FE Environmental formula sheet

Calculus and Differential Equations

Derivatives as rates and integrals as accumulated mass, maxima and minima, the fundamental theorem, and first- and second-order linear ODEs for decay, reactor, and mixing models.

Definition of the derivative

$$\dfrac{dy}{dx} = \lim_{\Delta x \to 0}\dfrac{f(x + \Delta x) - f(x)}{\Delta x}$$

Instantaneous rate of change / slope of $y = f(x)$. In process terms, a flux or reaction rate.

Chain and product rules

$$\dfrac{d}{dx}f(u) = f'(u)\dfrac{du}{dx}, \qquad \dfrac{d}{dx}(uv) = u\dfrac{dv}{dx} + v\dfrac{du}{dx}$$

Differentiation of composites and products; the chain rule drives every $e^{-kt}$ derivative.

Max/min/inflection test

$$f'(x^\ast) = 0; \quad f''<0 \Rightarrow \max,\;\; f''>0 \Rightarrow \min,\;\; f''=0\text{ (sign change)} \Rightarrow \text{inflection}$$

First derivative locates stationary points; second derivative classifies them.

Fundamental theorem of calculus

$$\int_a^b f(x)\,dx = F(b) - F(a), \qquad \dfrac{d}{dx}\int_a^x f(t)\,dt = f(x)$$

Integration and differentiation are inverse operations; $F$ is any antiderivative of $f$.

Accumulated mass (loading)

$$M = \int_0^T Q(t)\,C(t)\,dt$$

Total mass discharged over time. $Q$ = flow, $C$ = concentration; mind unit conversion to kg or lb.

First-order decay

$$\dfrac{dC}{dt} = -kC \;\Rightarrow\; C(t) = C_0 e^{-kt}$$

Decay/BOD/disinfection model. $k$ = rate constant (time$^{-1}$); time constant $\tau = 1/k$.

Half-life

$$t_{1/2} = \dfrac{\ln 2}{k}$$

Time for a first-order quantity to halve; independent of $C_0$.

CSTR transient (first-order nonhomogeneous)

$$C(t) = C_{in}\big(1 - e^{-t/\tau}\big) + C_0\,e^{-t/\tau}, \qquad \tau = \dfrac{V}{Q}$$

Step response of a completely mixed tank; approaches $C_{in}$ as $t \to \infty$. $\tau$ = hydraulic residence time.

Second-order characteristic equation

$$y'' + ay' + by = 0 \;\Rightarrow\; r^2 + ar + b = 0$$

Roots $r$ set the modes of the response; sign of $a^2 - 4b$ gives over/critical/underdamped.

Underdamped solution

$$y = e^{\alpha x}\big(C_1\cos\beta x + C_2\sin\beta x\big), \quad \alpha = -\tfrac{a}{2}$$

Complex-root case ($a^2 < 4b$): decaying oscillation, with $\beta = \tfrac{1}{2}\sqrt{4b - a^2}$.

Where it lives: NCEES FE Reference Handbook — Mathematics (Differential and Integral Calculus, Differential Equations) · Metcalf & Eddy, Wastewater Engineering: Treatment and Resource Recovery · Davis & Cornwell, Introduction to Environmental Engineering

Hypothesis Testing

Frame null and alternative hypotheses, run one- and two-tailed z and t tests, control Type I and II error, and decide with critical values and p-values.

z test statistic (σ known)

$$Z_0=\dfrac{\bar{X}-\mu_0}{\sigma/\sqrt{n}}$$

Standardized distance of the sample mean from $\mu_0$ when the population $\sigma$ is known.

t test statistic (σ unknown)

$$t_0=\dfrac{\bar{X}-\mu_0}{s/\sqrt{n}}$$

Same form using the sample $s$; compare to $t_{\alpha,\,n-1}$ or $t_{\alpha/2,\,n-1}$.

Two-tailed rejection (z)

$$\text{reject } H_0 \text{ if } |Z_0|>z_{\alpha/2}$$

$H_1:\mu\neq\mu_0$. At $\alpha=0.05$, $z_{\alpha/2}=1.960$.

One-tailed rejection (z)

$$Z_0>z_{\alpha}\ \ \text{or}\ \ Z_0<-z_{\alpha}$$

Upper- or lower-tail $H_1$. At $\alpha=0.05$, $z_{\alpha}=1.645$.

Type I and Type II error

$$\alpha=P(\text{reject }H_0\mid H_0\ \text{true}), \quad \beta=P(\text{accept }H_0\mid H_0\ \text{false})$$

$\alpha$ = false alarm (significance level); $\beta$ = missed detection.

Power of the test

$$\text{power}=1-\beta$$

Probability of correctly rejecting a false $H_0$; rises with $n$ and effect size.

P-value decision rule

$$\text{reject } H_0 \iff p\le\alpha$$

$p$ = probability of a statistic at least as extreme as observed, given $H_0$. Double the one-tail area for two-tailed tests.

Pooled variance (two means)

$$s_p^2=\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$$

Combined variance estimate for comparing two means with equal variances; df $=n_1+n_2-2$.

