FE Chemical formula sheet

Numerical Methods, Error and Significant Figures

Newton-Raphson roots, trapezoidal and Simpson integration, Taylor truncation, relative-error propagation, and significant-figure discipline for FE-Chem.

Newton-Raphson iteration

$$x_{j+1} = x_j - \dfrac{f(x_j)}{f'(x_j)}$$

Next root estimate; $f'$ is the derivative at the current guess. Quadratic convergence near a simple root; needs $f'\neq 0$ and a good seed.

Taylor series about a

$$f(x) = \sum_{k=0}^{\infty}\dfrac{f^{(k)}(a)}{k!}(x-a)^k$$

Basis of truncation-error analysis. Truncating after the linear term gives the Newton tangent. $a=0$ is the Maclaurin series.

Trapezoidal rule (n strips)

$$\int_a^b f\,dx \approx \dfrac{\Delta x}{2}\Big[f_0 + 2f_1 + 2f_2 + \cdots + 2f_{n-1} + f_n\Big]$$

Straight-chord approximation; error $\propto \Delta x^2$. $\Delta x=(b-a)/n$.

Simpson's 1/3 rule

$$\int_a^b f\,dx \approx \dfrac{\Delta x}{3}\Big[f_0 + 4f_1 + 2f_2 + \cdots + 4f_{n-1} + f_n\Big]$$

Parabolic fit; exact for cubics, error $\propto \Delta x^4$. $n$ must be even (odd number of points).

Step width

$$\Delta x = \dfrac{b-a}{n}$$

Uniform spacing for $n$ subintervals between limits $a$ and $b$.

Relative error

$$\varepsilon_{rel} = \dfrac{|x_{approx}-x_{true}|}{|x_{true}|}$$

Dimensionless; multiply by 100 for percent. Use the previous iterate as the reference when the true value is unknown.

Error propagation (products)

$$\left|\dfrac{\Delta y}{y}\right| = \sum_i |a_i|\left|\dfrac{\Delta x_i}{x_i}\right|$$

Worst-case relative error for $y=\prod x_i^{a_i}$; fractional errors add, weighted by exponents.

Error propagation (RSS)

$$\dfrac{\Delta y}{y} = \sqrt{\sum_i a_i^2\left(\dfrac{\Delta x_i}{x_i}\right)^2}$$

Probable error when inputs are independent and random; always smaller than the linear sum.

Error propagation (sums)

$$y = \sum_i c_i x_i \;\Rightarrow\; \Delta y = \sqrt{\sum_i c_i^2\,(\Delta x_i)^2}$$

For additive combinations the absolute errors (not relative) combine.

Where it lives: NCEES FE Reference Handbook — Mathematics (Numerical Methods, Numerical Integration, Taylor's Series) · NCEES FE Reference Handbook — Units and Conversion Factors (Significant Figures, p.2) · NCEES FE Reference Handbook — Engineering Probability and Statistics (Measurement Uncertainty / Propagation of Error, RSS form) · Chapra & Canale, Numerical Methods for Engineers

Distributions, Confidence Intervals and Control Limits

Discrete and continuous distributions, the binomial and normal, mean/median/mode and spread, the confidence interval for a mean, and X-bar/R control limits.

Sample mean

$$\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i$$

Arithmetic mean (balance point) of $n$ observations; the statistic every interval and chart centers on.

Sample variance and standard deviation

$$s^2=\frac{1}{n-1}\sum_{i=1}^{n}\left(X_i-\bar{X}\right)^2,\qquad s=\sqrt{s^2}$$

Spread of a sample; divide by $n-1$ (use $N$ for a full population). Units of $s$ match the data; $s^2$ is squared units.

Binomial mass function

$$P(x)=C(n,x)\,p^{x}(1-p)^{\,n-x}$$

Probability of exactly $x$ successes in $n$ independent trials, each with success probability $p$. $C(n,x)=n!/[x!(n-x)!]$.

Binomial mean and variance

$$\mu=np,\qquad \sigma^{2}=npq,\quad q=1-p$$

Expected count and its variance (a count, not a fraction). Standard deviation $\sigma=\sqrt{npq}$.

Normal probability density

$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}$$

Gaussian curve, mean $\mu$, standard deviation $\sigma$. Never integrated by hand on the FE; standardize instead.

Standard-normal transform

$$z=\frac{x-\mu}{\sigma}$$

Converts any normal value to the unit normal so the handbook table gives $F(z)$ (left area), $R(z)$ (right area), $W(z)$ (within $\pm z$).

Within-band probability

$$P(-z\le Z\le z)=W(z)=2F(z)-1$$

Fraction of a normal population inside $\mu\pm z\sigma$; e.g. $W(1.96)\approx0.95$.

Standard error of the mean

$$\sigma_{\bar X}=\frac{\sigma}{\sqrt{n}}\;\;(\text{or }s/\sqrt{n})$$

Spread of the sample mean; shrinks only as $\sqrt{n}$. Drives both the confidence interval and the X-bar chart width.

Confidence interval, sigma known

$$\bar{X}\pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}$$

Interval for $\mu$ when the population $\sigma$ is known; $z_{0.025}=1.960$ for 95%, $1.645$ for 90%.

Confidence interval, sigma unknown

$$\bar{X}\pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}$$

Interval for $\mu$ using sample $s$; $t$ has $n-1$ degrees of freedom and is wider than $z$ for small samples.

X-bar chart limits

$$\text{UCL}=\overline{\overline{X}}+A_2\bar{R},\qquad \text{LCL}=\overline{\overline{X}}-A_2\bar{R}$$

Three-sigma limits on subgroup averages; $\overline{\overline{X}}$ is the grand mean, $\bar{R}$ the mean range, $A_2$ from the handbook table for subgroup size $n$.

R chart limits and process sigma

$$\text{UCL}_R=D_4\bar{R},\quad \text{LCL}_R=D_3\bar{R},\qquad \hat{\sigma}=\frac{\bar{R}}{d_2}$$

Limits on subgroup range; $D_3,D_4,d_2$ from the table. $\hat{\sigma}=\bar{R}/d_2$ estimates the process standard deviation.

Where it lives: NCEES FE Reference Handbook — Engineering Probability and Statistics · NCEES FE Reference Handbook — Statistical Quality Control (Average and Range Charts) · Montgomery & Runger, Applied Statistics and Probability for Engineers

Newtonian Dynamics, Work, Energy and Power

Newton's second law with friction, the work-energy theorem, and impulse-momentum for particles — the three bookkeeping methods that solve almost every FE dynamics question.

Newton's second law (particle)

$$\sum \mathbf{F} = m\mathbf{a} = m\dfrac{d\mathbf{v}}{dt}$$

Net external force equals mass times acceleration. SI: $F$ in N, $m$ in kg, $a$ in m/s$^2$. Apply per axis from a free-body diagram.

Newton's law in USCS (gc form)

$$F = \dfrac{m\,a}{g_c}, \quad g_c = 32.174\ \tfrac{\text{lbm}\cdot\text{ft}}{\text{lbf}\cdot\text{s}^2}$$

Required whenever mass is in lbm and force in lbf; $g_c$ reconciles the two. Weight $W = mg/g_c$.

Coulomb friction

$$F_f \le \mu_s N, \qquad F_f = \mu_k N \ (\text{sliding})$$

Friction opposes relative motion. $N$ = actual normal force ($mg\cos\theta$ on an incline). $\mu_s,\mu_k$ dimensionless.

Constant-acceleration kinematics

$$v = v_0 + at, \quad v^2 = v_0^2 + 2a(x-x_0)$$

Links velocity, acceleration, time, and displacement for constant $a$. Use with $\sum F = ma$ to bridge force and motion.

Kinetic energy

$$T = \tfrac{1}{2}mv^{2}$$

Energy of motion of a particle. $v$ = speed (m/s), result in J. Note $v$ is squared — never linear.

Work of a force

$$U = \int \mathbf{F}\cdot d\mathbf{r} = Fs\cos\theta\ (\text{constant } F)$$

$\theta$ = angle between force and displacement; forces perpendicular to motion do zero work.

Work-energy theorem

$$U_{1\to 2} = \tfrac{1}{2}mv_2^{2} - \tfrac{1}{2}mv_1^{2}$$

Net work equals change in kinetic energy. Time-free — ideal for force-and-distance to speed problems.

Conservation of energy

$$T_1 + V_1 + U_{1\to 2}^{nc} = T_2 + V_2$$

$V = mgh + \tfrac{1}{2}ks^2$. $U_{1\to 2}^{nc}$ = work of nonconservative forces only (negative for friction; zero for conservative systems). Distinct from the work-energy theorem $U_{1\to 2}$, which is the net work of all forces — do not equate the two.

Linear impulse-momentum

$$m\mathbf{v}_2 = m\mathbf{v}_1 + \int_{t_1}^{t_2}\mathbf{F}\,dt$$

Impulse of net force equals change in momentum. For constant $F$: $F\,\Delta t = m\,\Delta v$.

Conservation of momentum (impact)

$$m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'$$

Holds when external impulse is negligible during impact. KE conserved only if $e=1$.

Coefficient of restitution

$$e = \dfrac{(v_2' - v_1')_n}{(v_1 - v_2)_n}, \ 0\le e\le 1$$

$e=1$ perfectly elastic, $e=0$ perfectly plastic (bodies stick). Applied along the line of impact.

Power and efficiency

$$P = \mathbf{F}\cdot\mathbf{v}, \qquad \eta = \dfrac{P_{out}}{P_{in}}$$

Power in W (or ft·lbf/s); $1\,\text{hp}=746\,\text{W}$. Force and velocity must be collinear.

Where it lives: NCEES FE Reference Handbook — Dynamics · NCEES FE Reference Handbook — Dynamics (Laws of Friction) · Hibbeler, Engineering Mechanics: Dynamics

Material Properties and Corrosion Control

Read the tensile curve, apply Hooke's law and the elastic constants, account for temperature and time, and set up the electrochemical cell that drives corrosion.

Engineering stress and strain

$$\sigma = \dfrac{F}{A_0}, \qquad \varepsilon = \dfrac{\Delta L}{L_0}$$

Based on ORIGINAL area $A_0$ and length $L_0$. $\sigma$ in Pa (or MPa), $\varepsilon$ dimensionless.

Hooke's law

$$\sigma = E\,\varepsilon$$

Linear elastic region only. $E$ = elastic (Young's) modulus, Pa; slope of the tensile curve's straight portion.

Axial deformation

$$\delta = \dfrac{PL}{AE}$$

Elastic elongation of a bar of length $L$, area $A$, modulus $E$ under axial load $P$. Units m.

Shear and bulk modulus

$$G = \dfrac{E}{2(1+\nu)}, \qquad K = \dfrac{E}{3(1-2\nu)}$$

Isotropic elastic links. $\nu$ = Poisson's ratio ($\approx 0.3$ for steel). Only two of $E,G,K,\nu$ are independent.

Poisson's ratio

$$\nu = -\dfrac{\varepsilon_{\text{lateral}}}{\varepsilon_{\text{longitudinal}}}$$

Negative ratio of transverse to axial strain. Dimensionless; $0 \le \nu \le 0.5$ for stable isotropic solids.

True stress and true strain

$$\sigma_T = \dfrac{F}{A}, \qquad \varepsilon_T = \ln(1+\varepsilon)$$

Use instantaneous area $A$; needed in plastic/necking analysis. Differs from engineering values beyond yield.

Thermal deformation and stress

$$\delta_t = \alpha L\,(T - T_0), \qquad \sigma_{\text{thermal}} = E\,\alpha\,\Delta T$$

Free expansion (first) vs fully restrained stress (second). $\alpha$ = expansion coefficient, 1/K or 1/°C.

Thermal expansion coefficient

$$\alpha = \dfrac{\varepsilon}{\Delta T}$$

Engineering strain per degree of temperature change. Units 1/K (equivalently 1/°C for a difference).

Fracture toughness

$$K_{IC} = Y\,\sigma\sqrt{\pi a}$$

Critical stress intensity; $a$ = crack length, $Y$ = geometry factor ($1$ interior, $1.1$ surface). Units MPa·m$^{1/2}$.

Hardness–strength estimate

$$\text{TS (MPa)} \approx 3.5\,\text{BHN}$$

Plain-carbon steel approximation (or TS(psi) ≈ 500·BHN). BHN = Brinell hardness number.

Resistivity and capacitance

$$\rho = \dfrac{RA}{L}, \qquad C = \dfrac{\varepsilon A}{d}$$

$\rho$ = resistivity (Ω·m); $C$ = parallel-plate capacitance, $\varepsilon=\kappa\varepsilon_0$, $\varepsilon_0=8.85\times10^{-12}\,$F/m.

Diffusion coefficient (Arrhenius)

$$D = D_0\,e^{-Q/(RT)}$$

Temperature dependence of diffusion/creep. $Q$ = activation energy, $R=8.314\,$J/(mol·K), $T$ in K.

Galvanic cell EMF

$$E_{cell} = E^\circ_{ox,\,\text{anode}} + E^\circ_{red,\,\text{cathode}}$$

Positive $E_{cell}$ means the couple corrodes spontaneously. Use handbook oxidation potentials (reverse sign for the cathode).

Nernst equation (25 °C)

$$E = E^\circ - \dfrac{0.0592}{n}\log_{10} Q$$

Concentration correction to the standard potential; $n$ = electrons transferred, $Q$ = reaction quotient.

Where it lives: NCEES FE Reference Handbook — Materials Science/Structure of Matter · NCEES FE Reference Handbook — Mechanics of Materials · NCEES FE Reference Handbook — Chemistry and Biology · Callister & Rethwisch, Materials Science and Engineering: An Introduction

Solutions, pH, Buffers and Electrochemistry

Concentration units, weak-acid and buffer pH, the solubility product, and the Nernst equation — the aqueous-chemistry workhorses the FE rewards for speed.

Molarity and normality

$$M = \dfrac{n_{solute}}{V_{soln}(\text{L})}, \qquad N = M\,z$$

$M$ in mol/L; $z$ = equivalents per mole (protons for acids, electrons for redox). Normality is solution-context dependent.

