GATE CH Formula Sheet

$N_A = -D_{AB}\dfrac{dC_A}{dz} + x_A(N_A+N_B)$

Fick's first law with the bulk-flow (convective) term, reduces to simple diffusion only when NA + NB = 0

$N_A = \dfrac{D_{AB}P}{RTz(1-y_{A2})_{lm}}\left(y_{A1}-y_{A2}\right)$

Diffusion of A through stagnant B, flux enhanced by the log-mean mole fraction of inert B, (1−yA)lm

$(1-y_A)_{lm} = \dfrac{(1-y_{A2})-(1-y_{A1})}{\ln\left[\frac{1-y_{A2}}{1-y_{A1}}\right]}$

Log-mean mole fraction of stagnant component B, used in the stagnant-film flux equation

$N_A = -N_B = \dfrac{D_{AB}}{RTz}(p_{A1}-p_{A2})$

Equimolar counterdiffusion flux, a simpler linear form, no log-mean correction needed

$Sh = \dfrac{k_c L}{D_{AB}}$

Sherwood number, ratio of convective to diffusive mass transfer, analogous to the Nusselt number

$Sc = \dfrac{\mu}{\rho D_{AB}}$

Schmidt number, ratio of momentum to mass diffusivity, analogous to the Prandtl number

$Sh = 0.023\,Re^{0.8}Sc^{1/3}$

Mass-transfer analog of the Dittus-Boelter correlation, for turbulent flow in a pipe or duct

$\dfrac{k_c}{v} = St_m = \dfrac{f}{2}\,Sc^{-2/3}$

Chilton-Colburn analogy connecting the mass transfer Stanton number to the Darcy/Fanning friction factor

$P = P_0 + \rho g h$

Hydrostatic pressure at depth h below a free surface

$A_1 v_1 = A_2 v_2$

Continuity equation for incompressible flow

$\dfrac{P}{\rho g} + \dfrac{v^2}{2g} + z = \text{const.}$

Bernoulli's equation along a streamline (energy per unit weight)

$Re = \dfrac{\rho v D}{\mu}$

Reynolds number, Re < 2100 laminar, 2100–4000 transitional, > 4000 turbulent (circular pipe)

Also in: Reynolds

$\Delta P = \dfrac{32\mu L v}{D^2}$

Hagen-Poiseuille equation, laminar pressure drop in a pipe (Q = πΔPD⁴/128μL)

Also in: Reynolds

$f = \dfrac{64}{Re}$

Darcy friction factor for laminar flow

$\Delta P = f\left(\dfrac{L}{D}\right)\left(\dfrac{\rho v^2}{2}\right)$

Darcy-Weisbach equation for frictional pressure drop (use the Moody chart for turbulent f)

Also in: Friction Factor, Pressure Drop

$f_{Darcy} = 4 \times f_{Fanning}$

Conversion between the two common friction-factor conventions, a frequent source of factor-of-4 errors

$\dfrac{\Delta P}{L} = \dfrac{150(1-\varepsilon)^2 \mu v}{\varepsilon^3 d_p^2} + \dfrac{1.75(1-\varepsilon)\rho v^2}{\varepsilon^3 d_p}$

Ergun equation for pressure drop through a packed bed (ε = voidage, dp = particle diameter)

$Q = C_d A_2 \sqrt{\dfrac{2\Delta P}{\rho\left[1-(A_2/A_1)^2\right]}}$

Venturi/orifice meter flow rate from measured pressure drop

$NPSH_a = \dfrac{P_{atm}-P_{vapor}}{\rho g} + h_s - h_f$

Available Net Positive Suction Head, must exceed the pump's required NPSH to avoid cavitation

$\text{Number of } \pi \text{ groups} = n - m$

Buckingham Pi theorem, n variables, m fundamental dimensions

$(\Delta P)\pi r^2 = -\mu\dfrac{dv}{dr}(2\pi r L)$

Derivation step: Deriving Hagen-Poiseuille from a Force Balance

$v(r) = \dfrac{\Delta P}{4\mu L}(R^2 - r^2), \qquad Q = \dfrac{\pi \Delta P R^4}{8\mu L} = \dfrac{\pi \Delta P D^4}{128\mu L}$

Derivation step: Deriving Hagen-Poiseuille from a Force Balance

$A = \dfrac{Q}{U \times \text{LMTD}}$

Required heat transfer area, given duty Q and overall coefficient U, apply an F-factor correction for multi-pass exchangers

Also in: Heat Exchanger Area, Lmtd

$q = -kA\dfrac{dT}{dx}$

Fourier's law of conduction; k = thermal conductivity, A = area normal to flow

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

Steady conduction through a plane wall of thickness L

$Q = \dfrac{2\pi k L (T_1-T_2)}{\ln(r_2/r_1)}$

Steady radial conduction through a cylindrical wall

$R_{cond} = \dfrac{L}{kA}, \quad R_{conv} = \dfrac{1}{hA}$

Thermal resistances, add in series for composite walls

$q = hA(T_s - T_\infty)$

Newton's law of cooling for convective heat flux

$r_c = \dfrac{k}{h}$

Critical radius of insulation for a cylinder, adding insulation below rc increases heat loss

