Fixed-Film Equation with Recycle

Environmental Engineering · FE Reference Handbook section

Environmental Engineering

11 formulas

10 exam-style examples

~60 min

All Environmental Engineering lectures

Learning objectives

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

This chapter section covers Fixed-Film Equation with Recycle 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 fixed-film equation with recycle describes physically and when it applies.
State every one of the 11 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. Fixed-Film Equation with Recycle 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: fixed-film equation with recycle.
Capstone Studio instructional photograph

Environmental Engineering — Fixed-Film Equation with Recycle: 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 11 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

Se	Quantity produced by "Se = effluent BOD5 (mg/L)" — read its definition and unit from the handbook line directly above the equation.
S0	Quantity produced by "S0 = influent BOD5 (mg/L)" — read its definition and unit from the handbook line directly above the equation.
R	Quantity produced by "R = recycle ratio = QR/Q0" — read its definition and unit from the handbook line directly above the equation.
QR	Quantity produced by "QR = recycle flowrate" — read its definition and unit from the handbook line directly above the equation.
Sa	Quantity produced by "Sa = o1 + R e" — read its definition and unit from the handbook line directly above the equation.
D	Quantity produced by "D = depth of biotower media (m)" — read its definition and unit from the handbook line directly above the equation.
q	Quantity produced by "q = hydraulic loading [m3/(m2 • min)] (Q0 + RQ0 )/Aplan (with recycle)" — read its definition and unit from the handbook line directly above the equation.
k	Quantity produced by "k = treatability constant; functions of wastewater and medium (min−1); range 0.01−0.1; for municipal" — read its definition and unit from the handbook line directly above the equation.
kT	Quantity produced by "kT = k20(1.035)T−20" — read its definition and unit from the handbook line directly above the equation.
n	Quantity produced by "n = coefficient relating to media characteristics; modular plastic, n = 0.5" — 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.

Se e- kD q
_1 + Ri - R c e- kD q m
where
S + RS
wastewater and modular plastic media 0.06 min−1 @ 20°C

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

BOD exerted after t days — Fixed-Film Equation with Recycle

A wastewater has ultimate BOD L₀ = 161 mg/L and k = 0.19 /day (base e). How much BOD is exerted in 7 days?

Given

L₀ = 161 mg/L
k = 0.19 /day
t = 7 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
$k t = 0.19 (7) = 1.33$
Exponential
$e^{(} - 1.33) = 0.2645$
Substituting
$B O D_{t} = 161 (1 - 0.2645) = 118.4 m g / L$
Remaining
$L_{0} - B O D_{t} = 42.6 m g / L$

Answer: BOD_7 ≈ 118.4 mg/L

Why the other options are there

43 mg/L (remaining reported as exerted)
214.1 mg/L (linear decay assumed)

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 2

Hydraulic detention time in a tank — Fixed-Film Equation with Recycle

A treatment tank holds 118,684 gal and treats 4.0 MGD. Find the hydraulic detention time.

Given

V = 118,684 gal
Q = 4.0 MGD

Find

Detention time θ

Start with the thinking

θ = V/Q with consistent volume units.
Express the answer in hours for design comparison.

Step-by-step solution

Detention
$θ = V / Q$
Flow in gal/day
$4.0 \times 1 0^{6} = 4, 000, 000 g a l / d a y$
Substituting
$θ = 118684/4, 000, 000 = 0.0297 d a y$
Convert
$θ = 0.0297 (24) = 0.71 h r$

Answer: θ ≈ 0.71 hr

Why the other options are there

0.030 hr (day/hour conversion missed)
33.70 hr (ratio inverted)

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 3

Chemical feed rate from dosage — Fixed-Film Equation with Recycle

A plant treating 1.5 MGD requires a 3.5 mg/L dose. What is the daily chemical demand in pounds?

Given

Q = 1.5 MGD
Dose = 3.5 mg/L
8.34 lb/gal

Find

lb/day of chemical

Start with the thinking

The 8.34 factor converts mg/L·MGD to lb/day.
Adjust for purity if the product is not 100% active.

Step-by-step solution

Feed
$l b / d a y = Q (M G D) \times d ose (m g / L) \times 8.34$
Substituting
$l b / d a y = 1.5 (3.5) (8.34)$
Evaluate — 43.8 lb/day
At 65% available strength — 67.4 lb/day of product

Answer: ≈ 44 lb/day

Why the other options are there

5.3 lb/day (8.34 omitted)
0.63 lb/day (factor divided)

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 4

Converting ppm to mg/m³ — Fixed-Film Equation with Recycle

A stack gas contains 11.0 ppm of a compound with molecular weight 64 g/mol at 25 °C and 1 atm. Express the concentration in mg/m³.

Given

C = 11.0 ppm
MW = 64 g/mol
Molar volume = 24.45 L/mol

Find

Concentration in mg/m³

Start with the thinking

ppm by volume needs the molar volume to become a mass concentration.
24.45 L/mol applies at 25 °C; use 22.4 at 0 °C.

Step-by-step solution

Conversion
$m g / m^{3} = pp m \times M W /24.45$
Substituting
$m g / m^{3} = 11.0 (64) /24.45$
Evaluate — 28.79 mg/m³
Reverse check
$pp m = 28.79 (24.45) /64 = 11.0 ✓$

Answer: ≈ 28.8 mg/m³

Why the other options are there

4.2 mg/m³ (ratio inverted)
31.4 mg/m³ (0 °C molar volume used)

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 5

First-order removal in a CSTR versus a plug-flow reactor — Fixed-Film Equation with Recycle

A reactor of volume 625 m³ treats 0.10 m³/s carrying 256 mg/L of a contaminant that decays first-order with k = 0.60 h⁻¹. Compute the hydraulic residence time and the effluent concentration if the tank behaves as a CSTR and as a plug-flow reactor.

Given

V = 625 m³
Q = 0.10 m³/s
C₀ = 256 mg/L
k = 0.60 h⁻¹

Find

τ, CSTR effluent and PFR effluent

Start with the thinking

The mass balance for steady state is: in − out − reaction = 0.
For the same volume, plug flow always outperforms a single completely mixed tank for first-order kinetics.

Step-by-step solution

Formula
$τ = V / Q$
Substituting
$τ = 625/0.10 = 6, 250 s = 1.736 h$
Formula (CSTR)
$C = C_{0} / (1 + k τ)$
Substituting
$C = 256/ (1 + 0.60 \times 1.736) = 125.4 m g / L$
Formula (PFR)
$C = C_{0} e^{(} - k τ)$
Substituting
$C = 256 e^{(} - 0.60 \times 1.736) = 90.33 m g / L$
Comparison — plug flow removes 64.7% versus 51.0% for the CSTR

Answer: τ = 1.74 h; C_CSTR = 125.4 mg/L, C_PFR = 90.3 mg/L

Why the other options are there

-10.7 mg/L (linear decay assumed)
125.4 mg/L for both reactors

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 6

Activated sludge F/M ratio, aeration time and sludge production — Fixed-Film Equation with Recycle

An activated sludge plant treats 15,489 m³/d with an influent BOD of 329 mg/L, effluent BOD 14 mg/L, MLSS 3,554 mg/L, and an aeration basin volume of 4,406 m³. With Y = 0.45 mg VSS/mg BOD, k_d = 0.07 d⁻¹ and an SRT of 5 days, compute the F/M ratio, hydraulic detention time and daily sludge production.

Given

Q = 15,489 m³/d
S₀ = 329 mg/L, S = 14 mg/L
X = 3,554 mg/L, V = 4,406 m³
Y = 0.45, k_d = 0.07 d⁻¹, SRT = 5 d

Find

F/M ratio, detention time τ and sludge wasted per day

Start with the thinking

F/M compares the food entering daily with the microbial mass held in the basin; conventional plants run near 0.2–0.5 d⁻¹.
Endogenous decay reduces net sludge yield as the SRT lengthens.