Titration equivalence

$$N_1 V_1 = N_2 V_2$$

Equal equivalents at the endpoint. Volumes in any consistent unit; valid only in normality, not molarity (unless 1:1). Not printed in the FE Reference Handbook 10.6.

pH and water ion product

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

At $25\,^\circ\mathrm{C}$. $[\mathrm{H^+}]$ in mol/L. Acids pH < 7, bases pH > 7.

Weak-acid dissociation

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

Equilibrium constant for $\mathrm{HA}\rightleftharpoons \mathrm{H^+}+\mathrm{A^-}$. Smaller $K_a$ (larger $\mathrm{p}K_a$) = weaker acid.

Weak-acid pH approximation

$$[\mathrm{H^+}] \approx \sqrt{K_a\,C}$$

Valid when dissociation $<5\%$ of formal concentration $C$ (mol/L). Verify the fraction $[\mathrm{H^+}]/C$ afterward.

Henderson-Hasselbalch

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

Buffer pH from the conjugate-base to acid ratio. Independent of dilution; $\mathrm{pH}=\mathrm{p}K_a$ when the ratio is 1.

Solubility product

$$\mathrm{A}_m\mathrm{B}_n \rightleftharpoons m\mathrm{A}^{n+}+n\mathrm{B}^{m-}, \quad K_{sp} = [\mathrm{A}^{n+}]^m[\mathrm{B}^{m-}]^n$$

Solids and pure liquids omitted (activity = 1). Compare $Q$ to $K_{sp}$ to test precipitation.

Equilibrium constant (general)

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

For $a\mathrm{A}+b\mathrm{B}\rightleftharpoons c\mathrm{C}+d\mathrm{D}$. Brackets are activities; = molar concentration in dilute solution, partial pressure for gases, 1 for pure solids/liquids.

Standard cell potential

$$E^\circ_{cell} = E^\circ_{cathode} - E^\circ_{anode}$$

Both values are reduction potentials (V). $E^\circ_{cell}>0$ means spontaneous as written.

Nernst equation (25 °C)

$$E = E^\circ - \dfrac{0.0592}{n}\log_{10}Q$$

$n$ = electrons transferred; $Q$ = reaction quotient (products over reactants). Full form: $E=E^\circ-(RT/nF)\ln Q$, $F=96{,}485\,\mathrm{C/mol}$.

Faraday's law of electrolysis

$$m = \dfrac{Q\,M}{zF}$$

$m$ = mass deposited (g); $Q$ = charge (C = A·s); $M$ = molar mass (g/mol); $z$ = electrons per ion; $F=96{,}485\,\mathrm{C/mol}$.

Where it lives: NCEES FE Reference Handbook — Chemistry and Biology · NCEES FE Reference Handbook — Environmental Engineering · Felder & Rousseau, Elementary Principles of Chemical Processes

The Mechanical Energy Balance and Bernoulli

Write extended Bernoulli as heads, add pipe, valve and fitting friction losses, account for elevation, and size pump head or pressure drop with confidence.

Bernoulli equation (no friction, no machine)

$$\frac{P_1}{\gamma}+\frac{v_1^{2}}{2g}+z_1=\frac{P_2}{\gamma}+\frac{v_2^{2}}{2g}+z_2$$

Conservation of mechanical energy per unit weight along a streamline; each term in meters (or feet). $\gamma=\rho g$.

Extended (mechanical energy) balance

$$\frac{P_1}{\gamma}+\frac{v_1^{2}}{2g}+z_1+h_p=\frac{P_2}{\gamma}+\frac{v_2^{2}}{2g}+z_2+h_f$$

Adds pump head $h_p$ (energy in) and total head loss $h_f$ (energy out, always positive). The form used for pump and pressure-drop sizing.

Continuity

$$\dot{m}=\rho Q=\rho A v,\qquad Q=A_1 v_1=A_2 v_2$$

Mass and (for constant density) volumetric flow conserved; $A=\tfrac{\pi}{4}D^2$ for round pipe. Units $\mathrm{m^3/s}$, $\mathrm{m/s}$.

Darcy-Weisbach major loss

$$h_f=f\,\frac{L}{D}\,\frac{v^{2}}{2g}$$

Straight-pipe friction head; $f$ from Moody chart vs $\mathrm{Re}$ and $\varepsilon/D$. $L$, $D$ in m, $v$ in m/s.

Fanning–Darcy relation

$$f_{\text{Darcy}}=4\,f_{\text{Fanning}}$$

Chemical-engineering charts often give Fanning $f_F$; the Darcy form needs the factor of four (or use $h_f=4f_F\tfrac{L}{D}\tfrac{v^2}{2g}$).

