \frac{dD}{dt} = 0 ]
Time to critical point t_c solves a transcendental equation — Handbook may give simplified form or ask qualitative which factor lowers minimum DO.
Factors lowering minimum DO:
- Higher L₀ (more BOD)
- Lower k_a (slow reaeration — sluggish deep pools)
- Higher k_d (faster exertion)
- Low initial DO or high temperature (lower C_s)
Reaeration Coefficient k_a
Empirical formulas relate k_a to depth, velocity, wind — exam supplies k_a or asks trend. Riffles and dams increase k_a.
Standards and Wasteload Allocation
State water quality standards set minimum DO (e.g., 5 mg/L for cold water fisheries). Wasteload allocation limits effluent BOD/CBOD so modeled sag complies.
Extensions (Conceptual)
- Nitrogenous BOD — nitrification adds delayed oxygen demand.
- Photosynthesis — daytime O₂ production in shallow eutrophic streams.
- Salinity — affects C_s.
FE usually tests classic carbonaceous BOD sag.
Modeling Steps on Exam
- Mix effluent and river → L₀ mix, DO_mix, D₀.
- Get k_d at river temperature; use given k_a.
- Apply Streeter-Phelps at distance or time of interest.
- Compare DO to standard.
Exam trap: Using BOD₅ as L₀ without conversion — L₀ exceeds BOD₅ unless problem states they are equal.
Exam trap: Forgetting saturation DO decreases with temperature — warmer river lowers C_s.
Streeter-Phelps connects treatment design to permit limits — know the sag shape and what deepens the trough.
Carbonaceous vs. Nitrogenous BOD
CBOD exertion dominates initial sag; NBOD from nitrification adds second oxygen demand plateau days downstream. nBOD control with longer SRT in WWTP reduces river oxygen debt.
Temperature and k_d
[ k_{d,T} = k_{d,20} \theta^{T-20} ]
Use stem θ. Warmer streams exert BOD faster — deeper sag if load unchanged.
Reaeration Enhancement
Weirs, cascades, and riffles increase k_a. Dams may reduce k_a in impoundments while improving settling — tradeoffs in river restoration design.
Permit Linkage
Effluent limits on CBOD5 and ammonia protect receiving water DO. Model with mixed-flow L₀ and temperature-corrected constants before costly plant upgrades.
Dissolved Oxygen Saturation
C_s decreases with temperature and salinity — estuarine modeling uses salinity-adjusted saturation tables. Percent saturation = DO/C_s × 100% for biological assessment.
Assimilative Capacity
Assimilative capacity is remaining pollutant load a stream can accept without violating standards — basis for wasteload allocation in TMDLs linking directly to Streeter-Phelps results. On the FE Environmental exam, confirm units before substituting into handbook equations; distractors often differ only by conversion factors. Practice locating handbook sections by keyword during timed drills so lookup takes under one minute.
Full Mixing + Deficit Example
River: Q_r = 30 m³/s, DO_r = 8.2 mg/L, T = 20°C, C_s = 9.1 mg/L. Effluent: Q_w = 1.5 m³/s, L₀,w = 80 mg/L, DO_w = 1.0 mg/L, k_d = 0.23 day⁻¹, k_a = 0.4 day⁻¹.
Mixed L₀ = (1.5×80)/31.5 = 3.81 mg/L. Mixed DO = (30×8.2 + 1.5×1)/31.5 = 7.85 mg/L. D₀ = 9.1 - 7.85 = 1.25 mg/L.
At t = 2 days downstream (u = 0.5 m/s → x = 86.4 km if needed), plug into Streeter-Phelps for deficit D(t) and convert DO = C_s - D. Minimum DO typically occurs where deoxygenation rate equals reaeration rate — exam may supply graph and ask which discharge scenario violates 5 mg/L standard.
BOD₅ to L₀ Conversion
If problem states BOD₅ = 60 mg/L at k_d = 0.2 day⁻¹:
[ L_0 = \frac{BOD_5}{1 - e^{-5k_d}} = \frac{60}{1 - e^{-1}} = \frac{60}{0.632} = 95\text{ mg/L} ]
Using BOD₅ directly as L₀ underestimates sag depth by ~37% in this case.
In Streeter-Phelps modeling, the dissolved oxygen sag minimum occurs when:
Ultimate BOD L0 represents:
Effluent mixing in a river immediately downstream of an outfall is commonly modeled as:
Increasing stream temperature tends to: