Streeter-Phelps Dissolved-Oxygen Sag: 22-Agric-B11 Principles of Waste Management, Undated paper, Q5

Question 5: Streeter-Phelps Dissolved-Oxygen Sag (25 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Quantity	Lagoon effluent	Upstream river
Flow	10 m³/h	50 m³/h
BOD_u (ultimate)	50 mg/L	2 mg/L
DO	2 mg/L	7 mg/L
Temperature	20 °C	20 °C

Mixed-flow kinetics: deoxygenation rate $k_1=0.15\ \text{d}^{-1}$; reaeration rate $k_2=0.20\ \text{d}^{-1}$ (both natural-log/e-based); saturation DO at 20°C $=9.1\ \text{mg/L}$.

Find. The minimum downstream DO and whether it falls below the 5.0 mg/L regulatory limit.

Approach. Both streams share the same 20°C temperature, so mix by simple flow-weighted averaging to get the initial ultimate BOD (L₀) and initial oxygen deficit (D₀) at the point of discharge, then apply the Streeter-Phelps sag equation to find the critical time and the minimum DO.

Mixed-flow characteristics. $Q_{mix}=10+50=60\ \text{m}^3/\text{h}$. Flow-weighted ultimate BOD: $$L_0 = \frac{(10)(50)+(50)(2)}{60} = \boxed{10.0\ \text{mg/L}}$$ Flow-weighted DO: $$DO_{mix} = \frac{(10)(2)+(50)(7)}{60} = 6.17\ \text{mg/L}$$ Initial deficit: $$D_0 = DO_{sat}-DO_{mix} = 9.1-6.17 = \boxed{2.93\ \text{mg/L}}$$
Critical time (point of minimum DO). $$t_c = \frac{1}{k_2-k_1}\ln\left[\frac{k_2}{k_1}\left(1-\frac{D_0(k_2-k_1)}{k_1 L_0}\right)\right] = \frac{1}{0.05}\ln\left[1.333\left(1-\frac{(2.93)(0.05)}{(0.15)(10.0)}\right)\right] = \boxed{3.70\ \text{d}}$$
Critical (maximum) deficit and minimum DO. BOD remaining at $t_c$: $L(t_c)=L_0 e^{-k_1 t_c}=10.0\,e^{-(0.15)(3.70)}=5.74\ \text{mg/L}$. At the critical point the deoxygenation and reaeration rates balance ($k_1 L(t_c)=k_2 D_c$), so $$D_c = \frac{k_1}{k_2}L(t_c) = \frac{0.15}{0.20}(5.74) = \boxed{4.31\ \text{mg/L}}$$ $$DO_{min} = DO_{sat}-D_c = 9.1-4.31 = \boxed{4.79\ \text{mg/L}}$$

Downstream dissolved-oxygen sag curve from the Streeter-Phelps equation. The minimum DO (4.79 mg/L at t_c ≈ 3.70 d) falls below the 5.0 mg/L regulatory limit (dashed line).

Quantity	Result
Mixed initial ultimate BOD, L₀	10.0 mg/L
Mixed initial DO / deficit, D₀	6.17 mg/L / 2.93 mg/L
Critical time, t_c	3.70 d downstream
Critical deficit, D_c	4.31 mg/L
Minimum downstream DO	4.79 mg/L — falls BELOW the 5.0 mg/L regulatory limit

The predicted sag bottoms out at 4.79 mg/L, about 3.7 days (roughly 90 hours) downstream of the discharge — 0.21 mg/L below the 5.0 mg/L requirement. The lagoon discharge as described would therefore NOT meet the regulatory dissolved-oxygen standard, and some combination of higher-level treatment (BOD reduction before discharge), effluent aeration, or a permit-limited discharge flow/timing restriction would be needed to close this 0.21 mg/L gap.

2) Basic assumptions of the Streeter-Phelps equation (5 marks). (i) The river is treated as an ideal steady-state plug-flow channel — constant cross-section, constant velocity, and no dispersion/back-mixing along the direction of flow, so that "time downstream" is simply distance divided by a single representative velocity. (ii) BOD exertion (deoxygenation) and reaeration are each modelled as independent, first-order reactions with constant rate coefficients (k₁, k₂) that do not change with distance or time. (iii) The only oxygen SINK is carbonaceous BOD decay (no separate accounting for nitrogenous BOD, sediment oxygen demand, or algal respiration) and the only oxygen SOURCE is atmospheric reaeration (no photosynthetic oxygen production is credited). (iv) There are no additional point or non-point BOD/oxygen inputs between the discharge and the point of interest — the river receives no other loads over the reach being modelled. (v) Temperature (and therefore k₁, k₂ and DO_sat) is taken as constant over the reach and over the time of travel, consistent with using a single stated 20°C value throughout this calculation.

Topic: Streeter-Phelps dissolved-oxygen sag modelling for a manure-lagoon discharge to a river.

Key relations: $D(t)=\dfrac{k_1 L_0}{k_2-k_1}\left(e^{-k_1t}-e^{-k_2t}\right)+D_0e^{-k_2t}$; critical time $t_c=\dfrac{1}{k_2-k_1}\ln\left[\dfrac{k_2}{k_1}\left(1-\dfrac{D_0(k_2-k_1)}{k_1L_0}\right)\right]$; at $t_c$, $k_1L(t_c)=k_2D_c$.

Why this works: the sag curve exists because deoxygenation (fast at first, since BOD concentration is highest right after discharge) and reaeration (proportional to how far DO already is below saturation) work against each other at different, changing relative rates as the plume moves downstream; the deficit grows until the two rates exactly balance at t_c, then shrinks as reaeration takes over once the bulk of the BOD has been exerted. This is exactly why the worst DO point is NOT at the discharge itself but some travel-time downstream — a critical, easy-to-miss feature when a regulator specifies a compliance point.

Common pitfall: checking DO only at the point of discharge (where the deficit is small, D₀=2.93 mg/L, DO=6.17 mg/L, apparently compliant) rather than at the critical time downstream where the deficit is actually largest; and forgetting to flow-weight-average BOD_u and DO across the two mixing streams before starting the sag calculation.

Source: Tchobanoglous, Burton & Stensel, Metcalf & Eddy Wastewater Engineering (Streeter-Phelps DO sag, stream self-purification); Sommer & Christensen (eds.), Animal Manure Recycling (lagoon-effluent discharge impacts).

Question 5 of 5: Streeter-Phelps Dissolved-Oxygen Sag

Question 5: Streeter-Phelps Dissolved-Oxygen Sag (25 marks)