Minor (fitting) losses

$$h_{f,\text{minor}}=\sum C\,\frac{v^{2}}{2g}$$

$C$ (or $K$) is the dimensionless loss coefficient per fitting/valve/entrance/exit; sum over the run.

Hydrostatic pressure

$$P_2-P_1=-\gamma(z_2-z_1)=\rho g h$$

Static-fluid pressure variation with depth $h$; basis of manometers. Use absolute or gauge consistently.

Pressure drop in a straight horizontal pipe

$$P_1-P_2=\gamma\,h_f=f\,\frac{L}{D}\,\frac{\rho v^{2}}{2}$$

Special case with $z_1=z_2$, $v_1=v_2$, no pump; converts head loss directly to a pressure drop in Pa.

Hydraulic (fluid) power

$$\dot{W}_{\text{fluid}}=\rho g Q h_p=\gamma Q h_p$$

Useful power delivered to fluid by a pump of head $h_p$; in watts when $Q$ in $\mathrm{m^3/s}$, $h_p$ in m.

Brake and electrical power

$$\dot{W}_{\text{brake}}=\frac{\dot{W}_{\text{fluid}}}{\eta_{\text{pump}}},\qquad \dot{W}_{\text{elec}}=\frac{\dot{W}_{\text{brake}}}{\eta_{\text{motor}}}$$

Shaft power and electrical draw; efficiencies are fractions $\le 1$.

Torricelli (frictionless efflux)

$$v=\sqrt{2gh}$$

Free jet velocity from a head $h$ when friction is negligible; the friction case puts $1+f\tfrac{L}{D}+\sum C$ in the denominator inside the root (the leading $1$ is the exit velocity head).

Where it lives: NCEES FE Reference Handbook — Fluid Mechanics · NCEES FE Reference Handbook — Fluid Mechanics (Energy Equation, Bernoulli, Darcy-Weisbach, Minor Losses) · Felder & Rousseau, Elementary Principles of Chemical Processes

First and Second Law: Processes, Cycles and Efficiency

Closed- and open-system energy balances, entropy and the second law, the isothermal/adiabatic/isentropic process family, and Carnot ceilings on efficiency and COP.

Closed-system first law

$$Q - W = \Delta U + \Delta KE + \Delta PE$$

Energy balance with no mass flow. $Q$ positive when added to the system, $W$ positive when done by the system; energies in kJ or Btu.

Steady-flow energy equation

$$\dot{Q}_{in} - \dot{W}_{net} = \sum_e \dot{m}_e\!\left(h_e + \tfrac{V_e^2}{2} + gZ_e\right) - \sum_i \dot{m}_i\!\left(h_i + \tfrac{V_i^2}{2} + gZ_i\right)$$

Open-system rate balance at steady state. $h$ = specific enthalpy (kJ/kg), $\dot m$ = mass flow (kg/s); flow work is already inside $h = u + Pv$.

Ideal-gas relations

$$Pv = RT, \quad \Delta u = c_v\,\Delta T, \quad \Delta h = c_p\,\Delta T, \quad c_p - c_v = R$$

$R$ = specific gas constant (kJ/kg·K). Property changes depend only on $\Delta T$ for an ideal gas, independent of path.

Isentropic (reversible adiabatic) relations

$$Pv^{k} = \text{const}, \qquad \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} = \left(\frac{v_1}{v_2}\right)^{k-1}$$

$k = c_p/c_v$. Use absolute $T$ and $P$. Valid for an ideal gas with constant specific heats.

Isentropic ideal-gas work

$$w_{closed} = \frac{R(T_2 - T_1)}{1 - k}, \qquad w_{open} = \frac{kR(T_2 - T_1)}{1 - k}$$

Closed-system (boundary) work versus open-system (shaft) work for an isentropic process; the two differ by the factor $k$.

Isothermal ideal-gas work

$$w = RT\ln\frac{v_2}{v_1} = RT\ln\frac{P_1}{P_2}$$

Constant-temperature ideal-gas process; for a closed system $q = w$ since $\Delta u = 0$.

Ideal-gas entropy change

$$\Delta s = c_p\ln\frac{T_2}{T_1} - R\ln\frac{P_2}{P_1} = c_v\ln\frac{T_2}{T_1} + R\ln\frac{v_2}{v_1}$$

Entropy change between two states of an ideal gas (kJ/kg·K). Absolute $T$ throughout.

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. Sets the direction of spontaneous change.

Thermal efficiency

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

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

Carnot efficiency

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

Maximum efficiency between reservoirs at absolute temperatures $T_H$, $T_L$ (K or °R). Upper bound no real engine exceeds.

Coefficients of performance

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

Cooling values $Q_L$; heating values $Q_H$. The two differ by exactly one because $Q_H = Q_L + W$.

Carnot COP ceilings

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

Maximum COP of a reversed-Carnot cycle between the two reservoirs; absolute temperatures only.

