Mathematics Applications for Environmental FE

Quick Answer: Environmental FE math is a workflow: audit units → write balance or geometry → search Handbook → compute → sanity-check magnitude. Master MGD conversions, mass balances, and log/exponential forms.

Unit

Conversion

Table

Worked example: 8 MGD to ft³/s: $8 \times 1.547 = 12.4 \text{ ft}^3/\text{s}$.

Mass and

Volume

Balances

Steady-state control volume: inputs = outputs + accumulation (zero at steady state).

Air dilution: $C_{stack} \times Q_{stack} = C_{ambient} \times Q_{total}$ for conservative tracers at downwind receptor (simplified).

Worked example: 2 m³/s at 30 mg/L mixes with 3 m³/s at 10 mg/L → $C = (60+30)/5 = 18 \text{ mg/L}$.

Geometry for

Channels and

Basins

Trapezoidal channel: $A = (b_1+b_2)y/2$, $P = b_1 + b_2 + 2y\sqrt{1+m^2}$ for side slope m horizontal:1 vertical.

Circular pipe full: $A=\pi D^2/4$, $R_h = D/4$.

Logarithms and

Exponentials

pH: $\text{pH} = -\log_{10}[\text{H}^+]$. Decay: $t = \ln(C_0/C_t)/k$.

Dimensional

Analysis

If result units do not match expected (mg/L·day vs. mg/L), revisit algebra before selecting from multiple choice.

Multi-Step

Problem

Template

1. List givens with units.
2. Identify governing relation (balance, Manning, decay).
3. Convert to consistent set.
4. Solve; compare to order-of-magnitude estimate.

Exam

Traps

• ppm in air ≠ mg/L in water without molecular weight and pressure.
• Percent by weight vs. by volume in chemical solutions.
• Adding concentrations instead of mass flows in blend problems.

Integration with

Other

Topics

Math supports hydrology (runoff depth), treatment (detention time τ=V/Q), and risk (unit conversions for dose). Weak algebra slows every knowledge area — drill conversions daily.

Exam trap: Handbook constants may be in US customary while the stem uses SI — convert once at the start, not mid-calculation.

FE Exam

Integration

Environmental FE items on this topic often combine regulatory classification with a quantitative step. Read the stem for the governing law (CWA, CAA, RCRA, OSHA) before selecting equations. Flag multi-step problems and return after your first pass — average time is under three minutes per question across 110 items.

Practice locating handbook relationships by keyword during timed drills. Confirm units on every constant: mg/L versus μg/m³, ft³/s versus MGD, and days versus seconds in decay and pumping problems are frequent error sources.

FE Exam Integration

Extended Unit Conversion Drill

Worked example: Pump 500 gpm against 80 ft head, efficiency 70%. Water horsepower WHP = QH/3960 = 500×80/3960 = 10.1 hp. Brake hp = 10.1/0.70 = 14.4 hp.

Three-Stream Mass Balance

River (Q=40 m³/s, C=8 mg/L) + Plant effluent (Q=2, C=30) + Tributary (Q=8, C=5):

$$C_{mix} = \frac{40(8)+2(30)+8(5)}{50} = \frac{320+60+40}{50} = 8.4\text{ mg/L}$$

Percent Removal and Effluent

If influent BOD = 220 mg/L and removal = 92%:

$$C_{eff} = 220(1-0.92) = 17.6\text{ mg/L}$$

Reverse problem: permit 15 mg/L at 220 mg/L influent requires 93.2% removal minimum.

Geometry — Circular Clarifier

Diameter D = 60 ft, sidewater depth = 10 ft. Volume V = π(30)²(10) = 28,274 ft³. At Q = 5 MGD = 3.47 cfs, detention (t = V/Q = 8140) s ≈ 2.26 h.

Log-Transform for Skewed Data

Geometric mean for n values: $GM = \exp(\frac{1}{n}\sum \ln x_i)$. Three fecal coliform samples: 100, 200, 800 cfu/100 mL.

$$GM = \exp\left(\frac{\ln 100 + \ln 200 + \ln 800}{3}\right) = \exp(5.30) \approx 200\text{ cfu/100 mL}$$

Arithmetic mean = 367 — geometric mean is the compliance statistic for bacteria.

Reynolds Number Link

$$Re = \frac{vD}{\nu}$$

Transition from laminar to turbulent in pipes near Re = 4000. Darcy-Weisbach headloss rises with friction factor. Environmental FE ties Re to mixing, settling, and aeration basin hydraulics conceptually.

Percent by Weight vs. Volume

10% NaCl by weight in solution differs from 10% by volume. Always read the stem. Converting w/w to mg/L needs solution density — exam usually supplies density or uses dilute aqueous ≈ 1 g/mL.

Hazen-Williams vs. Manning Decision Tree

Hazen-Williams example: C=120, L=3000 ft, D=12 in, Q=3 cfs → head loss from handbook form; compare to pump head available.

Detention Time and Loading

$$\tau = \frac{V}{Q} \quad ; \quad \text{Surface loading} = \frac{Q}{A}$$