Where it lives: NCEES FE Reference Handbook — Engineering Probability and Statistics · Montgomery & Runger, Applied Statistics and Probability for Engineers · Davis & Cornwell, Introduction to Environmental Engineering

Codes of Ethics and Public Welfare

How the NCEES Model Rules put public health, safety, and welfare above client loyalty, demand competence, forbid conflicts of interest, and frame ethical dilemmas.

Hierarchy of obligation

$$\text{Public H, S \& W} \succ \text{Employer / Client} \succ \text{Other Licensees} \succ \text{Self}$$

Lexical priority of the three Model Rules tiers. When duties conflict, the higher tier controls; the public's health, safety, and welfare is paramount.

Paramount-duty trigger

$$\text{judgment overruled} \;\wedge\; \text{public endangered} \;\Rightarrow\; \text{notify employer/client + proper authority}$$

Action rule from Section 240.15(A)(3): if your professional judgment is overruled in a way that endangers the public, you must notify your employer or client and an appropriate authority.

Responsible-charge / sealing rule

$$\text{may seal} \iff (\text{prepared by you } \vee \text{ under your responsible charge}) \;\wedge\; \text{within competence}$$

You may sign and seal a document only if it was prepared by you or under your responsible charge and lies within your area of competence. Plan-stamping fails this test.

Conflict-of-interest test

$$\text{appears to influence judgment} \;\Rightarrow\; \text{disclose in writing to all parties}$$

The appearance standard: any interest that could influence or appear to influence professional judgment must be disclosed; gifts/gratuities from connected parties are prohibited outright.

Acceptable carcinogenic risk band

$$10^{-6} \le \text{Risk} \le 10^{-4}$$

EPA range of acceptable added individual lifetime cancer risk; the quantitative anchor for a risk-vs-benefit ethical tradeoff. Risk is dimensionless (probability).

Added cancer risk

$$\text{Risk} = CDI \times CSF$$

Carcinogenic risk = chronic daily intake $CDI$ (mg/(kg·day)) times cancer slope factor $CSF$ ((mg/(kg·day))$^{-1}$); Safety chapter tool used to argue 'how safe is safe enough.'

Noncarcinogen hazard index

$$HI = \dfrac{CDI_{\text{noncarc}}}{RfD}$$

Hazard index = chronic daily intake divided by the reference dose $RfD$ (mg/(kg·day)); $HI \le 1$ is considered acceptable. Both terms share units, so $HI$ is dimensionless.

Where it lives: NCEES FE Reference Handbook — Ethics and Professional Practice (Model Rules Section 240.15) · NCEES FE Reference Handbook — Safety: Risk Assessment / Toxicology · NCEES Model Rules and Model Law (Rules of Professional Conduct)

Time Value of Money and Life-Cycle Cost

The interest-factor family (P/F, F/P, P/A, A/P, A/F, F/A), gradients, nominal vs effective rates, and putting capital and annual O&M on one basis for life-cycle cost.

Single-payment compound amount (F/P)

$$F = P\,(1+i)^{n}$$

Grows a present lump sum $P$ to its future value $F$ after $n$ periods at periodic rate $i$. Factor symbol $(F/P,i\%,n)$.

Single-payment present worth (P/F)

$$P = F\,(1+i)^{-n}$$

Discounts a future amount $F$ back to the present. Reciprocal of $(F/P)$; the building block of all present-worth analysis.

Uniform-series present worth (P/A)

$$P = A\,\dfrac{(1+i)^{n}-1}{i\,(1+i)^{n}}$$

Present value of an end-of-period annuity $A$ for $n$ periods. Used to capitalize annual O&M or benefits.

Capital recovery (A/P)

$$A = P\,\dfrac{i\,(1+i)^{n}}{(1+i)^{n}-1}$$

Spreads a present cost $P$ into $n$ equal payments $A$; this annualizes capital cost. Reciprocal of $(P/A)$.

Sinking fund (A/F)

$$A = F\,\dfrac{i}{(1+i)^{n}-1}$$

Equal deposits $A$ that accumulate to a future amount $F$ (e.g., a replacement reserve). Note $(A/P)=(A/F)+i$.

Uniform-series compound amount (F/A)

$$F = A\,\dfrac{(1+i)^{n}-1}{i}$$

Future value of an annuity $A$ after $n$ periods. Reciprocal of $(A/F)$.

Gradient present worth (P/G)

$$P = G\left[\dfrac{(1+i)^{n}-1}{i^{2}(1+i)^{n}} - \dfrac{n}{i\,(1+i)^{n}}\right]$$

Present worth of an arithmetic gradient $0,G,2G,\dots,(n-1)G$. $G$ = constant yearly increase.

Gradient to uniform series (A/G)

$$A = G\left[\dfrac{1}{i} - \dfrac{n}{(1+i)^{n}-1}\right]$$

Equivalent flat annuity of an arithmetic gradient. Add to a base $A_1$ to levelize a rising cost.