$Bi = \dfrac{hL_c}{k}$

Biot number, ratio of internal conductive to external convective resistance

$Fo = \dfrac{\alpha t}{L_c^2}$

Fourier number, dimensionless time for unsteady conduction problems

$Nu = \dfrac{hL}{k}$

Nusselt number, ratio of convective to conductive heat transfer

$Nu = 0.023\,Re^{0.8}Pr^{n}$

Dittus-Boelter correlation for turbulent flow in a pipe (n = 0.4 heating, 0.3 cooling)

$\Delta T_{lm} = \dfrac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$

Log mean temperature difference (LMTD) between hot and cold streams

$\varepsilon = \dfrac{Q_{actual}}{Q_{max}}, \quad NTU = \dfrac{UA}{C_{min}}$

Effectiveness-NTU method, an alternative to LMTD when outlet temperatures are unknown

$E_b = \sigma T^4$

Stefan-Boltzmann law for black-body emissive power (σ = 5.67×10⁻⁸ W/m²K⁴)

$Q = \varepsilon \sigma A (T_1^4 - T_2^4)$

Radiative heat exchange between a grey surface and its surroundings

$dQ = -\dot m_h c_{p,h}\,dT_h = \pm\dot m_c c_{p,c}\,dT_c = U(T_h - T_c)\,dA$

Derivation step: Deriving LMTD from a Differential Energy Balance

$Q = UA \cdot \dfrac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)} = UA \cdot \Delta T_{lm}$

Derivation step: Deriving LMTD from a Differential Energy Balance

$\log_{10}(P^{*}) = A - \dfrac{B}{C+T}$

Antoine equation for pure-component vapor pressure (T in °C, constants from Perry's Handbook)

Also in: Antoine

$\sum x_i P_i^{*} = P$

Bubble point condition for a liquid mixture at total pressure P

Also in: Antoine

$N_A = -D_{AB}\dfrac{dC_A}{dz}$

Fick's first law of molecular diffusion (add bulk-flow term for diffusion through a stagnant film)

$p_A = x_A P_A^{*}$

Raoult's law, partial pressure of component A over an ideal liquid mixture

$\alpha = \dfrac{y_A/x_A}{y_B/x_B}$

Relative volatility, the key parameter driving distillation separability

$\sum \dfrac{y_i P}{P_i^{*}} = 1$

Dew point condition for a vapor mixture at total pressure P

$N_A = K_y(y_A - y_A^{*}) = K_x(x_A^{*} - x_A)$

Two-film theory, flux written in terms of overall gas- or liquid-phase driving force

$\dfrac{1}{K_y} = \dfrac{1}{k_y} + \dfrac{m}{k_x}$

Overall mass transfer coefficient combining individual gas- and liquid-film resistances (m = local slope of equilibrium line); Derivation step: Building the Overall Mass Transfer Coefficient from Two Films

$q = \dfrac{q_{max} K C}{1 + K C}$

Langmuir adsorption isotherm, monolayer equilibrium loading q vs. fluid concentration C

$q = K C^{1/n}$

Freundlich adsorption isotherm, empirical power-law equilibrium relationship

$t_c = \dfrac{(W_1 - W_c) L_s}{A R_c}$

Constant-rate drying time, from initial moisture W1 down to critical moisture Wc

$t_f = \dfrac{L_s}{A R_c} W_c \ln\!\left(\dfrac{W_c}{W_2}\right)$

Falling-rate drying time (linear falling-rate assumption), from Wc down to final moisture W2

$H = \dfrac{18}{29}\cdot\dfrac{p_w}{P - p_w}$

Humidity (kg water vapor per kg dry air) from water vapor partial pressure pw and total pressure P

$F = C - P + 2$

Gibbs phase rule, degrees of freedom F for C components and P phases, used to check VLE problem consistency; Gibbs' phase rule, degrees of freedom F for C components and P phases

Also in: Thermodynamics

$N_A = k_y(y - y_i) = k_x(x_i - x)$

Derivation step: Building the Overall Mass Transfer Coefficient from Two Films

$y_{n+1} = \dfrac{L}{V}x_n + \dfrac{D}{V}x_D = \dfrac{R}{R+1}x_n + \dfrac{x_D}{R+1}$

Derivation step: Why the McCabe-Thiele Operating Line Is Linear

$\phi_s = \dfrac{\text{surface area of a sphere of equal volume}}{\text{actual surface area of particle}}$

Sphericity, a shape factor equal to 1 for a perfect sphere

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

Stokes' law terminal settling velocity (valid for particle Reynolds number < 1)