Step-by-step solution

Formula
$F / M = Q S_{0} / (V X)$
Substituting
$F / M = 15489 (329) / (4406 \times 3554) = 0.325 d^{- 1}$
Formula
$τ = V / Q$
Substituting
$τ = 4406/15489 = 0.2845 d = 6.83 h$
Formula
$P_{x} = Y Q (S_{0} - S) / (1 + k_{d} \cdot S R T)$
Substituting
$P_{x} = 0.45 (15489) (329 - 14) / (1 + 0.07 \times 5) m g / d$
Evaluate
$P_{x} = 1, 626 k g V S S / d a y$

Answer: F/M = 0.33 d⁻¹, τ = 6.8 h, sludge = 1,626 kg/d

Why the other options are there

F/M = 1,434 (basin volume omitted)
P_x = 2,196 kg/d (decay term ignored)

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 7

Circular clarifier overflow rate, detention time and Stokes settling — Fixed-Film Equation with Recycle

A circular clarifier 14 m in diameter and 4.5 m deep treats 7,769 m³/d. Compute the surface overflow rate, the detention time and the weir loading rate, then check whether a 60 µm particle (SG = 2.65) settles out by Stokes' law.

Given

Q = 7,769 m³/d
D = 14 m, depth = 4.5 m
d_p = 60 µm, ρ_s = 2650 kg/m³
μ = 0.001 Pa·s

Find

SOR, detention time, weir loading and the Stokes settling velocity

Start with the thinking

An ideal settling basin removes every particle whose settling velocity exceeds the overflow rate — depth does not matter for removal, only for detention.
Stokes' law applies to the laminar (small particle) regime.

Step-by-step solution

Surface area
$A = (π /4) (14)^{2} = 153.9 m^{2}$
Formula
$S O R = Q / A$
Substituting — SOR = 7769/153.9 = 50.47 m³/m²·d
Formula — τ = V/Q = A·h/Q
Substituting
$τ = 692.7/7769 = 2.14 h$
Formula
$w e i r l o a d in g = Q / (π D)$
Substituting — 7769/(π × 14) = 176.6 m³/m·d
Formula
$v_{s} = g (ρ_{s} - ρ) d^{2} / (18 μ)$
Substituting
$v_{s} = 9.81 (1650) (6.0 e - 5)^{2} / (18 \times 0.001) = 279.7 m / d$
Compare
$v_{s} = 279.7 m / d > S O R = 50.47 m / d \to p a r t i c l e i sr e m o v e d$

Answer: SOR = 50.5 m/d, τ = 2.1 h, weir loading = 176.6 m³/m·d, v_s = 279.7 m/d

Why the other options are there

SOR = 39.3 m/d (side area used)
v_s computed with the diameter unsquared

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 8

Series particulate control: overall collection efficiency and emission rate — Fixed-Film Equation with Recycle

A stack gas stream of 31 m³/s carries 21 g/m³ of particulate. It passes a cyclone at 80% efficiency followed by a fabric filter at 94.5% efficiency. Compute the concentration after each device, the overall efficiency, and the emission rate in kg/h.

Given

Q = 31 m³/s
C_in = 21 g/m³
η₁ = 0.80
η₂ = 0.945

Find

Intermediate and final concentrations, overall η and kg/h emitted

Start with the thinking

Efficiencies in series multiply as penetrations (1 − η), they never simply add.
The overall penetration is the product of the individual penetrations.