Effective annual interest rate

$$i_e = \left(1 + \dfrac{r}{m}\right)^{m} - 1$$

Converts nominal annual rate $r$ with $m$ compoundings/yr to the effective annual rate $i_e$. $m\to\infty$ gives $i_e=e^{r}-1$.

Equivalent uniform annual cost (EUAC)

$$\text{EUAC} = P\,(A/P,i,n) + A_{\text{O\&M}} - S\,(A/F,i,n)$$

Levelized annual life-cycle cost: capital recovery plus annual O&M less salvage credit. $S$ = salvage at year $n$.

Capitalized cost (perpetual)

$$P = \dfrac{A}{i}$$

Present worth of a uniform cost continuing forever ($n\to\infty$); used for permanent works and replacement-in-perpetuity comparisons.

Where it lives: NCEES FE Reference Handbook — Engineering Economics · Newnan, Lavelle & Eschenbach, Engineering Economic Analysis · Davis & Cornwell, Introduction to Environmental Engineering

Ideal Reactors: Batch, CSTR, and Plug Flow

The three ideal reactors, the general mass balance, hydraulic residence time, and why a PFR always beats a CSTR for first-order conversion.

General mass balance (constant V)

$$V\dfrac{dC}{dt} = Q\,C_0 - Q\,C + r_A V$$

Accumulation = in − out + generation. $r_A<0$ for a consumed reactant; set the left side to zero for steady state.

Batch reactor, first order

$$C_A = C_{A0}\,e^{-kt}$$

No flow; concentration decays with reaction time $t$. $k$ in $\text{time}^{-1}$.

CSTR design equation

$$V = \dfrac{Q\,(C_0 - C)}{-r_A}$$

$-r_A$ evaluated at EXIT (tank) conditions. $V$ = volume, $Q$ = volumetric flow.

CSTR, first order

$$\dfrac{C}{C_0} = \dfrac{1}{1 + k\tau}$$

Fraction remaining for a single completely-mixed tank; $\tau=V/Q$.

PFR design integral

$$\tau = -\int_{C_0}^{C}\dfrac{dC}{-r_A}$$

Each slice acts as a batch element; reduces to $e^{-k\tau}$ for first order.

PFR, first order

$$\dfrac{C}{C_0} = e^{-k\tau}$$

Fraction remaining for plug flow; identical to a batch reactor with $t=\tau$.

Hydraulic residence time

$$\tau = \dfrac{V}{Q}$$

Mean detention time (space-time). Reciprocal is space-velocity $SV=1/\tau$.

Zero-order conversion

$$C = C_0 - k\tau \quad (\text{CSTR or PFR})$$

For a zero-order reaction the rate is constant, so CSTR and PFR give the SAME effluent; $k$ in mass/(vol·time).

Damköhler number

$$Da = k\tau$$

Dimensionless ratio of reaction rate to throughput; governs conversion in both continuous reactors.

Where it lives: NCEES FE Reference Handbook — Chemical Engineering (Reactor Design; Flow Reactors, Steady State) · NCEES FE Reference Handbook — Environmental Engineering (hydraulic residence time) · Metcalf & Eddy, Wastewater Engineering: Treatment and Resource Recovery · Davis & Cornwell, Introduction to Environmental Engineering

Chemical Equilibrium, Acids–Bases, and pH

Balance reactions and find limiting reactants, then use the equilibrium constant, K_w, and pK_a to predict pH, buffering, and as-CaCO3 equivalents.

Moles from mass

$$n = \dfrac{m}{M}$$

$n$ = moles, $m$ = mass (g), $M$ = molar mass (g/mol). The starting point of every stoichiometry problem.

Equilibrium constant

$$K_{eq} = \dfrac{[C]^{c}[D]^{d}}{[A]^{a}[B]^{b}}$$

Ratio of product to reactant activities for $aA+bB\rightleftharpoons cC+dD$; pure solids/liquids count as 1. Dimensionless in concentration form here.

Ion product of water

$$K_w = [\text{H}^+][\text{OH}^-] = 1.0\times10^{-14}$$

At $25^\circ\text{C}$. Fixes $[\text{OH}^-]$ once $[\text{H}^+]$ is known (mol/L).

pH and pOH

$$\text{pH} = -\log_{10}[\text{H}^+], \quad \text{pH}+\text{pOH}=14$$

$[\text{H}^+]$ in mol/L. Acids: pH < 7; bases: pH > 7 at $25^\circ\text{C}$.

Acid dissociation constant

$$K_a = \dfrac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}, \quad \text{p}K_a = -\log_{10}K_a$$

Strength of a weak acid HA. Larger $K_a$ (smaller $\text{p}K_a$) = stronger acid.

Weak-acid pH approximation

$$[\text{H}^+] \approx \sqrt{K_a\,C_a}$$

$C_a$ = analytical acid concentration (mol/L). Valid when dissociation is < ~5% of $C_a$.

Henderson–Hasselbalch

$$\text{pH} = \text{p}K_a + \log_{10}\!\dfrac{[\text{A}^-]}{[\text{HA}]}$$

Buffer pH from the conjugate base-to-acid ratio. $\text{pH}=\text{p}K_a$ when the two are equal.

