BOD5 for Mixed Lagoons in Series

Environmental Engineering · FE Reference Handbook section

Environmental Engineering

5 formulas

10 exam-style examples

~55 min

All Environmental Engineering lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers BOD5 for Mixed Lagoons in Series within Environmental Engineering. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what bod5 for mixed lagoons in series describes physically and when it applies.
State every one of the 5 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: mg/L × MGD × 8.34 = lb/day is the single most used conversion.

Lecture

Why this section exists. BOD5 for Mixed Lagoons in Series is the part of Environmental Engineering that lets you connect a treatment unit or receiving water body to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a mass balance across one reactor or one unit process. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. mg/L × MGD × 8.34 = lb/day is the single most used conversion. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Aeration basin at a wastewater treatment plant with churning aerated water and walkways. — Photo 1. Where this shows up in practice: bod5 for mixed lagoons in series.
Capstone Studio instructional photograph

Environmental Engineering — BOD5 for Mixed Lagoons in Series: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a treatment unit or receiving water body. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 5 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

S0	Quantity produced by "S0 = Inlet total BOD5" — read its definition and unit from the handbook line directly above the equation.
S	Quantity produced by "S = Outlet total BOD5" — read its definition and unit from the handbook line directly above the equation.
θ	Quantity produced by "θ = Fresh-feed residence time" — read its definition and unit from the handbook line directly above the equation.
kp	Quantity produced by "kp = Kinetic constant (time–1)" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

S 1
S0 1 + kp i
where

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Remaining BOD after five days

A wastewater has BOD_u = 320 mg/L with k = 0.23 /day (base e). What is BOD₅ and the remaining oxygen demand at day 5?

Given

BOD_u = 320 mg/L
k = 0.23 /day
t = 5 days

Find

BOD₅ and the remaining demand

Start with the thinking

BOD₅ is the amount exerted, not what remains.
Exponential decay uses base e with this k.

Step-by-step solution

Decay factor
$e^{- k t} = e^{- 0.23 (5)} = e^{- 1.15} = 0.3166$
Exerted demand
$B O D_{5} = B O D_{u} (1 - e^{- k t}) = 320 (1 - 0.3166)$
Evaluate
$B O D_{5} = 320 (0.6834) = 219 m g / L$
Remaining
$320 - 219 = 101 m g / L$

Answer: BOD₅ = 219 mg/L; 101 mg/L remains

Why the other options are there

101 mg/L (remaining reported as BOD₅)
320 mg/L (ultimate reported)

Reference: FE Reference Handbook — Environmental — BOD kinetics

Example 2

Sedimentation basin overflow rate

A rectangular clarifier is 25 m long and 8 m wide, treating 6,000 m³/day. Find the surface overflow rate and, for a 3.0 m depth, the detention time.

Given

L = 25 m, W = 8 m
Q = 6,000 m³/day
Depth = 3.0 m

Find

Overflow rate and detention time

Start with the thinking

Overflow rate uses plan area only — depth does not appear.
Detention time uses the full volume.

Step-by-step solution

Plan area
$A = 25 (8) = 200 m^{2}$
Overflow rate
$v_{o} = Q / A = 6, 000/200 = 30.0 m / d a y$
Volume
$V = 200 (3.0) = 600 m^{3}$
Detention time
$t = V / Q = 600/6, 000 = 0.100 d a y$
Result — v_o = 30 m³/(m²·day), t = 2.4 hours

Answer: v_o = 30 m/day; t = 2.4 h

Why the other options are there

10 m/day (depth divided in)
t = 24 h (day-to-hour conversion missed)

Reference: FE Reference Handbook — Environmental — Sedimentation

Example 3

BOD exerted after t days — BOD5 for Mixed Lagoons in Series

A wastewater has ultimate BOD L₀ = 227 mg/L and k = 0.29 /day (base e). How much BOD is exerted in 5 days?

Given

L₀ = 227 mg/L
k = 0.29 /day
t = 5 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.29 (5) = 1.45$
Exponential
$e^{(} - 1.45) = 0.2346$
Substituting
$B O D_{t} = 227 (1 - 0.2346) = 173.8 m g / L$
Remaining
$L_{0} - B O D_{t} = 53.2 m g / L$

Answer: BOD_5 ≈ 173.8 mg/L

Why the other options are there

53 mg/L (remaining reported as exerted)
329.2 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

Example 4

BOD exerted after t days — BOD5 for Mixed Lagoons in Series (2)

A wastewater has ultimate BOD L₀ = 263 mg/L and k = 0.28 /day (base e). How much BOD is exerted in 6 days?

Given

L₀ = 263 mg/L
k = 0.28 /day
t = 6 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.28 (6) = 1.68$
Exponential
$e^{(} - 1.68) = 0.1864$
Substituting
$B O D_{t} = 263 (1 - 0.1864) = 214.0 m g / L$
Remaining
$L_{0} - B O D_{t} = 49.0 m g / L$

Answer: BOD_6 ≈ 214.0 mg/L

Why the other options are there

49 mg/L (remaining reported as exerted)
441.8 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

Example 5

BOD exerted after t days — BOD5 for Mixed Lagoons in Series (3)

A wastewater has ultimate BOD L₀ = 332 mg/L and k = 0.26 /day (base e). How much BOD is exerted in 5 days?