$Re_p = \dfrac{d_p v_t \rho_f}{\mu}$

Particle Reynolds number, checks whether Stokes' law (creeping flow) applies

$\dfrac{dV}{dt} = \dfrac{A^2 \Delta P}{\mu(\alpha c V + A R_m)}$

Cake filtration rate equation (α = specific cake resistance, c = mass solids/volume filtrate, Rm = medium resistance)

$\dfrac{t}{V} = \dfrac{\mu \alpha c}{2A^2\Delta P}V + \dfrac{\mu R_m}{A\Delta P}$

Linearized constant-pressure filtration equation, plot t/V vs. V to extract α and Rm

$\dfrac{dE}{dL} = -K L^{-n}$

General size-reduction energy law; n = 2 (Rittinger), n = 1 (Kick), n = 1.5 (Bond)

$W = W_i\left(\dfrac{1}{\sqrt{d_{p2}}} - \dfrac{1}{\sqrt{d_{p1}}}\right)$

Bond's law for grinding work, using the work index Wi

$v_{mf}: \; \Delta P_{Ergun}(v_{mf}) = (\rho_p-\rho_f)g(1-\varepsilon_{mf})L$

Minimum fluidization velocity, found by equating Ergun-equation pressure drop to the bed's net weight per unit area

$N_p = \dfrac{P}{\rho N^3 D^5}$

Power number for a stirred-tank agitator (P = power, N = impeller speed, D = impeller diameter)

$\eta_{screen} = \dfrac{\text{oversize recovered in overflow}}{\text{oversize in feed}} \times \dfrac{\text{undersize recovered in underflow}}{\text{undersize in feed}}$

Overall screen effectiveness combining oversize and undersize recovery

$\dfrac{\pi}{6}d_p^3 \rho_p g = \dfrac{\pi}{6}d_p^3 \rho_f g + 3\pi\mu d_p v_t$

Derivation step: Deriving Stokes' Law from a Force Balance on a Settling Sphere

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

Derivation step: Deriving Stokes' Law from a Force Balance on a Settling Sphere

$p_A = H x_A$

Henry's law, equilibrium partial pressure of dilute solute A, H = Henry's constant

$y = \dfrac{L}{G}x + \left(y_2 - \dfrac{L}{G}x_2\right)$

Absorption operating line from an overall mass balance (L = liquid, G = gas molar flow rate)

$\left(\dfrac{L}{G}\right)_{min}: \text{operating line touches equilibrium curve}$

Minimum liquid-to-gas ratio, analogous to minimum reflux in distillation

$Z = HTU \times NTU$

Packed column height from the height and number of transfer units

$NTU = \int_{y_2}^{y_1} \dfrac{dy}{y - y^{*}}$

Number of transfer units, a driving-force integral over the tower

$HTU = \dfrac{G}{K_y a S}$

Height of a transfer unit, using the overall gas-phase mass transfer coefficient Ky and packing interfacial area a

$m = \dfrac{y_{solute}}{x_{solute}}$

Distribution (partition) coefficient for liquid-liquid extraction, analogous to Henry's constant

$E = \dfrac{m V_{solvent}}{L_{feed}}$

Extraction factor, analogous to the absorption/stripping factor, governs recovery in a multistage cascade

$Gy + Lx_2 = Gy_2 + Lx \;\Rightarrow\; y = \dfrac{L}{G}(x - x_2) + y_2$

Derivation step: Deriving the Absorption Operating Line from an Overall Balance

$\alpha_{AB} = \dfrac{y_A/x_A}{y_B/x_B}$

Relative volatility, the driving parameter for how easily a binary mixture separates by distillation

$y = \dfrac{\alpha x}{1+(\alpha-1)x}$

Equilibrium curve for a binary system with constant relative volatility α

$y_{n+1} = \dfrac{R}{R+1}x_n + \dfrac{x_D}{R+1}$

Rectifying-section operating line, R = reflux ratio = L/D

$y_m = \dfrac{L'}{V'}x_{m-1} - \dfrac{W}{V'}x_W$

Stripping-section operating line (L', V' = liquid/vapor flow below the feed stage)

$q = \dfrac{\text{heat to vaporize 1 mol of feed at feed-plate conditions}}{\text{molar latent heat of feed}}$

q-line parameter, q = 1 for saturated liquid feed, q = 0 for saturated vapor feed

$y = \dfrac{q}{q-1}x - \dfrac{x_F}{q-1}$

q-line equation on the y-x diagram, passing through (xF, xF)

$\ln\left(\dfrac{x_1}{x_2}\right)\bigg/\left(\alpha-1\right) \;+\; \ln\left(\dfrac{1-x_2}{1-x_1}\right) = \ln\dfrac{L_1}{L_2}$

Rayleigh equation for batch (differential) distillation with constant relative volatility

$N_{min} = \dfrac{\ln\left[\dfrac{x_D}{1-x_D}\cdot\dfrac{1-x_W}{x_W}\right]}{\ln\alpha}$