Reservoir entropy change

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

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

Where it lives: NCEES FE Reference Handbook — Thermodynamics · Smith, Van Ness, Abbott & Swihart, Introduction to Chemical Engineering Thermodynamics · Moran, Shapiro, Boettner & Bailey, Fundamentals of Engineering Thermodynamics

Material Balances and Degrees of Freedom

The steady-state total and component mass balance, degrees-of-freedom analysis, tie components, and a disciplined recipe for solving multi-unit flowsheets.

General material balance

$$\dfrac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out} + \dot{m}_{gen}$$

Accumulation = in − out + generation. Per unit time for a defined control volume; $\dot m$ in kg/s (or mol/s for a species).

Steady-state total balance

$$\sum \dot{m}_{in} = \sum \dot{m}_{out}$$

Total mass is conserved even with reaction; accumulation and net generation of total mass are zero.

Component balance (no reaction)

$$\sum_{\text{in}} F_k\,x_{j,k} = \sum_{\text{out}} P_l\,x_{j,l}$$

For each species $j$: mass (or mole) of $j$ in = out. $x_{j,k}$ is the fraction of $j$ in stream $k$.

Composition closure

$$\sum_j x_j = 1$$

Mass or mole fractions in any single stream sum to unity — one usable equation per stream.

Tie-component relation

$$F\,x_{\text{tie},F} = P\,x_{\text{tie},P}$$

An inert that enters and leaves in one stream each directly links the two stream flows $F$ and $P$.

Degrees of freedom

$$DOF = N_{\text{unknowns}} - N_{\text{indep. equations}}$$

$DOF=0$ solvable; $>0$ underspecified (set a basis); $<0$ over-specified. Independent equations = balances + specs + relations.

Independent balances

$$N_{\text{indep. balances}} = S$$

For $S$ species, only $S$ of the (total + $S$ component) balances are independent.

Mass–mole conversion

$$n_j = \dfrac{m_j}{M_j}, \qquad y_j = \dfrac{n_j}{\sum_k n_k}$$

Convert a stream between mass basis ($m_j$, mass fraction) and mole basis ($n_j$, mole fraction $y_j$) using molar masses $M_j$.

Where it lives: NCEES FE Reference Handbook — Chemical Engineering · NCEES FE Reference Handbook — Thermodynamics · Felder & Rousseau, Elementary Principles of Chemical Processes

Conduction, Thermal Resistance and the Overall U

Fourier conduction through walls and cylinders, the series-parallel resistance network, fouling, and the overall heat-transfer coefficient U.

Fourier's law (1-D)

$$\dot{Q} = -kA\,\dfrac{dT}{dx}$$

Conduction heat rate (W). $k$ = thermal conductivity [W/(m·K)], $A$ = area normal to flow (m²); minus sign = flow down the gradient.

Plane-wall conduction

$$\dot{Q} = \dfrac{kA}{L}\,(T_1 - T_2)$$

Steady 1-D slab of thickness $L$ (m); $T_1>T_2$ are the two face temperatures (K or °C — a difference is the same in both).

Thermal-resistance (Ohm) form

$$\dot{Q} = \dfrac{\Delta T_{\text{total}}}{\sum R}$$

Heat rate equals total temperature drop over the sum of series resistances (K/W). The same $\dot{Q}$ flows through each.

Plane-wall resistance

$$R = \dfrac{L}{kA}$$

Conduction resistance of a slab (K/W). $L$ = thickness, $A$ = face area.

Convection resistance

$$R = \dfrac{1}{hA}$$

Film resistance (K/W). $h$ = convection coefficient [W/(m²·K)], $A$ = wetted area on that side.

Cylindrical-wall resistance

$$R = \dfrac{\ln(r_2/r_1)}{2\pi k L}$$

Radial conduction through a tube of length $L$ between radii $r_1$ (inner) and $r_2$ (outer).

Critical insulation radius

$$r_{cr} = \dfrac{k_{\text{ins}}}{h_o}$$

Outer radius that maximizes heat loss from an insulated cylinder; below it, adding insulation increases loss.

Fouling resistance

$$R_{\text{foul}} = \dfrac{R_f}{A}$$

$R_f$ is the tabulated per-area fouling factor [(m²·K)/W]; divide by the fouled-side area to get K/W.

Overall coefficient (UA)

$$\dfrac{1}{UA} = \dfrac{1}{h_i A_i} + \dfrac{R_{f,i}}{A_i} + \dfrac{\ln(D_o/D_i)}{2\pi k L} + \dfrac{R_{f,o}}{A_o} + \dfrac{1}{h_o A_o}$$

Series sum of all tube-side resistances. $UA$ (W/K) carries the area, so $\dot{Q}=UA\,\Delta T$ needs no reference-area choice.