Step-by-step solution

Formula
$C_{1} = C_{i} n (1 - η_{1})$
Substituting
$C_{1} = 21 (1 - 0.80) = 4.200 g / m^{3}$
Formula
$C_{2} = C_{1} (1 - η_{2})$
Substituting
$C_{2} = 4.200 (1 - 0.945) = 0.2310 g / m^{3}$
Formula
$η_{o} v er a l l = 1 - (1 - η_{1}) (1 - η_{2})$
Substituting
$η = 1 - 0.200 \times 0.0550 = 0.98900 = 98.900$
Formula
$e mi ss i o n = C_{2} Q$
Substituting
$0.2310 g / m^{3} \times 31 m^{3} / s \times 3600 s / h \div 1000 = 25.780 k g / h$

Answer: C_out = 0.2310 g/m³, η = 98.90%, emission = 25.78 kg/h

Why the other options are there

η = 174.5% (efficiencies added)
2,344 kg/h (uncontrolled rate)

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 9

Landfill volume required for a community's solid waste — Fixed-Film Equation with Recycle

A community of 182,092 people generates 2.9 kg/person/day of MSW and diverts 15% through recycling. Compacted in place at 849 kg/m³ with 20% additional daily cover volume, find the airspace needed for 29 years.

Given

Population = 182,092
Generation = 2.9 kg/cap/d
Diversion = 0.15
Compacted density = 849 kg/m³
Cover = 20%, design life 29 yr

Find

Annual and total landfill airspace

Start with the thinking

Only the landfilled fraction consumes airspace — diverted material is subtracted first.
Daily cover soil is real volume and must be added to the waste volume.

Step-by-step solution

Formula
$l an df i l l e d ma ss = p o p u l a t i o n \times r a t e \times (1 - d i v er s i o n)$
Substituting
$182092 (2.9) (1 - 0.15) = 448, 857 k g / d a y$
Formula
$V = ma ss / d e n s i t y$
Substituting
$V = 448, 857/849 = 528.7 m^{3} / d a y$
Annual with cover
$528.7 \times 365 \times 1.20 = 231, 566 m^{3} / y r$
Design life
$231, 566 \times 29 = 6, 715, 405 m^{3}$

Answer: ≈ 231,566 m³/yr, or 6,715,405 m³ over 29 years

Why the other options are there

6,583,730 m³ (diversion and cover ignored)
4,751,149,016 m³ (mass reported as volume)

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

Example 10

Chlorine dose, CT value and daily chemical demand — Fixed-Film Equation with Recycle

A water plant treating 21,879 m³/d applies a chlorine dose of 3.5 mg/L against a demand of 2.2 mg/L, with 108 minutes of contact time. Compute the free residual, the CT value, the daily chlorine mass required, and the concentration remaining after a 2.5-log pathogen reduction of an initial 10⁶ organisms/100 mL.

Given

Q = 21,879 m³/d
Dose = 3.5 mg/L
Demand = 2.2 mg/L
t = 108 min
Target = 2.5 log

Find

Residual, CT, kg/day of chlorine and surviving organisms

Start with the thinking

Residual = dose − demand; the residual, not the dose, drives disinfection credit.
Each log of removal divides the surviving organism count by ten.

Step-by-step solution

Formula
$r es i d u a l = d ose - d e man d$
Substituting
$3.5 - 2.2 = 1.30 m g / L$
Formula
$C T = C \times t$
Substituting — CT = 1.30(108) = 140.4 mg·min/L
Formula
$ma ss = d ose \times Q$
Substituting
$3.5 g / m^{3} \times 21879 m^{3} / d \div 1000 = 76.58 k g / d a y$
Log removal
$N = 1 0^{6} /1 0^{2} .5 = 3.16 e + 3 or g ani s m s /100 m L$

Answer: Residual = 1.30 mg/L, CT = 140.4 mg·min/L, 76.6 kg Cl₂/day, survivors 3.2e+3/100 mL

Why the other options are there

CT = 378.0 (dose used instead of residual)
76,577 kg/day (unit conversion missed)

Reference: FE Reference Handbook — Environmental Engineering → Fixed-Film Equation with Recycle

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

Fixed-Film Equation with Recycle contains 11 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.