Given

L₀ = 332 mg/L
k = 0.26 /day
t = 5 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.26 (5) = 1.30$
Exponential
$e^{(} - 1.30) = 0.2725$
Substituting
$B O D_{t} = 332 (1 - 0.2725) = 241.5 m g / L$
Remaining
$L_{0} - B O D_{t} = 90.5 m g / L$

Answer: BOD_5 ≈ 241.5 mg/L

Why the other options are there

90 mg/L (remaining reported as exerted)
431.6 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

Example 6

BOD exerted after t days — BOD5 for Mixed Lagoons in Series (4)

A wastewater has ultimate BOD L₀ = 341 mg/L and k = 0.16 /day (base e). How much BOD is exerted in 10 days?

Given

L₀ = 341 mg/L
k = 0.16 /day
t = 10 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.16 (10) = 1.60$
Exponential
$e^{(} - 1.60) = 0.2019$
Substituting
$B O D_{t} = 341 (1 - 0.2019) = 272.2 m g / L$
Remaining
$L_{0} - B O D_{t} = 68.8 m g / L$

Answer: BOD_10 ≈ 272.2 mg/L

Why the other options are there

69 mg/L (remaining reported as exerted)
545.6 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

Example 7

BOD exerted after t days — BOD5 for Mixed Lagoons in Series (5)

A wastewater has ultimate BOD L₀ = 141 mg/L and k = 0.33 /day (base e). How much BOD is exerted in 10 days?

Given

L₀ = 141 mg/L
k = 0.33 /day
t = 10 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.33 (10) = 3.30$
Exponential
$e^{(} - 3.30) = 0.0369$
Substituting
$B O D_{t} = 141 (1 - 0.0369) = 135.8 m g / L$
Remaining
$L_{0} - B O D_{t} = 5.2 m g / L$

Answer: BOD_10 ≈ 135.8 mg/L

Why the other options are there

5 mg/L (remaining reported as exerted)
465.3 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

Example 8

BOD exerted after t days — BOD5 for Mixed Lagoons in Series (6)

A wastewater has ultimate BOD L₀ = 191 mg/L and k = 0.23 /day (base e). How much BOD is exerted in 6 days?

Given

L₀ = 191 mg/L
k = 0.23 /day
t = 6 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.23 (6) = 1.38$
Exponential
$e^{(} - 1.38) = 0.2516$
Substituting
$B O D_{t} = 191 (1 - 0.2516) = 142.9 m g / L$
Remaining
$L_{0} - B O D_{t} = 48.1 m g / L$

Answer: BOD_6 ≈ 142.9 mg/L

Why the other options are there

48 mg/L (remaining reported as exerted)
263.6 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

Example 9

BOD exerted after t days — BOD5 for Mixed Lagoons in Series (7)

A wastewater has ultimate BOD L₀ = 354 mg/L and k = 0.27 /day (base e). How much BOD is exerted in 3 days?

Given

L₀ = 354 mg/L
k = 0.27 /day
t = 3 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.27 (3) = 0.81$
Exponential
$e^{(} - 0.81) = 0.4449$
Substituting
$B O D_{t} = 354 (1 - 0.4449) = 196.5 m g / L$
Remaining
$L_{0} - B O D_{t} = 157.5 m g / L$

Answer: BOD_3 ≈ 196.5 mg/L

Why the other options are there

157.5 mg/L (remaining reported as exerted)
286.7 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

Example 10

BOD exerted after t days — BOD5 for Mixed Lagoons in Series (8)

A wastewater has ultimate BOD L₀ = 135 mg/L and k = 0.16 /day (base e). How much BOD is exerted in 3 days?

Given

L₀ = 135 mg/L
k = 0.16 /day
t = 3 days

Find

BOD_t

Start with the thinking

BOD exertion is first-order and approaches L₀ asymptotically.
Check whether k is base e or base 10.

Step-by-step solution

First-order
$B O D_{t} = L_{0} (1 - e^{(} - k t))$
Exponent
$k t = 0.16 (3) = 0.48$
Exponential
$e^{(} - 0.48) = 0.6188$
Substituting
$B O D_{t} = 135 (1 - 0.6188) = 51.5 m g / L$
Remaining
$L_{0} - B O D_{t} = 83.5 m g / L$

Answer: BOD_3 ≈ 51 mg/L

Why the other options are there

84 mg/L (remaining reported as exerted)
65 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → BOD5 for Mixed Lagoons in Series

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a treatment unit or receiving water body, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

BOD5 for Mixed Lagoons in Series contains 5 relations; you must be able to find this page in under 15 seconds.
Exam style: a mass balance across one reactor or one unit process.
Unit rule: mg/L × MGD × 8.34 = lb/day is the single most used conversion.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

mg/L × MGD × 8.34 = lb/day is the single most used conversion
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.