Overall U vs reference area

$$U_i A_i = U_o A_o = \dfrac{1}{\sum R}$$

$U$ must be quoted against a stated area (inside or outside); only the product $UA$ is unambiguous.

Interface temperature

$$T_{n+1} = T_n - \dot{Q}\,R_n$$

March the same $\dot{Q}$ through each resistance to recover any intermediate temperature.

Biot number (lumped check)

$$Bi = \dfrac{h V}{k A_s} = \dfrac{h L_c}{k}$$

If $Bi < 0.1$ a body is nearly isothermal and the lumped-capacitance transient model applies; $L_c=V/A_s$.

Where it lives: NCEES FE Reference Handbook — Heat Transfer · Incropera & DeWitt, Fundamentals of Heat and Mass Transfer · Perry's Chemical Engineers' Handbook — Heat Transfer

Molecular Diffusion and Convective Mass Transfer

Fick's law, equimolar counter-diffusion versus diffusion through stagnant B, the two-film model and its coefficients, and the momentum-heat-mass analogy.

Fick's first law

$$J_A = -D\,\dfrac{dC_A}{dx}$$

Steady 1-D molar flux ($\text{mol·m}^{-2}\text{s}^{-1}$) of A; $D$ = diffusivity ($\text{m}^2/\text{s}$), $C_A$ = concentration ($\text{mol/m}^3$). Minus sign: flux goes down the gradient.

Fick's second law

$$\dfrac{\partial C}{\partial t} = D\,\dfrac{\partial^2 C}{\partial x^2}$$

Unsteady 1-D diffusion (constant $D$). The mass-transfer twin of the transient heat-conduction equation.

Semi-infinite solution (erf)

$$\dfrac{C - C_s}{C_0 - C_s} = \operatorname{erf}\!\left(\dfrac{x}{2\sqrt{Dt}}\right)$$

Surface held at $C_s$, bulk initially $C_0$. Form $z = x/(2\sqrt{Dt})$, then read $\operatorname{erf}(z)$ from the handbook table.

Equimolar counter-diffusion (gases)

$$N_A = \dfrac{D_m}{RT\,\Delta z}\,(p_{A1}-p_{A2})$$

Use when $N_B=-N_A$ (no net molar flow), e.g., binary distillation. $\Delta z$ = path length.

Diffusion through stagnant B

$$N_A = \dfrac{D_m\,P}{RT\,\Delta z\,(p_B)_{lm}}\,(p_{A1}-p_{A2})$$

Use when B is inert and motionless (gas absorption). The $(p_B)_{lm}$ term adds the Stefan drift; always gives a larger flux than equimolar.

Log-mean partial pressure of B

$$(p_B)_{lm} = \dfrac{p_{B2}-p_{B1}}{\ln(p_{B2}/p_{B1})}, \quad p_B = P - p_A$$

Log mean of the inert's partial pressures at the two film faces.

Two-film flux

$$N_A = k_G'(p_{AG}-p_{Ai}) = k_L'(C_{Ai}-C_{AL})$$

Gas-film and liquid-film rate expressions; equal at steady state. $k_G'$, $k_L'$ are individual coefficients, interface in equilibrium.

Overall coefficients

$$\dfrac{1}{K_G'} = \dfrac{1}{k_G'} + \dfrac{H}{k_L'}, \quad \dfrac{1}{K_L'} = \dfrac{1}{H k_G'} + \dfrac{1}{k_L'}$$

Series resistances. Large $H$ (insoluble) -> liquid film controls; small $H$ (soluble) -> gas film controls.

Henry's law (interface)

$$p_A^{*} = H\,C_{AL}, \qquad C_A^{*} = p_{AG}/H$$

Links a liquid concentration to the equilibrium gas partial pressure; $H$ has units of pressure per concentration.

Sherwood correlation (turbulent tube)

$$\mathrm{Sh} = \dfrac{k_m D}{D_m} = 0.023\,\mathrm{Re}^{0.8}\,\mathrm{Sc}^{1/3}$$

Here $D$ = tube diameter (NOT the diffusivity $D_m$); $\mathrm{Sc}=\mu/(\rho D_m)$, $\mathrm{Re}=\rho V D/\mu$. Mass-transfer twin of Dittus-Boelter.

Chilton-Colburn analogy

$$j_H = \dfrac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}^{1/3}} = j_M = \dfrac{\mathrm{Sh}}{\mathrm{Re}\,\mathrm{Sc}^{1/3}} = \dfrac{f}{8}$$

Estimates $k_m$ from a friction factor or heat coefficient when a direct correlation is missing. The $f/8$ form uses the DARCY friction factor (with the Fanning factor it is $f/2$). FE Reference Handbook 10.6 prints it without the name, p. 255 (Use of Friction Factor to Predict Heat-Transfer and Mass-Transfer Coefficients): $j_H = (\mathrm{Nu}/\mathrm{Re\,Pr})\,\mathrm{Pr}^{2/3} = f/8$ and $j_M = (\mathrm{Sh}/\mathrm{Re\,Sc})\,\mathrm{Sc}^{2/3} = f/8$.