Solubility product

$$K_{sp} = [\text{A}^{+}]^{m}[\text{B}^{-}]^{n}$$

For $\text{A}_m\text{B}_n(s)\rightleftharpoons m\text{A}^{+}+n\text{B}^{-}$. Precipitation occurs when the ion product exceeds $K_{sp}$.

Equivalent weight and normality

$$EW = \dfrac{\mathrm{MW}}{z}, \qquad N = M\,z$$

$\mathrm{MW}$ = molar mass (g/mol); $M$ = molarity (mol/L); $z$ = charge or protons exchanged per mole. Equivalent weight (g/eq) is the basis for the as-CaCO3 system; normality $N = M\times z$.

Conversion to as-CaCO3

$$C_{\text{as CaCO}_3} = C_{species}\times\dfrac{50}{EW_{species}}$$

$EW(\text{CaCO}_3)=50$ g/eq. Use to add hardness/alkalinity ions on a common basis (mg/L as CaCO3).

Alkalinity (carbonate system)

$$\text{Alk} = [\text{HCO}_3^-] + 2[\text{CO}_3^{2-}] + [\text{OH}^-] - [\text{H}^+]$$

Acid-neutralizing capacity in eq/L; multiply by $50{,}000$ to express as mg/L as CaCO3.

Where it lives: NCEES FE Reference Handbook — Chemistry and Biology · NCEES FE Reference Handbook — Environmental Engineering (equivalent-weight / common radicals table) · Davis & Cornwell, Introduction to Environmental Engineering · Sawyer, McCarty & Parkin, Chemistry for Environmental Engineering and Science · Snoeyink & Jenkins, Water Chemistry

Dose–Response Toxicity (Carcinogens and Noncarcinogens)

How a contaminant dose becomes a quantified health risk along two NCEES tracks: linear-no-threshold cancer risk (Risk = CDI×CSF) and the noncancer hazard quotient (HI = CDI/RfD).

Cancer risk (linear, no threshold)

$$\text{Risk} = \text{CDI}\times\text{CSF}$$

Excess lifetime cancer probability (dimensionless). CDI = chronic daily intake in $\text{mg}/(\text{kg}\cdot\text{day})$; CSF = cancer slope factor in $[\text{mg}/(\text{kg}\cdot\text{day})]^{-1}$. Acceptable EPA range $10^{-6}$ to $10^{-4}$.

Hazard quotient

$$\text{HQ} = \dfrac{\text{CDI}}{\text{RfD}}$$

Fraction of the safe dose received for one noncarcinogen and pathway (dimensionless). RfD = reference dose in $\text{mg}/(\text{kg}\cdot\text{day})$.

Hazard index

$$\text{HI} = \sum_i \dfrac{\text{CDI}_i}{\text{RfD}_i}$$

Sum of hazard quotients over all noncarcinogens and routes. $\text{HI}>1.0$ indicates possible adverse effect.

Reference dose from NOAEL

$$\text{RfD} = \dfrac{\text{NOAEL}}{\text{UF}}$$

Chronic safe daily dose, $\text{mg}/(\text{kg}\cdot\text{day})$. NOAEL = no-observed-adverse-effect level from the animal dose-response curve; UF = total uncertainty factor (often $10$–$10{,}000$).

Safe human dose

$$\text{SHD} = \text{RfD}\cdot W = \dfrac{\text{NOAEL}\cdot W}{\text{UF}}$$

Allowable daily intake in $\text{mg}/\text{day}$ for a person of body weight $W$ (kg).

Median lethal dose / concentration

$$\text{LD}_{50},\ \text{LC}_{50}$$

Dose ($\text{mg}/\text{kg}$) or air concentration killing $50\%$ of test animals. Smaller value = more toxic. $\text{LD}_{10}/\text{LC}_{10}$ defined at $10\%$.

Hazard index pass/fail

$$\text{HI} \le 1.0 \ \Rightarrow\ \text{acceptable}, \qquad \text{HI} > 1.0 \ \Rightarrow\ \text{possible adverse effect}$$

EPA decision rule for noncarcinogenic risk characterization.

Acceptable carcinogenic risk band

$$10^{-6} \le \text{Risk} \le 10^{-4}$$

EPA range for incremental lifetime cancer risk used in site cleanup decisions.

Where it lives: NCEES FE Reference Handbook — Safety (Risk Assessment/Toxicology) · EPA, Risk Assessment Guidance for Superfund, Vol. I (Human Health Evaluation Manual, Part A) · Davis & Cornwell, Introduction to Environmental Engineering

Continuity, Bernoulli, and the Energy Equation

Conservation of mass and energy along a streamline, the head form with pump, turbine, and friction terms, EGL/HGL, and the impulse-momentum principle.

Continuity (volumetric)

$$Q = A_1 v_1 = A_2 v_2$$

Incompressible steady flow; $Q$ in m$^3$/s, $A$ in m$^2$, $v$ in m/s. Velocity rises where area shrinks.