Fenske equation, minimum theoretical stages at total reflux (Nmin + 1 including the reboiler)

$R_{min} = \dfrac{1}{\alpha - 1}\left[\dfrac{x_D}{x_F} - \alpha\dfrac{1-x_D}{1-x_F}\right]$

Underwood shortcut estimate for minimum reflux ratio (saturated liquid feed case)

$H = \dfrac{18}{29}\cdot\dfrac{p_w}{P-p_w}$

Absolute humidity from water vapor partial pressure pw and total pressure P

Also in: Psychrometrics

$\%RH = \dfrac{p_w}{p_w^{*}} \times 100$

Relative humidity, ratio of actual to saturation vapor pressure at the same temperature

$\%H = \dfrac{H}{H_s} \times 100$

Percentage humidity, ratio of actual to saturation humidity at the same temperature

$v_H = \left(\dfrac{1}{29} + \dfrac{H}{18}\right)RT/P$

Humid volume, volume of moist air per unit mass of dry air

$T_{db} - T_{wb} \approx \dfrac{k_y \lambda}{h}(H_s - H)\Big/k_y \;\text{(air-water system: } T_{wb} \approx T_{as}\text{)}$

Wet-bulb depression relates to humidity driving force via the psychrometric ratio h/(ky·cs), ≈ 1 for air-water

$R_c = \dfrac{L_s(W_1-W_c)}{A\,t_c}$

Constant-rate drying, rate Rc from initial moisture W1 to critical moisture Wc over time tc

$t_c = \dfrac{L_s(W_1-W_c)}{AR_c}$

Time to complete the constant-rate period

$t_f = \dfrac{L_s W_c}{AR_c}\ln\!\left(\dfrac{W_c}{W_2}\right)$

Time for the (linear) falling-rate period, from Wc down to final moisture W2

$-\dfrac{L_s}{A}\dfrac{dW}{dt} = \dfrac{R_c}{W_c}W \;\Rightarrow\; t_f = \dfrac{L_s W_c}{AR_c}\ln\!\left(\dfrac{W_c}{W_2}\right)$

Derivation step: Why the Falling-Rate Drying Time Uses a Logarithmic Form

$S\lambda_s = V\lambda_v + \text{(sensible heat terms)}$

Single-effect evaporator enthalpy balance, steam S (latent heat λs) supplies the latent heat to vaporize V (latent heat λv), plus any feed preheating

$\text{Steam economy} = \dfrac{V_{total}}{S}$

Kilograms of total vapor produced per kilogram of live steam fed, approaches N (number of effects) for well-designed multiple-effect trains

$T_{b,solution} = T_{b,solvent} + BPE$

Boiling point elevation, read from Dühring's chart or correlation for the specific solute-solvent system

$y = mx + c\ \text{(Dühring line)}$

Dühring's rule, solution boiling point plotted linearly against solvent boiling point at constant concentration

$F x_F = L x_L + C x_C$

Overall solute mass balance for a crystallizer, feed F, mother liquor L (at the solubility limit xL), crystals C (composition xC, may include water of hydration)

$C = F\cdot\dfrac{x_F - x_L}{x_C - x_L}\ \text{(simplified, anhydrous crystals)}$

Crystal yield from a mass balance when the crystals are anhydrous (no water of crystallization)

$-r_A = k C_A^n$

Power-law rate expression; n = reaction order, k = rate constant

$k = A e^{-E_a/RT}$

Arrhenius equation, temperature dependence of the rate constant

$V_{CSTR} = \dfrac{F_{A0}X}{-r_A}$

CSTR design equation, exit conditions apply throughout the whole reactor volume

$V_{PFR} = F_{A0}\int_0^X \dfrac{dX}{-r_A}$

PFR design equation, conditions change continuously along the reactor length

$t_{batch} = N_{A0}\int_0^X \dfrac{dX}{(-r_A)V}$

Batch reactor design equation, time required for a given conversion

$S_{D/U} = \dfrac{r_D}{r_U}$

Instantaneous selectivity of a desired product D over an undesired product U in parallel/series reactions

$\tau = \dfrac{V}{v_0}$

Space time, the ideal residence time based on volumetric flow rate v0

$\bar t = \int_0^{\infty} t\,E(t)\,dt$

Mean residence time from the residence time distribution E(t)

$\phi = L\sqrt{\dfrac{k}{D_{eff}}}$

Thiele modulus, compares reaction rate to internal (pore) diffusion rate in a catalyst pellet

$\eta = \dfrac{\text{actual rate with pore diffusion}}{\text{rate at surface conditions}}$

Catalyst effectiveness factor, approaches 1 at low Thiele modulus, 1/φ at high Thiele modulus

$C_{A} = C_{A0}(1-X)$

Concentration in terms of conversion for a constant-volume (or constant-density) system

$\text{In} - \text{Out} + \text{Generation} - \text{Consumption} = \text{Accumulation}$

