Your FREE PE Mechanical Flashcards 2026 – 300+ Cards

PE Mechanical Flashcards

First law of thermodynamics

Energy is conserved: Q − W = ΔU for a closed system. Heat added minus work done by the system equals the change in internal energy.

Bernoulli's equation

Along a streamline (incompressible, inviscid, steady): P/ρ + V²/2 + gz = constant. Pressure, kinetic, and potential energy per unit mass are conserved.

Reynolds number

Re = ρVD/μ = VD/ν. Laminar pipe flow Re < ~2300; turbulent Re > ~4000. It is the ratio of inertial to viscous forces.

Factor of safety

FS = (failure load or strength) / (allowable or applied load), e.g., FS = S_y / σ_allow. A value > 1 indicates a margin against failure.

Coefficient of performance (COP)

Refrigeration COP = Q_cold / W_in = useful cooling / work input. Heat pump COP = Q_hot / W_in = COP_refrig + 1.

Fourier's law of conduction

q = −kA(dT/dx). Heat flow is proportional to thermal conductivity k, area A, and temperature gradient; it flows from hot to cold.

Bending stress (σ = Mc/I)

σ = Mc/I, where M = bending moment, c = distance from neutral axis to extreme fiber, I = area moment of inertia.

Carnot efficiency

η_Carnot = 1 − T_cold/T_hot, with temperatures absolute (K or °R). It is the maximum efficiency of any heat engine between two reservoirs.

Natural frequency of a SDOF system

ωn = √(k/m) in rad/s; fn = ωn/(2π) in Hz. k = stiffness, m = mass. Resonance occurs when forcing frequency approaches ωn.

Darcy-Weisbach equation

h_L = f (L/D)(V²/2g). f = Darcy friction factor, L = pipe length, D = diameter, V = velocity, g = 9.81 m/s².

Ideal gas law

PV = mRT or Pv = RT, where R is the specific gas constant. Also PV = nR_u T with universal R_u = 8.314 J/(mol·K).

Newton's law of cooling

q = hA(T_s − T_∞). Convective heat transfer is proportional to the convection coefficient h, surface area A, and the surface-to-fluid temperature difference.

Second law of thermodynamics

Entropy of an isolated system never decreases: ΔS_universe ≥ 0. Heat flows spontaneously from hot to cold, never the reverse without work input.

Entropy change (general)

dS = δQ_rev/T. For a reversible process the entropy change equals the heat transferred divided by absolute temperature.

Enthalpy definition

h = u + Pv. Enthalpy combines internal energy with flow work; it is convenient for open (control-volume) analysis.

Specific heats relation (ideal gas)

c_p − c_v = R and k = c_p/c_v. c_p is constant-pressure, c_v is constant-volume specific heat; k is the specific heat ratio.

Internal energy change (ideal gas)

Δu = c_v ΔT. For an ideal gas internal energy depends only on temperature, regardless of process path.

Enthalpy change (ideal gas)

Δh = c_p ΔT. Like internal energy, ideal-gas enthalpy is a function of temperature alone.

Steady-flow energy equation (SFEE)

q − w = Δh + ΔV²/2 + gΔz. Per unit mass for a control volume at steady state; often kinetic and potential terms are negligible.

Quality of a two-phase mixture

x = m_vapor / m_total. Property: y = y_f + x(y_g − y_f), e.g., h = h_f + x·h_fg, valid between saturated liquid (x=0) and saturated vapor (x=1).

Isentropic process (ideal gas)

Pvᵏ = constant; T₂/T₁ = (P₂/P₁) raised to (k−1)/k = (v₁/v₂) raised to (k−1). Reversible and adiabatic, so entropy is constant.

Isentropic efficiency of a turbine

η_t = (h₁ − h₂_actual)/(h₁ − h₂s). It compares actual work output to the ideal isentropic work for the same pressure drop.

Isentropic efficiency of a compressor

η_c = (h₂s − h₁)/(h₂_actual − h₁). It compares ideal isentropic work to the larger actual work input.

Reversible work in a closed system

W = ∫P dV. Boundary (moving-boundary) work equals the area under the process curve on a P–V diagram.

Polytropic process

PVⁿ = constant. n=0 isobaric, n=1 isothermal (ideal gas), n=k isentropic, n=∞ isochoric. Work = (P₂V₂ − P₁V₁)/(1 − n) for n ≠ 1.

Carnot COP (refrigeration)

COP = T_cold/(T_hot − T_cold), temperatures absolute. This is the maximum COP for a refrigerator between two reservoirs.

Carnot COP (heat pump)

COP_HP = T_hot/(T_hot − T_cold), temperatures absolute. It equals the refrigeration COP plus one.

Thermal efficiency of a heat engine

η = W_net/Q_in = 1 − Q_out/Q_in. It is the fraction of supplied heat converted to net work output.

Back work ratio

BWR = W_compressor/W_turbine. High in gas-turbine (Brayton) cycles because compression of gas takes a large share of turbine output.

Mean effective pressure (MEP)

MEP = W_net / V_displacement. A fictitious constant pressure that, acting over the displacement volume, would produce the cycle net work.

Otto cycle efficiency

η = 1 − 1/r raised to (k−1), where r = compression ratio and k = c_p/c_v. Models the ideal spark-ignition engine.

Diesel cycle efficiency

η = 1 − [1/r raised to (k−1)]·[(r_c raised to k − 1)/(k(r_c − 1))], with cutoff ratio r_c. Lower than Otto at the same r due to constant-pressure heat addition.

Clausius statement (2nd law)

Heat cannot spontaneously flow from a colder body to a hotter body without external work input.

Kelvin-Planck statement (2nd law)

No heat engine can convert all absorbed heat into work; some heat must be rejected to a lower-temperature reservoir.

Gibbs free energy

g = h − Ts. A minimum in Gibbs energy indicates equilibrium at constant temperature and pressure.

Helmholtz free energy

a = u − Ts. Useful for processes at constant temperature and volume.

Availability / exergy (closed system)

Φ = (u − u₀) + P₀(v − v₀) − T₀(s − s₀). The maximum useful work obtainable as a system reaches dead-state equilibrium with the surroundings.

Entropy generation

S_gen = ΔS_system + ΔS_surroundings ≥ 0. It quantifies irreversibility; zero only for ideal reversible processes.

Throttling process (valve)

Adiabatic with negligible work and kinetic change so h₁ = h₂. For an ideal gas temperature is unchanged; real fluids show the Joule-Thomson effect.

Joule-Thomson coefficient

μ_JT = (∂T/∂P)_h. Positive means the gas cools on throttling; negative means it warms. Zero at the inversion temperature.

Saturation temperature/pressure

The temperature and pressure at which liquid and vapor coexist in equilibrium. They are uniquely linked along the saturation curve.

Latent heat (h_fg)

h_fg = h_g − h_f. The enthalpy absorbed or released during phase change at constant pressure, with no temperature change.

Compressibility factor Z

Pv = ZRT. Z = 1 for an ideal gas; deviations measure real-gas behavior, found from generalized charts using reduced T_r and P_r.

Dalton's law of partial pressures

P_total = ΣP_i. Each gas in a mixture exerts the pressure it would alone in the same volume and temperature.

Universal vs specific gas constant

R = R_u/M, where R_u = 8.314 J/(mol·K) and M = molar mass. For air R ≈ 287 J/(kg·K).

Adiabatic vs isothermal process

Adiabatic: Q = 0 (no heat exchange). Isothermal: ΔT = 0 (constant temperature). They coincide only in the trivial no-process case.

Heat addition in a Rankine boiler

q_in = h₃ − h₂. Energy added per unit mass from compressed liquid (state 2) to superheated/saturated vapor (state 3).

Continuity equation

ṁ = ρAV = constant. For incompressible flow A₁V₁ = A₂V₂; mass is conserved through a control volume at steady state.

Hydrostatic pressure

P = ρgh (gauge). Pressure increases linearly with depth h below a free surface in a static fluid.

Moody chart purpose

It gives the Darcy friction factor f as a function of Reynolds number and relative roughness ε/D for pipe flow.

Laminar friction factor

f = 64/Re for fully developed laminar flow in a circular pipe. Independent of pipe roughness.

Minor (local) head loss

h_L = K (V²/2g), where K is a loss coefficient for fittings, valves, entrances, and bends.

Pump head added

h_pump = (P₂ − P₁)/(ρg) + (V₂² − V₁²)/2g + (z₂ − z₁) + h_L. The energy per unit weight the pump supplies to the fluid.

Pump hydraulic power

P_hyd = ρgQh = γQh. Brake power = ρgQh/η_pump, where Q = volumetric flow, h = head, η = efficiency.

Net positive suction head (NPSH)

NPSH_avail = (P_atm − P_vapor)/(ρg) + z − h_L,suction. It must exceed NPSH_required to avoid cavitation.

Cavitation

Formation and collapse of vapor bubbles when local pressure drops below the fluid's vapor pressure. It causes noise, vibration, and pump erosion.

Mach number

Ma = V/c, where c = √(kRT) for an ideal gas. Ma < 1 subsonic, Ma = 1 sonic, Ma > 1 supersonic.

Speed of sound (ideal gas)

c = √(kRT), where k = c_p/c_v, R = specific gas constant, T = absolute temperature. For air at 20°C c ≈ 343 m/s.

Hydraulic diameter

D_h = 4A/P, where A = cross-sectional flow area and P = wetted perimeter. Used for noncircular ducts in Re and friction.

Drag force

F_D = C_D (ρV²/2) A. C_D = drag coefficient, A = reference area, ρ = density, V = relative velocity.

Lift force

F_L = C_L (ρV²/2) A. Lift acts perpendicular to flow; C_L is the lift coefficient and A is the planform area.

Dynamic vs kinematic viscosity

Kinematic ν = μ/ρ. Dynamic μ has units Pa·s; ν has units m²/s. ν appears directly in Reynolds number.

Manometer equation

ΔP = ρgΔh. A height difference of fluid in a manometer column measures a pressure difference.

Venturi / orifice flow

Q = C_d A₂ √[2ΔP/(ρ(1 − (A₂/A₁)²))]. Flow rate is found from the measured pressure drop across the restriction.

Pitot tube (stagnation pressure)

V = √(2(P_0 − P_static)/ρ). The dynamic pressure P_0 − P_static gives the local velocity.

Affinity laws (pumps/fans)

Q ∝ N, H ∝ N², P ∝ N³ at constant impeller diameter. Flow scales with speed, head with speed squared, power with speed cubed.

Specific weight

γ = ρg. Weight per unit volume; for water γ ≈ 9.81 kN/m³ (62.4 lbf/ft³).

Buoyant force (Archimedes)

F_B = ρ_fluid g V_displaced. The upward force equals the weight of fluid displaced by the submerged volume.

Momentum equation (control volume)

ΣF = ṁ(V_out − V_in). The net force equals the rate of change of momentum; used for thrust and pipe-bend forces.

Friction factor: Darcy vs Fanning

f_Darcy = 4 f_Fanning. Darcy is used in h_L = f(L/D)(V²/2g); always confirm which convention a chart uses.

Head vs pressure conversion

h = P/(ρg) = P/γ. Head (m or ft) is pressure expressed as an equivalent fluid column height.

Boundary layer

The thin near-wall region where velocity rises from zero (no-slip) to free-stream. It governs viscous drag and separation.

Volumetric flow vs mass flow

Q = AV (volumetric, m³/s); ṁ = ρAV = ρQ (mass, kg/s). Related by the fluid density.

Series vs parallel pipes

Series: same flow, head losses add. Parallel: same head loss across each branch, flows add.

Total dynamic head (system)

TDH = static lift + friction losses + velocity head + pressure head. The total head a pump must overcome.

Surface tension pressure (droplet)

ΔP = 2σ/r for a droplet; ΔP = 4σ/r for a soap bubble (two surfaces). σ = surface tension, r = radius.

Reynolds transport theorem

It relates the rate of change of a system property to control-volume storage plus net flux across the control surface.

Thermal resistance (conduction, plane wall)

R = L/(kA). Heat flow q = ΔT/R_total; resistances in series add like electrical resistors.

Thermal resistance (convection)

R = 1/(hA). The film resistance at a surface; larger h gives smaller resistance and more heat transfer.

Conduction through a cylindrical wall

R = ln(r₂/r₁)/(2πkL). Resistance for radial heat flow through a pipe or cylinder of length L.

Overall heat transfer coefficient U

1/(UA) = ΣR = 1/(h_i A_i) + R_wall + 1/(h_o A_o). Then q = UA·ΔT for a heat exchanger.

Stefan-Boltzmann radiation law

q = εσA(T_s⁴ − T_surr⁴), with σ = 5.67×10⁻⁸ W/(m²·K⁴), ε = emissivity, T in Kelvin.

Log mean temperature difference (LMTD)

LMTD = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂). Used in q = UA·LMTD·F for heat exchangers; F is a correction factor.

Effectiveness-NTU method

ε = Q_actual/Q_max, NTU = UA/C_min. Preferred when outlet temperatures are unknown; ε depends on NTU and C_min/C_max.

Biot number

Bi = hL_c/k_solid, L_c = V/A_s. Bi < 0.1 justifies the lumped-capacitance assumption (uniform internal temperature).

Lumped-capacitance transient

T(t) − T_∞ = (T_i − T_∞)·exp(−t/τ), τ = ρVc/(hA_s). Valid when internal conduction is fast relative to surface convection (Bi < 0.1).

Nusselt number

Nu = hL/k. The dimensionless convection coefficient; a higher Nu means convection dominates over pure conduction.

Prandtl number

Pr = ν/α = μc_p/k. The ratio of momentum to thermal diffusivity; ~0.7 for air, ~7 for water near room temperature.

Thermal diffusivity

α = k/(ρc_p). It measures how fast a material conducts heat relative to storing it; units m²/s.

Fin efficiency

η_fin = q_actual/q_ideal, where q_ideal is the heat if the whole fin were at base temperature. Approaches 1 for short, high-conductivity fins.

Fin effectiveness

ε_fin = q_with fin/q_without fin. A fin is worthwhile only when ε_fin > 1 (typically ε > 2 to justify).

Heat exchanger duty (one stream)

q = ṁ c_p ΔT for the fluid (single-phase). For phase change use q = ṁ·h_fg at constant temperature.

Heat capacity rate

C = ṁ c_p. In a heat exchanger the stream with the smaller C (C_min) experiences the larger temperature change.

Radiation view factor reciprocity

A₁F₁₂ = A₂F₂₁. The fraction of radiation leaving surface 1 reaching surface 2, weighted by area, equals the reverse.

Blackbody vs gray body

A blackbody has ε = 1 and absorbs all incident radiation. A gray body has constant ε < 1 across wavelengths.

Wien's displacement law

λ_max·T = 2898 μm·K. The wavelength of peak blackbody emission shifts inversely with absolute temperature.

Parallel-flow vs counter-flow exchanger

Counter-flow gives a larger LMTD and higher effectiveness for the same area; it can also raise cold-outlet above hot-outlet temperature.

Fouling factor

R_f adds thermal resistance from deposits: 1/U_dirty = 1/U_clean + R_f. It degrades heat-exchanger performance over time.

Critical radius of insulation

r_cr = k/h for a cylinder. Below r_cr, adding insulation increases heat loss; above it, insulation reduces loss.

Contact resistance

An added interfacial resistance R_c = 1/(h_c A) at imperfect surface contacts; reduced by smoother surfaces or thermal paste.

Conduction shape factor

q = Sk·ΔT, where S is a geometry-dependent shape factor for multidimensional steady conduction (e.g., buried pipe).