Where it lives: NCEES FE Reference Handbook — Chemical Engineering (Mass Transfer) · Welty, Wicks, Wilson & Rorrer, Fundamentals of Momentum, Heat, and Mass Transfer · Geankoplis, Transport Processes and Separation Process Principles

Particle Size, Distributions and Comminution

How to describe a particle-size distribution, compute the right mean diameter, relate surface area to size, and apply the Rittinger/Kick/Bond grinding-energy laws and crystallization basics.

Number-length (mean-length) diameter

$$X_{ML}=\int_0^1 x\,dF_N=\sum_{i=1}^{m} x_i\,\Delta F_{Ni}$$

Plain number-average particle size. $x_i$ = mean size of sub-range $i$ ($\mu$m or m); $\Delta F_{Ni}$ = number fraction in that sub-range (dimensionless, summing to 1).

Sauter (surface-mean) diameter

$$X_{SM}=\dfrac{\sum_i x_i^{3}\,\Delta F_{Ni}}{\sum_i x_i^{2}\,\Delta F_{Ni}}=\dfrac{1}{\sum_i \frac{1}{x_i}\,\Delta F_{Vi}}$$

Diameter of a sphere with the same surface-to-volume ratio as the sample; use for surface-rate processes (dissolution, drying, catalysis). $\Delta F_{Vi}$ = volume/mass fraction in sub-range $i$.

Mean-volume diameter

$$X_{MV}=\left(\sum_i x_i^{3}\,\Delta F_{Ni}\right)^{1/3}$$

Volume-weighted mean; governs mass distribution. Always $X_{ML}\le X_{MV}\le X_{SM}$ for the same powder.

Specific surface area of a sphere

$$a_v=\dfrac{6}{d}, \qquad a_m=\dfrac{6}{\rho_p\,d}$$

Surface area per unit volume ($a_v$, $\text{m}^2/\text{m}^3$) or per unit mass ($a_m$, $\text{m}^2/\text{kg}$). Scales as $1/d$: halving size doubles area. $\rho_p$ = particle density.

Specific surface area with sphericity

$$a_v=\dfrac{6}{\Phi_s\,d}$$

Non-spherical correction. $\Phi_s$ = sphericity (0–1): area of equal-volume sphere divided by actual surface area; the same $\Phi_s$ used in the Ergun equation.

General comminution law

$$\dfrac{dE}{dx}=-C\,x^{-n}$$

Energy per unit mass to reduce size; $n=2$ Rittinger, $n=1$ Kick, $n=1.5$ Bond. $x$ = particle size; $C$ = material constant. Not printed in the FE Reference Handbook 10.6.

Rittinger's law

$$E=K_R\left(\dfrac{1}{x_2}-\dfrac{1}{x_1}\right)$$

Energy $\propto$ new surface created; best for fine grinding. $x_1$ feed, $x_2$ product size; $K_R$ Rittinger constant. Not printed in the FE Reference Handbook 10.6.

Kick's law

$$E=K_K\,\ln\dfrac{x_1}{x_2}$$

Energy depends only on size reduction ratio; best for coarse crushing. $K_K$ = Kick constant. Not printed in the FE Reference Handbook 10.6.

Bond's law (size form)

$$E=K_B\left(\dfrac{1}{\sqrt{x_2}}-\dfrac{1}{\sqrt{x_1}}\right)$$

Intermediate law for rod/ball mills. $K_B$ = Bond constant tied to the work index. Not printed in the FE Reference Handbook 10.6.

Bond work-index equation

$$W = 10\,W_i\left(\dfrac{1}{\sqrt{P_{80}}}-\dfrac{1}{\sqrt{F_{80}}}\right)$$

Specific energy in kWh/short ton; $W_i$ = Bond work index (kWh/short ton), $F_{80},P_{80}$ = 80%-passing feed and product sizes in $\mu$m. Mill power $=W\times$ mass throughput. Not printed in the FE Reference Handbook 10.6.

Supersaturation

$$S=\dfrac{C}{C^{*}}, \qquad \Delta C=C-C^{*}$$

Crystallization driving force. $C$ = actual concentration, $C^{*}$ = equilibrium solubility at the operating temperature. $S>1$ required for nucleation and growth.

Stokes terminal settling velocity

$$v_t=\dfrac{g\,(\rho_p-\rho_f)\,d^{2}}{18\,\mu}$$

Laminar ($Re\le1$) settling/classification of a particle of diameter $d$; $\rho_p,\rho_f$ = particle and fluid density, $\mu$ = fluid viscosity. Ties particle size to separation.