Continuity (mass)

$$\dot m = \rho A v = \text{constant}$$

General steady form; use when density changes (gas/air streams), $\dot m$ in kg/s.

Energy equation (head form)

$$\frac{p_1}{\gamma} + \frac{v_1^2}{2g} + z_1 + h_p = \frac{p_2}{\gamma} + \frac{v_2^2}{2g} + z_2 + h_t + h_f$$

Each term a length; $h_p$ pump head added, $h_t$ turbine head removed, $h_f$ friction + minor losses.

Bernoulli equation

$$\frac{p}{\gamma} + \frac{v^2}{2g} + z = \text{constant}$$

Frictionless, no machine, along a streamline. Use for nozzles, venturis, tank efflux.

Velocity head

$$h_v = \frac{v^2}{2g}$$

Kinetic energy per unit weight; the vertical gap between EGL and HGL (m or ft).

Pressure-head / pressure relation

$$p_1 - p_2 = \gamma\,h_f = \rho g\,h_f$$

Pressure drop in a horizontal constant-area pipe equals the friction head times specific weight.

Pump head and fluid power

$$\dot W_{\text{fluid}} = \gamma Q h_p, \qquad \dot W_{\text{brake}} = \frac{\gamma Q h_p}{\eta}$$

Fluid (hydraulic) power vs. shaft/brake power; $\eta$ is pump efficiency (0-1).

Impulse-momentum

$$\Sigma F = \rho Q\,(v_{\text{out}} - v_{\text{in}})$$

Per direction; gives force on bends, nozzles, plates. Include $pA$ pressure forces on bends.

Force of a jet on a fixed plate

$$F = \rho Q v$$

Normal flat plate stops the jet ($v_{\text{out}}=0$ in flow direction); $F$ in N.

Where it lives: NCEES FE Reference Handbook — Fluid Mechanics · NCEES FE Reference Handbook — Civil Engineering · Munson, Young & Okiishi, Fundamentals of Fluid Mechanics

Thermodynamic Laws, Efficiency, and COP

The first and second laws, the Carnot absolute-temperature ceiling, refrigeration and heat-pump COP, and the one-way direction of entropy.

First law for a cycle

$$Q_H = W + Q_L$$

Over one complete cycle internal energy returns to its start; heat in equals net work plus heat rejected. Energies in consistent units (J, kJ, or W for rates).

Sensible-heat rate (first law, flowing stream)

$$\dot{Q} = \dot{m}\,c_p\,\Delta T$$

$\dot{m}$ = mass flow (kg/s), $c_p$ = specific heat (J/kg·K), $\Delta T = T_{out}-T_{in}$ (K or °C, same magnitude). Gives rate in W.

Thermal efficiency

$$\eta = \frac{W}{Q_H} = 1 - \frac{Q_L}{Q_H}$$

Fraction of input heat converted to net work. Denominator is the heat supplied $Q_H$, never $W$.

Carnot (maximum) efficiency

$$\eta_{Carnot} = 1 - \frac{T_L}{T_H}$$

Upper bound for any engine between reservoirs at absolute temperatures $T_H$ and $T_L$ (K or °R). No real engine can exceed it.

Second-law (exergetic) efficiency

$$\eta_{II} = \frac{\eta_{actual}}{\eta_{Carnot}}$$

How close a real device comes to the reversible ideal; dimensionless, always < 1 for real machines.

Refrigeration COP

$$\mathrm{COP}_{ref} = \frac{Q_L}{W}$$

Useful cooling (heat removed from cold space $Q_L$) per unit work input. For refrigerators and air conditioners.

Heat-pump COP

$$\mathrm{COP}_{HP} = \frac{Q_H}{W} = \mathrm{COP}_{ref} + 1$$

Useful heating (heat delivered to warm space $Q_H$) per unit work input.

Carnot COP ceilings

$$\mathrm{COP}_{HP,\max} = \frac{T_H}{T_H-T_L}, \quad \mathrm{COP}_{ref,\max} = \frac{T_L}{T_H-T_L}$$

Maximum COP from a reversed-Carnot cycle; absolute temperatures only.

Increase-of-entropy principle

$$\Delta S_{total} = \Delta S_{sys} + \Delta S_{surr} \ge 0$$

Total entropy never decreases; equality only for a reversible process. Fixes the hot-to-cold direction of spontaneous heat flow.

Reservoir entropy change

$$\Delta S_{res} = \frac{Q}{T_{res}}$$

Entropy change of a thermal reservoir exchanging heat $Q$ at constant absolute temperature $T_{res}$ (J/K).

Where it lives: NCEES FE Reference Handbook — Thermodynamics · Moran, Shapiro, Boettner & Bailey, Fundamentals of Engineering Thermodynamics · Davis & Cornwell, Introduction to Environmental Engineering

The Streeter–Phelps Dissolved-Oxygen Sag

Model the dissolved-oxygen sag below a BOD outfall: deoxygenation vs reaeration, the deficit equation, the critical point, temperature correction, and nutrient limitation.