Forced vs free convection

Forced: fluid motion driven externally (fan, pump); correlated by Re and Pr. Free (natural): motion from buoyancy; correlated by Grashof or Rayleigh number.

Rayleigh number

Ra = Gr·Pr = gβ(ΔT)L³/(να). Governs natural convection; exceeding a critical value triggers the onset of fluid motion.

Fourier number

Fo = αt/L_c². Dimensionless time for transient conduction; larger Fo means the body has had more time to respond thermally.

Thermal conductivity ranking

Metals (high, copper ~400 W/m·K) > liquids > gases > insulation (low, ~0.04 W/m·K). Higher k transmits heat more readily.

Axial (normal) stress

σ = P/A. Direct stress from an axial load P acting over cross-sectional area A.

Direct shear stress

τ = V/A. Shear stress from a transverse force V over the area resisting shear (e.g., bolts, pins).

Torsional shear stress

τ = Tr/J, where T = torque, r = radius, J = polar moment of inertia. Maximum at the outer surface of a shaft.

Polar moment of inertia (solid shaft)

J = πd⁴/32 for a solid circular shaft of diameter d. For a hollow shaft J = π(d_o⁴ − d_i⁴)/32.

Angle of twist

θ = TL/(JG), where T = torque, L = length, J = polar moment of inertia, G = shear modulus.

Mohr's circle

A graphical tool plotting normal stress (x-axis) vs shear stress (y-axis) to find principal stresses and maximum shear at any orientation.

Principal stresses (2D)

σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ_xy²]. Principal planes carry zero shear stress.

Maximum in-plane shear stress

τ_max = √[((σ_x − σ_y)/2)² + τ_xy²] = (σ₁ − σ₂)/2. It is the radius of Mohr's circle.

Endurance limit

S_e = the fully reversed stress amplitude a material can endure for infinite life. For steel S_e' ≈ 0.5 S_ut (up to ~700 MPa cap).

Goodman criterion (fatigue)

σ_a/S_e + σ_m/S_ut = 1/n. It relates alternating stress σ_a and mean stress σ_m to fatigue safety factor n.

Soderberg criterion (fatigue)

σ_a/S_e + σ_m/S_y = 1/n. More conservative than Goodman because it uses yield strength S_y instead of ultimate strength.

Stress concentration factor

σ_max = K_t σ_nom. K_t accounts for local stress rise at holes, notches, and fillets. Fatigue uses K_f from notch sensitivity.

Maximum shear stress theory (Tresca)

Failure when τ_max = S_y/2, i.e., σ₁ − σ₃ = S_y. A conservative yield criterion for ductile materials.

Distortion energy theory (von Mises)

σ' = √(σ₁² − σ₁σ₂ + σ₂²) (2D). Yielding occurs when σ' = S_y; the standard ductile-failure criterion.

Euler buckling load

P_cr = π²EI/(KL)². K depends on end conditions (1 pinned-pinned, 0.5 fixed-fixed, 2 fixed-free). Slender columns fail by buckling.

Slenderness ratio

λ = KL/r, where r = √(I/A) is the radius of gyration. High slenderness means buckling controls over crushing.

Beam deflection (simply supported, center load)

δ_max = PL³/(48EI) at the center for a point load P on a simply supported span L.

Beam deflection (cantilever, end load)

δ_max = PL³/(3EI) at the free end for a point load P on a cantilever of length L.

Transverse shear stress in a beam

τ = VQ/(Ib), where V = shear force, Q = first moment of area, I = moment of inertia, b = width. Maximum at the neutral axis.

Section modulus

S = I/c. Bending stress σ = M/S; a larger section modulus reduces bending stress for a given moment.

Spring rate (helical compression)

k = Gd⁴/(8D³N), where G = shear modulus, d = wire dia, D = mean coil dia, N = active coils.

Spring shear stress

τ = K_s·8FD/(πd³), where K_s is the Wahl/shear correction factor, F = force, D = mean dia, d = wire dia.