General balance equation; for steady, non-reactive systems this reduces to In = Out

$x_i = \dfrac{n_i}{\sum_j n_j}$

Mole (or mass) fraction of component i in a stream

$\text{Overall conversion} = \dfrac{\text{fresh feed reacted}}{\text{fresh feed in}}$

Overall conversion across a recycle system, always based on fresh feed, not reactor-inlet feed

$\text{Single-pass conversion} = \dfrac{\text{reacted per pass}}{\text{feed to reactor}}$

Per-pass conversion is always lower than overall conversion when unreacted feed is recycled

$\%\text{ excess air} = \dfrac{\text{air supplied} - \text{air theoretical}}{\text{air theoretical}} \times 100$

Excess air relative to the stoichiometric (theoretical) requirement for complete combustion

$C_xH_y + \left(x + \dfrac{y}{4}\right)O_2 \rightarrow xCO_2 + \dfrac{y}{2}H_2O$

Generic hydrocarbon combustion stoichiometry used to find theoretical O2/air

$DOF = N_{unknowns} - N_{independent\ equations}$

Degrees of freedom; DOF = 0 means the problem is exactly (and uniquely) solvable

$Q = \Delta H = \sum n_i \int_{T_{ref}}^{T} C_{p,i}\,dT$

Sensible heat change of a stream from reference temperature to T

$\Delta H_{rxn}^{\circ} = \sum \nu_p \Delta H_{f,p}^{\circ} - \sum \nu_r \Delta H_{f,r}^{\circ}$

Standard heat of reaction from standard enthalpies of formation (Hess's law)

$Purge\ fraction = \dfrac{\text{purge stream}}{\text{recycle stream before split}}$

Fraction of the recycle loop continuously bled off to control inert buildup

$G(s) = \dfrac{Y(s)}{X(s)} = \dfrac{K}{\tau s + 1}$

Standard first-order transfer function; K = steady-state gain, τ = time constant

$y(t) = KA\left(1 - e^{-t/\tau}\right)$

First-order step response to a step input of magnitude A

$G(s) = \dfrac{K}{\tau^2 s^2 + 2\zeta\tau s + 1}$

Standard second-order transfer function; ζ = damping ratio (ζ < 1 underdamped, oscillatory)

$G(s) = \dfrac{Ke^{-\theta s}}{\tau s + 1}$

FOPDT (first-order-plus-dead-time) model; θ = dead time / transport delay

$\dfrac{Y(s)}{Y_{sp}(s)} = \dfrac{G_cG_pG_f}{1+G_cG_pG_fG_m}$

Standard closed-loop (setpoint-tracking) transfer function for a unity-feedback control loop

$u(t) = K_c\left[e(t) + \dfrac{1}{\tau_I}\int_0^t e\,dt + \tau_D\dfrac{de}{dt}\right]$

PID controller action in the time domain (Kc = controller gain, τI = integral time, τD = derivative time)

$\text{Offset} = \lim_{t\to\infty}\left[y_{sp}(t) - y(t)\right]$

Steady-state offset, nonzero for proportional-only control under a sustained load change, zero once integral action is added

$K_{cu}, P_u \;\rightarrow\; K_c = 0.6K_{cu},\; \tau_I = P_u/2,\; \tau_D = P_u/8$

Ziegler-Nichols closed-loop (ultimate gain) PID tuning rule

$V\rho c_p\dfrac{dT}{dt} = F\rho c_p(T_i - T) \;\Rightarrow\; \tau\dfrac{dT^{\prime}}{dt} + T^{\prime} = T_i^{\prime}, \quad \tau = \dfrac{V}{F}$

Derivation step: From a Stirred-Tank Energy Balance to a First-Order Transfer Function

$\tau s\,T^{\prime}(s) + T^{\prime}(s) = T_i^{\prime}(s) \;\Rightarrow\; \dfrac{T^{\prime}(s)}{T_i^{\prime}(s)} = \dfrac{1}{\tau s + 1}$

Derivation step: From a Stirred-Tank Energy Balance to a First-Order Transfer Function

$F = C - P + 2$

Gibbs phase rule, degrees of freedom F for C components and P phases, used to check VLE problem consistency; Gibbs' phase rule, degrees of freedom F for C components and P phases

Also in: Mass Transfer

$\Delta U = Q - W$

First law for a closed system (W = work done by the system)

$\Delta H = \Delta U + \Delta(PV)$

Enthalpy, the natural energy variable for steady-flow open systems

$dS \geq \dfrac{\delta Q}{T}$

Second law, equality for reversible processes, strict inequality for irreversible ones

$PV = ZRT$

Real-gas equation of state with compressibility factor Z (Z = 1 for ideal gas)

$Z = 1 + \dfrac{BP}{RT}$

Truncated virial equation of state using the second virial coefficient B

$f_i = \phi_i P$

Fugacity of a pure gas via the fugacity coefficient φ (φ → 1 as P → 0)