BOD exertion (first-order)

$$\text{BOD}_t = L_o\left(1 - e^{-k t}\right)$$

Oxygen demand exerted by time $t$ (days); $L_o$ = ultimate BOD (mg/L), $k$ = base-$e$ decay constant (d$^{-1}$). A base-10 $K$ relates as $k = 2.303\,K$.

Mixed (initial) ultimate BOD

$$L_a = \dfrac{Q_r L_r + Q_w L_w}{Q_r + Q_w}$$

Flow-weighted ultimate BOD entering the reach; $Q$ = flow, $L$ = ultimate BOD, subscripts $r$ = river, $w$ = waste.

Initial DO deficit

$$D_a = \text{DO}_{sat} - \dfrac{Q_r\,\text{DO}_r + Q_w\,\text{DO}_w}{Q_r + Q_w}$$

Saturation DO minus the flow-weighted mixed DO (mg/L) at the outfall; the deficit the reach starts with.

Streeter–Phelps deficit

$$D = \dfrac{k_d L_a}{k_r - k_d}\left(e^{-k_d t} - e^{-k_r t}\right) + D_a\,e^{-k_r t}$$

DO deficit (mg/L) at travel time $t$ (days). $k_d$ = deoxygenation, $k_r$ = reaeration (both base-$e$, d$^{-1}$). Then $\text{DO} = \text{DO}_{sat} - D$.

Critical time

$$t_c = \dfrac{1}{k_r - k_d}\ln\!\left[\dfrac{k_r}{k_d}\left(1 - \dfrac{D_a\,(k_r - k_d)}{k_d\,L_a}\right)\right]$$

Travel time (days) to the minimum DO, where $dD/dt = 0$. Requires $k_r \ne k_d$.

Critical deficit

$$D_c = \dfrac{k_d}{k_r}\,L_a\,e^{-k_d t_c}$$

Maximum DO deficit (mg/L), at $t_c$. Minimum dissolved oxygen is $\text{DO}_{min} = \text{DO}_{sat} - D_c$.

Temperature correction

$$k_T = k_{20}\,\theta^{\,T-20}$$

Rate constant at stream temperature $T$ ($^\circ$C). $\theta_{k_d} = 1.135$ (4–20°C) or $1.056$ (21–30°C); $\theta_{k_r} = 1.024$.

Mass loading

$$M\,(\text{lb/day}) = C\,(\text{mg/L}) \times Q\,(\text{MGD}) \times 8.34$$

Converts a concentration and flow to a daily mass load; the $8.34$ factor carries units lb·L/(mg·MG). Used to set BOD load limits.

Where it lives: NCEES FE Reference Handbook — Environmental Engineering: Stream Modeling (Streeter–Phelps) · NCEES FE Reference Handbook — Environmental Engineering: Kinetic Temperature Corrections · Davis & Cornwell, Introduction to Environmental Engineering · Clean Water Act (CWA)

Darcy’s Law and Seepage Velocity

Darcy’s law, hydraulic conductivity, transmissivity and storativity, and the crucial gap between the Darcy flux and the seepage velocity a contaminant actually travels at.

Darcy’s law (discharge)

$$Q = -KA\,\dfrac{dh}{dx}$$

$Q$ = volumetric discharge (m³/s, ft³/s); $K$ = hydraulic conductivity (m/s, ft/s); $A$ = total cross-sectional area; $dh/dx$ = hydraulic gradient. Minus sign: flow goes down-gradient.

Specific discharge (Darcy flux)

$$q = \dfrac{Q}{A} = -K\,\dfrac{dh}{dx} = K\,i$$

Discharge per unit total area; units of velocity but NOT a particle speed. Use for flux/mass-loading. $i = -dh/dx$ is the gradient (dimensionless).

Seepage (pore) velocity

$$v = \dfrac{q}{n_e} = \dfrac{K\,i}{n_e}$$

Actual average water velocity through the interconnected pores; $n_e$ = effective porosity. This is the velocity for travel-time and plume problems.

Hydraulic gradient

$$i = -\dfrac{dh}{dx} = \dfrac{h_1 - h_2}{L}$$

Head drop per unit distance along the flow path, dimensionless. Use consistent length units for head and distance.

Transmissivity

$$T = K\,b$$

Conductivity times saturated thickness $b$; units $L^2/T$ (m²/d, ft²/s). Aquifer-scale measure of water-transmitting capacity.

Aquifer discharge through a width

$$Q = T\,i\,w = K\,b\,i\,w$$

Discharge through a vertical strip of aquifer of width $w$ and thickness $b$; folds thickness into $T$.

Storativity (storage coefficient)

$$S = S_s\,b \quad (\text{confined}), \qquad S \approx S_y \quad (\text{unconfined})$$

Volume released per unit area per unit head drop, dimensionless. Confined $S \sim 10^{-5}\text{–}10^{-3}$; unconfined $S \approx S_y \sim 0.05\text{–}0.30$.

Travel time

$$t = \dfrac{L}{v} = \dfrac{L\,n_e}{K\,i}$$

Time for water (or a non-sorbing tracer) to travel distance $L$ at the seepage velocity. Uses $n_e$, not $q$.