Springs in series vs parallel

Series: 1/k_eq = Σ1/k_i (softer). Parallel: k_eq = Σk_i (stiffer). Opposite of resistor rules.

Bearing L10 life

L10 = (C/P) raised to a, × 10⁶ revolutions; a = 3 for ball, 10/3 for roller. C = dynamic load rating, P = equivalent load. 90% survive L10.

Bolt preload / proof load

F_i ≈ 0.75 F_p (reused) or 0.90 F_p (permanent), F_p = A_t·S_p. Preload keeps the joint in compression and resists fatigue.

Bolt tightening torque

T = K·F_i·d, where K ≈ 0.2 (lubricated ~0.15), F_i = preload, d = nominal diameter.

Gear ratio

GR = N_out/N_in = ω_in/ω_out = T_out/T_in (ideal). Tooth counts set speed and torque ratios.

Spur gear tangential load

W_t = 2T/d = P/V. T = torque, d = pitch diameter, P = transmitted power, V = pitch-line velocity.

Lewis bending equation (gears)

σ = W_t·P_d/(F·Y), where P_d = diametral pitch, F = face width, Y = Lewis form factor. Estimates gear-tooth bending stress.

Key (shaft) shear failure

τ = F/(w·L) = 2T/(d·w·L). F = tangential force at shaft surface, w = key width, L = key length, d = shaft diameter.

Weld (fillet) shear stress

τ = F/(0.707·h·L), where h = weld leg size and L = weld length. The throat area governs strength.

Hoop (circumferential) stress, thin wall

σ_θ = Pr/t. For a thin-walled pressure vessel (r/t > 10); twice the longitudinal stress.

Longitudinal stress, thin-wall cylinder

σ_L = Pr/(2t). Half the hoop stress, so cylinders typically fail by longitudinal seams splitting.

Thin-wall sphere stress

σ = Pr/(2t), equal in all directions. Spheres carry pressure more efficiently than cylinders for the same r and t.

Combined axial and bending stress

σ = P/A ± Mc/I. Superpose direct and bending stresses; the maximum occurs where they add.

Press fit / interference pressure

Interference δ creates a contact pressure p, generating tangential (hoop) and radial stresses in the mating parts per thick-wall (Lamé) equations.

Power-torque-speed relation

P = Tω = 2πNT, where ω in rad/s and N in rev/s. Useful for sizing shafts, motors, and couplings.

Shaft sizing for combined load (ASME)

d³ = (16/π) √[(K_b M)² + (K_t T)²] / τ_allow. Combines bending and torsion with shock/fatigue factors.

Thermal stress (constrained bar)

σ = EαΔT for a fully restrained member. α = thermal expansion coefficient; heating a restrained bar induces compressive stress.

Castigliano's theorem

δ = ∂U/∂P. The deflection at a load equals the partial derivative of strain energy with respect to that load.

Strain energy in axial member

U = P²L/(2AE). Energy stored elastically; basis for energy methods like Castigliano's theorem.

Allowable stress design

σ_allow = S_y/FS (or S_ut/FS). Working stress kept below the strength divided by a factor of safety.

Hooke's law

σ = Eε in the elastic region. Stress is proportional to strain; E is Young's modulus (modulus of elasticity).

Young's modulus

E = σ/ε, the slope of the elastic stress-strain line. Steel ≈ 200 GPa, aluminum ≈ 70 GPa; measures stiffness, not strength.

Poisson's ratio

ν = −ε_lateral/ε_axial. Typically 0.25–0.35 for metals; relates transverse contraction to axial elongation.

Shear modulus relation

G = E/[2(1 + ν)]. Links the shear (rigidity) modulus to Young's modulus and Poisson's ratio.

Yield vs ultimate strength

Yield strength S_y = onset of permanent deformation; ultimate strength S_ut = maximum stress before fracture/necking.

0.2% offset yield

Yield defined by a line parallel to the elastic slope, offset 0.2% strain, intersecting the stress-strain curve. Used when no sharp yield point exists.