$\hat{f}_i^{L} = x_i \gamma_i f_i^{\circ}$

Fugacity of component i in a non-ideal liquid mixture using activity coefficient γ_i

$K_i = \dfrac{y_i}{x_i}$

Vapor-liquid equilibrium K-value for component i

$\Delta G_{rxn}^{\circ} = -RT\ln K$

Standard Gibbs free energy of reaction and the equilibrium constant K

$\dfrac{d\ln K}{dT} = \dfrac{\Delta H_{rxn}^{\circ}}{RT^2}$

Van't Hoff equation, how K shifts with temperature (integrate for a two-point estimate)

$PV^{n} = \text{const.}$

Polytropic process path for an ideal gas (n = 1 isothermal, n = γ isentropic, n = 0 isobaric)

$\sum \nu_i \mu_i = 0 \;\Rightarrow\; \Delta G_{rxn}^{\circ} = -RT\ln K, \qquad K = \prod_i (\hat a_i)^{\nu_i}$

Derivation step: From the Second Law to the Equilibrium Constant

$N_2 + 3H_2 \underset{Fe\ catalyst}{\overset{450°C,\ 200\ bar}{\rightleftharpoons}} 2NH_3$

Haber-Bosch ammonia synthesis, exothermic, so high pressure (not high temperature) favors conversion by Le Chatelier's principle

$2NaCl + 2H_2O \xrightarrow{\text{electrolysis}} Cl_2 + H_2 + 2NaOH$

Chlor-alkali electrolysis, the overall reaction across the membrane/diaphragm cell

$S + O_2 \rightarrow SO_2, \quad 2SO_2 + O_2 \xrightarrow{V_2O_5,\ 450°C} 2SO_3, \quad SO_3 + H_2SO_4 \rightarrow H_2S_2O_7 \xrightarrow{H_2O} 2H_2SO_4$

Contact process, SO3 is absorbed into concentrated acid (forming oleum) then diluted, not absorbed directly into water

$CaCO_3 + \text{clay} \xrightarrow{1450°C} \text{clinker (}Ca_3SiO_5, Ca_2SiO_4\text{)} + CO_2$

Cement clinkering reaction in the rotary kiln

$2NH_3 + CO_2 \rightarrow NH_2COONH_4 \xrightarrow{-H_2O} NH_2CONH_2\ (\text{urea})$

Urea synthesis via the ammonium carbamate intermediate

$nCH_2=CH_2 \xrightarrow{\text{catalyst}} (-CH_2-CH_2-)_n$

Addition polymerization of ethylene to polyethylene

$\det(A - \lambda I) = 0$

Characteristic equation, solve for eigenvalues λ of matrix A

$(A-\lambda I)v = 0$

Eigenvector equation, solve for the eigenvector v corresponding to eigenvalue λ

$\nabla f = \left(\dfrac{\partial f}{\partial x}, \dfrac{\partial f}{\partial y}, \dfrac{\partial f}{\partial z}\right)$

Gradient of a scalar field, points in the direction of steepest increase

$D_{\hat u}f = \nabla f \cdot \hat u$

Directional derivative of f in the direction of unit vector û

$z = re^{i\theta} = r(\cos\theta + i\sin\theta)$

Polar/exponential form of a complex number (Euler's formula)

$y'' + a y' + by = 0 \;\Rightarrow\; y = C_1e^{m_1x}+C_2e^{m_2x}$

General solution of a second-order linear homogeneous ODE with constant coefficients (distinct real roots m1, m2 of the auxiliary equation)

$x^2 y^{\prime\prime} + ax y^{\prime} + by = 0$

Euler-Cauchy equation, solved via the substitution y = x^m, giving an algebraic equation in m

$x_{n+1} = x_n - \dfrac{f(x_n)}{f^{\prime}(x_n)}$

Newton-Raphson iteration for root-finding, converges quadratically near a simple root

$\int_a^b f(x)\,dx \approx \dfrac{h}{2}\left[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\right]$

Trapezoidal rule for numerical integration, step size h = (b−a)/n

$P(A|B) = \dfrac{P(B|A)P(A)}{P(B)}$

Bayes' theorem for conditional probability

$\varepsilon_{n+1} \approx \dfrac{f^{\prime\prime}(x^{*})}{2f^{\prime}(x^{*})}\,\varepsilon_n^2$

Derivation step: Why Newton-Raphson Converges Quadratically Near a Simple Root

$\dfrac{C_2}{C_1} = \left(\dfrac{S_2}{S_1}\right)^{0.6}$

Six-tenths-factor rule for scaling equipment cost C with capacity S

$C_2 = C_1 \times \dfrac{I_2}{I_1}$

Cost index scaling, adjusts a historical cost C1 (index I1) to present-day cost C2 (index I2)

$D = \dfrac{C - S_v}{n}$

Straight-line depreciation, original cost C, salvage value Sv, useful life n years