Where it lives: NCEES FE Reference Handbook — Civil Engineering: Darcy's Law, Transmissivity, Storativity · Fetter, Applied Hydrogeology · Davis & Cornwell, Introduction to Environmental Engineering

Activated Sludge: F/M, SRT, and Clarifiers

The workhorse of biological wastewater treatment, sized by F/M and solids retention time, balanced across the aeration basin and coupled secondary clarifier.

Food-to-microorganism ratio

$$\text{F/M} = \dfrac{Q_0\,S_0}{V\,X_A}$$

Applied BOD load per unit biomass per day (kg BOD/(kg MLSS·d)). $Q_0$=influent flow, $S_0$=influent BOD, $V$=basin volume, $X_A$=MLSS. Conventional 0.2–0.4.

Solids retention time (SRT)

$$\theta_c = \dfrac{V\,X_A}{Q_w X_w + Q_e X_e}$$

Mean cell residence time / sludge age (d). Basin solids inventory divided by solids leaving in waste + effluent.

Hydraulic retention time

$$\theta = \dfrac{V}{Q}$$

Liquid residence time in the basin (h). Typically 4–8 h for conventional activated sludge; distinct from SRT.

Biomass concentration (kinetics)

$$X_A = \dfrac{\theta_c\,Y\,(S_0 - S_e)}{\theta\,(1 + k_d\,\theta_c)}$$

Design MLSS from kinetics. $Y$=yield (kg VSS/kg BOD), $k_d$=decay rate (d⁻¹), $S_e$=effluent BOD.

Clarifier solids mass balance

$$(Q_0 + Q_R)\,X_A = Q_e X_e + Q_R X_r + Q_w X_w$$

Steady-state solids around the secondary clarifier. $X_r$=RAS (underflow) solids, $X_w$=waste solids, $X_e$=effluent SS.

Recycle ratio

$$R = \dfrac{Q_R}{Q_0} = \dfrac{X_A}{X_r - X_A}$$

Return-sludge ratio to hold target MLSS (right form neglects effluent/waste solids). Typically 0.25–1.0.

Sludge volume index

$$\text{SVI} = \dfrac{V_{30}\,(\text{mL/L})\times 1000}{X_{\text{MLSS}}\,(\text{mg/L})}$$

Volume (mL) of 1 g settled solids after 30 min. <100 good settling, >150 bulking. Underflow X_r ≈ 10⁶/SVI.

Solids loading rate

$$\text{SLR} = \dfrac{(Q_0 + Q_R)\,X_A}{A}$$

Thickening criterion for the secondary clarifier (kg/(m²·d) or lb/(ft²·d)); often controls over overflow rate.

Volumetric organic loading

$$L_v = \dfrac{Q_0\,S_0}{V}$$

BOD load per basin volume (kg BOD/(m³·d)). Conventional ~0.3–0.6.

Where it lives: NCEES FE Reference Handbook — Environmental Engineering · Metcalf & Eddy, Wastewater Engineering: Treatment and Resource Recovery · Davis & Cornwell, Introduction to Environmental Engineering

The Gaussian Plume and Atmospheric Stability

Predict downwind ground-level concentrations from an elevated source using the Gaussian dispersion model, Pasquill stability classes, effective stack height, and the lapse rate.

Gaussian plume (general)

$$C = \dfrac{Q}{2\pi u \sigma_y \sigma_z}\exp\!\left(-\dfrac{y^2}{2\sigma_y^2}\right)\!\left[\exp\!\left(-\dfrac{(z-H)^2}{2\sigma_z^2}\right)+\exp\!\left(-\dfrac{(z+H)^2}{2\sigma_z^2}\right)\right]$$

Concentration $C$ ($\mu\text{g/m}^3$) at $(x,y,z)$ from a continuous elevated point source. $Q$ in $\mu\text{g/s}$, $u$ in m/s, $\sigma_y,\sigma_z$ in m, $H$ effective stack height in m.

Ground-level, centerline concentration

$$C(x,0,0) = \dfrac{Q}{\pi u \sigma_y \sigma_z}\exp\!\left(-\dfrac{H^2}{2\sigma_z^2}\right)$$

Set $z=0$ (ground reflection doubles the term) and $y=0$ (on centerline). The everyday working form for worst-case ground exposure.

Effective stack height

$$H = h + \Delta h$$

$h$ = physical stack height (m), $\Delta h$ = plume rise (m) from buoyancy and exit momentum. Always use $H$, not $h$, in the dispersion equation.

Maximum ground-level concentration

$$C_{\max} = \dfrac{Q}{\pi e\, u\, \sigma_y \sigma_z}, \qquad \text{at } \sigma_z = \dfrac{H}{\sqrt{2}}$$

Peak ground concentration downwind for neutral atmospheric conditions (the handbook's stated case); the exponential becomes $e^{-1}$ when $\sigma_z=H/\sqrt2$. Other stability classes: the handbook's $(Cu/Q)_{\max}$ chart. Evaluate $\sigma_y$ at that same $x$.