Ductile vs brittle material

Ductile materials yield and deform plastically with warning (high % elongation); brittle materials fracture suddenly with little plastic strain.

Toughness vs resilience

Toughness = total area under the stress-strain curve (energy to fracture). Resilience = elastic area up to yield.

Engineering vs true stress

Engineering stress uses the original area; true stress uses the instantaneous area, so true stress is higher after necking begins.

Hardness testing

Brinell, Rockwell, and Vickers measure resistance to indentation. Hardness correlates roughly with tensile strength (S_ut ≈ 3.45·HB MPa).

Strain hardening (cold work)

Plastic deformation increases strength and hardness while reducing ductility, by raising dislocation density.

Annealing

Heating then slow cooling to relieve stress, soften, and increase ductility by recrystallizing the grain structure.

Quenching and tempering

Rapid cooling (quench) forms hard martensite; tempering reheats to trade some hardness for toughness and reduced brittleness.

Iron-carbon phases

Ferrite (soft, BCC), austenite (FCC, high-temp), cementite (Fe₃C, hard), and pearlite (ferrite + cementite layers).

Eutectoid composition (steel)

0.76% carbon transforms fully to pearlite at ~727°C. Hypoeutectoid < 0.76%C, hypereutectoid > 0.76%C.

Creep

Slow, time-dependent plastic deformation under constant stress at high temperature (typically > 0.4 T_melt absolute).

Fatigue (S-N curve)

A plot of stress amplitude vs cycles to failure. Steels show an endurance limit (flat region); aluminum does not.

Stress relaxation vs creep

Creep = increasing strain at constant stress. Stress relaxation = decreasing stress at constant strain (e.g., loosening bolts).

Ductile-to-brittle transition

Some BCC metals (e.g., carbon steel) become brittle below a transition temperature; Charpy impact testing identifies it.

Thermal expansion coefficient

ΔL = αLΔT. α (per °C) measures length change with temperature; steel α ≈ 12×10⁻⁶/°C, aluminum ≈ 23×10⁻⁶/°C.

Corrosion (galvanic)

When dissimilar metals contact in an electrolyte, the more anodic (active) metal corrodes preferentially, protecting the cathodic metal.

Stress-strain curve regions

Elastic (linear) → yield → plastic (strain hardening) → ultimate (peak) → necking → fracture.

Composite rule of mixtures

E_c = E_f·V_f + E_m·V_m (longitudinal). Composite stiffness is the volume-weighted average of fiber and matrix moduli.

Modulus of resilience

U_r = σ_y²/(2E). The elastic strain energy per unit volume a material absorbs up to yielding.

Percent elongation / reduction in area

Measures ductility: %EL = (L_f − L_0)/L_0 ×100; %RA = (A_0 − A_f)/A_0 ×100. Higher values mean more ductile.

Fracture toughness

K_IC = critical stress intensity. Fracture when K = Yσ√(πa) reaches K_IC; a = crack length, Y = geometry factor.

Bulk modulus

K = −V(dP/dV) = E/[3(1 − 2ν)]. Resistance to uniform (volumetric) compression.

Alloying purpose

Adding elements (e.g., C, Cr, Ni) to a base metal to improve strength, hardness, corrosion resistance, or hardenability.

Newton's second law

ΣF = ma (translation); ΣM = Iα (rotation). Net force/moment equals mass/inertia times acceleration.

Kinematics (constant acceleration)

v = v₀ + at; s = s₀ + v₀t + ½at²; v² = v₀² + 2a(s − s₀). Valid only for constant acceleration.

Work-energy theorem

W_net = ΔKE = ½m(v₂² − v₁²). The net work done on a body equals its change in kinetic energy.

Conservation of mechanical energy

KE₁ + PE₁ = KE₂ + PE₂ when only conservative forces act. ½mv² + mgh is constant.

Impulse-momentum

F·Δt = Δ(mv). Impulse equals change in linear momentum; basis for impact and collision analysis.