$D_t = C(1-f)^{t-1}f$

Declining-balance depreciation in year t, with fixed depreciation rate f

$PW = \dfrac{FW}{(1+i)^n}$

Present worth of a future cash flow FW, n years away, at interest rate i

$\text{Payback period} = \dfrac{\text{Fixed capital investment}}{\text{Annual cash flow (avg.)}}$

Simple payback period, ignores the time value of money

$NPV = \sum_{t=0}^{n} \dfrac{CF_t}{(1+i)^t}$

Net present value, sum of all discounted cash flows, including the initial (negative) investment at t = 0

$NPV(IRR) = 0$

Internal rate of return (IRR), the discount rate that makes NPV exactly zero

$\text{Break-even capacity: } R(Q) = C_F + C_V(Q)$

Break-even point where total revenue R equals total cost (fixed CF plus variable CV) at production rate Q

$\log_{10}(P^{*}) = A - \dfrac{B}{C+T}$

Antoine equation for pure-component vapor pressure (T in °C, constants from Perry's Handbook)

Also in: Mass Transfer

$p_i = x_i P_i^{*}$

Raoult's law, partial pressure of component i in an ideal liquid mixture

$\dfrac{dP}{dT} = \dfrac{\Delta H_{vap}}{T\,\Delta V}$

Clausius-Clapeyron equation, exact form relating the vapor-pressure curve slope to the enthalpy and volume change of vaporization

$\dfrac{dP}{dT} = \dfrac{\Delta H_{vap}\,P}{RT^2}$

Clausius-Clapeyron equation with the ideal-gas approximation substituted for ΔV

$\ln(P^{*}) = -\dfrac{\Delta H_{vap}}{RT} + \text{constant}$

Integrated Clausius-Clapeyron equation, the basis for the Antoine equation’s log-linear vapor pressure fit

$\alpha = P_A^{*}/P_B^{*}$

Relative volatility of component A to B, estimated from their pure-component vapor pressures

$\sum x_i P_i^{*} = P$

Bubble point condition for a liquid mixture at total pressure P

Also in: Mass Transfer

$\Delta P = f\left(\dfrac{L}{D}\right)\left(\dfrac{\rho v^2}{2}\right)$

Darcy-Weisbach equation for frictional pressure drop (use the Moody chart for turbulent f)

Also in: Fluid Mechanics, Pressure Drop

$\dfrac{1}{\sqrt f} = -2\log_{10}\left(\dfrac{\varepsilon/D}{3.7}+\dfrac{2.51}{Re\sqrt f}\right)$

Colebrook equation for the Darcy friction factor in turbulent pipe flow (implicit in f, solved iteratively)

$f = 64/Re$

Darcy friction factor for fully-developed laminar pipe flow

Also in: Reynolds

$Q = U A \,\Delta T_{lm}$

Heat exchanger duty from the overall coefficient, area, and log mean temperature difference

$A = \dfrac{Q}{U \times \Delta T_{lm}}$

Required heat transfer area, solved from duty Q, overall coefficient U, and LMTD

$A = \dfrac{Q}{U \times \text{LMTD}}$

Required heat transfer area, given duty Q and overall coefficient U, apply an F-factor correction for multi-pass exchangers

Also in: Heat Transfer, Lmtd

$A = \dfrac{Q}{U \times \text{LMTD}}$

Required heat transfer area, given duty Q and overall coefficient U, apply an F-factor correction for multi-pass exchangers

Also in: Heat Exchanger Area, Heat Transfer

$dQ = U\,\Delta T\,dA$

Local heat duty across a differential exchanger area dA, driven by the local temperature difference

$dT_h = -dQ/C_h$

Differential temperature drop of the hot stream per unit heat transferred, C_h = hot-stream heat capacity rate

$dT_c = -dQ/C_c$

Differential temperature change of the cold stream per unit heat transferred, C_c = cold-stream heat capacity rate

$C_h = \dot{m}_h Cp_h$

Heat capacity rate of the hot stream (mass flow rate × specific heat)

$C_c = \dot{m}_c Cp_c$

Heat capacity rate of the cold stream (mass flow rate × specific heat)

$d(\Delta T) = -dQ\left(\dfrac{1}{C_h} - \dfrac{1}{C_c}\right)$

Change in the hot-cold temperature difference per unit heat transferred, combining both streams’ energy balances

$\dfrac{d(\Delta T)}{\Delta T} = -U\left(\dfrac{1}{C_h} - \dfrac{1}{C_c}\right)dA$

Separated form of the LMTD derivation, integrating this over the exchanger area produces the logarithmic mean

$Q = UAF \cdot \text{LMTD}_{cf}$

Heat duty for a multi-pass/cross-flow exchanger, counter-flow LMTD corrected by the F-factor

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

Sensible heat duty of a single stream from its flow rate, specific heat, and temperature change