Lapse rate

$$\Gamma = -\dfrac{\Delta T}{\Delta z}, \qquad \Gamma_{AD} = 0.98\,^\circ\text{C}/100\,\text{m}$$

Environmental lapse rate $\Gamma$ versus dry adiabatic $\Gamma_{AD}$. $\Delta T$ = temperature change, $\Delta z$ = elevation change.

Stability criterion

$$\Gamma > \Gamma_{AD}\ \Rightarrow\ \text{unstable}; \quad \Gamma = \Gamma_{AD}\ \Rightarrow\ \text{neutral}; \quad \Gamma < \Gamma_{AD}\ \Rightarrow\ \text{stable}$$

Inversion ($\Gamma<0$) is the strongly stable limit that caps mixing and traps the plume.

ppb / concentration conversion

$$\text{ppb} = \dfrac{(\mu\text{g/m}^3)\,RT}{P\,\text{MW}}$$

Convert mass concentration to volume mixing ratio. $R=0.0821\,\text{L·atm/(mol·K)}$, $T$ in K, $P$ in atm, MW in g/mol.

Where it lives: NCEES FE Reference Handbook — Environmental Engineering · Turner, Workbook of Atmospheric Dispersion Estimates · Davis & Cornwell, Introduction to Environmental Engineering

Landfills: Airspace, Leachate, and Gas

Size landfill airspace from compacted density and compaction ratio, estimate leachate by water balance, predict landfill gas from anaerobic stoichiometry, and value MSW as fuel.

Airspace from compacted density

$$V = \dfrac{M}{\rho_c}$$

Volume of airspace a waste mass $M$ consumes; $\rho_c$ = in-place compacted density ($\approx 1{,}000$–$1{,}400\,\text{lb/yd}^3$). Use compacted, never loose, density.

Compaction ratio

$$CR = \dfrac{V_{loose}}{V_{compacted}} = \dfrac{\rho_c}{\rho_{loose}}$$

How many loose volumes collapse into one compacted volume; about $4$–$8$ from curbside-loose to in-place landfill density. Dimensionless.

Volume reduction

$$VR = \dfrac{V_i - V_f}{V_i}\times 100\%$$

Percent volume saved by compaction or baling; $V_i$ initial (loose), $V_f$ final (compacted). Related to $CR$ by $VR = 1 - 1/CR$.

Overburden specific weight

$$SW_p = SW_i + \dfrac{p}{a + b\,p}$$

In-place specific weight ($\text{lb/yd}^3$) at overburden pressure $p$ ($\text{lb/in}^2$); $SW_i$ initial compacted ($\approx 1{,}000$), $a,b$ empirical constants.

Site life

$$L = \dfrac{V_{airspace}}{\dot V_{waste} + \dot V_{cover}}$$

Years of capacity = permitted airspace divided by annual airspace consumed by compacted waste plus cover soil.

Cover water balance (percolation)

$$PER_{sw} = P - R - ET - \Delta S_{LC}$$

Leachate percolation (in.) = precipitation $-$ runoff $-$ evapotranspiration $-$ change in cover-soil storage. At field capacity $\Delta S_{LC}\to 0$.

Clay-liner breakthrough time

$$t = \dfrac{d^{2}\,\eta}{K\,(d + h)}$$

Years for leachate to penetrate a clay liner; $d$ thickness (ft), $\eta$ porosity, $K$ hydraulic conductivity (ft/yr), $h$ leachate head (ft).

Anaerobic stabilization (Buswell)

$$m = \dfrac{4a + b - 2c - 3d}{8}, \quad s = \dfrac{4a - b + 2c + 3d}{8}$$

Methane ($m$) and carbon-dioxide ($s$) moles from complete stabilization of $\mathrm{C_aH_bO_cN_d}$. Ammonia coefficient $= d$; water $r = (4a - b - 2c + 3d)/4$.

First-order gas generation

$$Q_{CH_4} = k\,L_0\,M\,e^{-k\,t}$$

Methane rate from a waste mass $M$; $L_0$ = methane potential ($\text{m}^3/\text{Mg}$), $k$ = decay constant ($\text{yr}^{-1}$), $t$ = age. Basis of EPA LandGEM.

Gas flux through cover (Fick)

$$N_A = D\,\eta_{gas}^{4/3}\,\dfrac{C_{A,fill} - C_{A,atm}}{L}$$

Diffusive (outward) flux of gas $A$ through cover of depth $L$; $D$ diffusion coefficient, $\eta_{gas}$ gas-filled porosity (the $\eta_{gas}^{4/3}$ factor is the porosity (tortuosity) correction that converts the free-gas diffusion coefficient $D$ into the effective diffusivity $D\,\eta_{gas}^{4/3}$). Writing the driving force as $C_{fill}-C_{atm}$ makes outward flux positive.

Modified Dulong heating value

$$HHV = 145\,C + 610\left(H - \tfrac{O}{8}\right) + 40\,S + 10\,N$$

Higher heating value (Btu/lb) from ultimate-analysis mass percents $C,H,O,S,N$. The $O/8$ term removes hydrogen already bound as water.