$(\Delta T_1+\Delta T_2)/2$

Arithmetic mean temperature difference, a simpler (but less accurate) alternative to LMTD when ΔT₁ and ΔT₂ are close

$\text{LMTD} = \dfrac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$

Log mean temperature difference between the hot and cold streams of a heat exchanger

$\dfrac{P_{atm}}{\rho g} + z_{source} = \dfrac{P_{suction}}{\rho g} + z_{pump} + h_f$

Bernoulli energy balance from the suction source to the pump inlet, including friction loss h_f

$h_s = z_{source} - z_{pump}$

Static suction head, elevation of the liquid source above (or below) the pump centerline

$P_v/\rho g$

Vapor pressure head, the liquid's vapor pressure expressed as a head, subtracted in the NPSH available calculation

$NPSH_a = \dfrac{P_{suction}}{\rho g} - \dfrac{P_v}{\rho g} = \dfrac{P_{atm}-P_v}{\rho g} + h_s - h_f$

Net positive suction head available, expanded from suction-side pressure head down to atmospheric pressure, static head, and friction loss

$NPSH_a = \dfrac{P_{atm}-P_v}{\rho g} + h_s - h_f$

Net positive suction head available, the usable margin above vapor pressure at the pump suction, must exceed the pump’s required NPSH to avoid cavitation

$\Delta P = f\left(\dfrac{L}{D}\right)\left(\dfrac{\rho v^2}{2}\right)$

Darcy-Weisbach equation for frictional pressure drop (use the Moody chart for turbulent f)

Also in: Fluid Mechanics, Friction Factor

$H = \dfrac{18}{29}\cdot\dfrac{p_w}{P-p_w}$

Absolute humidity from water vapor partial pressure pw and total pressure P

Also in: Drying Humidification

$p_w = p_{ws}(T_{wb}) - A_p P (T_{db}-T_{wb})$

Modified psychrometer equation, actual water vapor partial pressure from wet-bulb and dry-bulb temperatures

$P_{hydraulic} = \dot{m} g H = \rho Q g H$

Hydraulic (fluid) power delivered by a pump, from mass or volumetric flow rate and total head H

$\eta = \dfrac{P_{hydraulic}}{P_{brake}} \;\Rightarrow\; P_{brake} = \dfrac{P_{hydraulic}}{\eta} = \dfrac{\rho Q g H}{\eta}$

Pump efficiency relates hydraulic power to shaft (brake) power, rearranged to solve for brake power

$P_{hydraulic} = \rho g Q H$

Hydraulic power in terms of fluid density, volumetric flow rate, and total head

$P_{brake} = P_{hydraulic}/\eta$

Brake (shaft) power required, accounting for pump efficiency losses

$Re = \dfrac{\rho v D}{\mu}$

Reynolds number, Re < 2100 laminar, 2100–4000 transitional, > 4000 turbulent (circular pipe)

Also in: Fluid Mechanics

$\Delta P = \dfrac{32\mu L v}{D^2}$

Hagen-Poiseuille equation, laminar pressure drop in a pipe (Q = πΔPD⁴/128μL)

Also in: Fluid Mechanics

$f = 64/Re$

Darcy friction factor for fully-developed laminar pipe flow

Also in: Friction Factor

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

Reynolds number in terms of kinematic viscosity ν (equivalent to ρvD/μ)

$D_h = 4A/P$

Hydraulic diameter for non-circular ducts, 4× cross-sectional area over wetted perimeter, used in place of D in Re for non-circular flow

$\rho\left(\dfrac{\partial v}{\partial t} + v\cdot\nabla v\right) = -\nabla p + \mu \nabla^2 v$

Navier-Stokes momentum equation for incompressible flow, the physical origin of the Reynolds number as a ratio of its inertial to viscous terms

$x^{*}=x/L,\ v^{*}=v/U,\ t^{*}=tU/L,\ p^{*}=p/(\rho U^2)$

Dimensionless scaling variables used to non-dimensionalize the Navier-Stokes equation

$\dfrac{\partial v^{*}}{\partial t^{*}} + v^{*}\cdot\nabla^{*} v^{*} = -\nabla^{*} p^{*} + \dfrac{\mu}{\rho U L}\nabla^{*2} v^{*}$

Non-dimensionalized Navier-Stokes equation, the coefficient μ/(ρUL) that appears is 1/Re

$\dfrac{\text{Inertial}}{\text{Viscous}} = \dfrac{\rho U^2/L}{\mu U/L^2} = \dfrac{\rho U L}{\mu} = Re$

Physical definition of the Reynolds number as the ratio of inertial to viscous forces in a flow

$\Delta P = \dfrac{128\mu L Q}{\pi D^4}$

Hagen-Poiseuille pressure drop for laminar flow in a circular pipe, in terms of volumetric flow rate Q

$Re = \rho v D/\mu$

Reynolds number, ratio of inertial to viscous forces, determines laminar vs. turbulent flow regime