Coefficient of restitution

e = (v₂' − v₁')/(v₁ − v₂), relative separation over approach velocity. e = 1 elastic, e = 0 perfectly plastic.

Rotational kinetic energy

KE = ½Iω². I = mass moment of inertia, ω = angular velocity. Adds to translational KE for rolling bodies.

Mass moment of inertia (disk)

I = ½mr² for a solid disk/cylinder about its central axis. A thin ring is mr²; a sphere is (2/5)mr².

Parallel axis theorem

I = I_cg + md². Moment of inertia about a parallel axis offset distance d from the centroidal axis.

Damping ratio

ζ = c/c_c, c_c = 2√(km) = 2mωn. ζ < 1 underdamped (oscillates), ζ = 1 critical, ζ > 1 overdamped.

Damped natural frequency

ω_d = ωn√(1 − ζ²). Damping lowers the oscillation frequency below the undamped natural frequency.

Resonance

Large amplitude growth when forcing frequency approaches ωn. Light damping yields very high peaks; controlled by adding damping or shifting ωn.

Transmissibility

TR = transmitted force/applied force. For isolation, TR < 1 requires the frequency ratio r = ω/ωn > √2.

Logarithmic decrement

δ = ln(x_n/x_{n+1}) = 2πζ/√(1 − ζ²). It estimates damping ratio from the decay of successive oscillation peaks.

Simple pendulum frequency

ωn = √(g/L), where L = pendulum length. Independent of mass for small-angle oscillation.

Torsional vibration frequency

ωn = √(k_t/I), where k_t = torsional stiffness (T/θ) and I = mass moment of inertia. Analogous to translational ωn = √(k/m).

Centripetal acceleration

a_c = v²/r = ω²r, directed toward the center. The radial acceleration that keeps a body in circular motion.

Normal and tangential acceleration

a_t = dv/dt (tangential, speed change); a_n = v²/ρ (normal, direction change). Total a = √(a_t² + a_n²).

Angular impulse-momentum

ΣM·Δt = Δ(Iω). Angular impulse equals the change in angular momentum about an axis.

Friction force

F_f ≤ μN. Static friction up to μ_s N prevents motion; once sliding, kinetic friction = μ_k N (usually μ_k < μ_s).

Rolling without slipping

v = ωr and a = αr at the contact point. The contact point is instantaneously at rest; no relative sliding occurs.

Critical speed of a rotating shaft

The speed where rotational frequency equals the shaft's lateral natural frequency, causing large whirl amplitudes. Operate away from it.

Forced response amplitude (SDOF)

X = (F₀/k)/√[(1 − r²)² + (2ζr)²], r = ω/ωn. The dynamic magnification factor peaks near resonance.

Static deflection and ωn

ωn = √(g/δ_st), where δ_st = mg/k is the static deflection. A quick estimate of natural frequency.

Vibration isolation principle

A soft mount (low ωn) reduces transmitted force when r = ω/ωn > √2; below √2 it can amplify vibration.

Equations of equilibrium (statics)

ΣF_x = 0, ΣF_y = 0, ΣM = 0 for a 2D body in static equilibrium. Three equations solve up to three unknown reactions.

D'Alembert's principle

Treat −ma as an inertial force so a dynamics problem can be solved as static equilibrium: ΣF − ma = 0.

Power in rotational motion

P = Tω. Instantaneous power equals torque times angular velocity (rad/s).

Sensible heat

q = ṁ c_p ΔT_dry-bulb. Heat that changes air temperature without changing moisture content.

Latent heat (air conditioning)

q = ṁ·h_fg·Δω, where Δω is the humidity-ratio change. Energy associated with adding or removing moisture (condensation).

Total cooling load (air)

q_total = ṁ Δh = q_sensible + q_latent. The enthalpy change of the air stream across a cooling coil.

Humidity ratio

ω = m_water/m_dry air = 0.622·P_v/(P − P_v). Mass of water vapor per mass of dry air.

Relative humidity

φ = P_v/P_sat at the air's temperature. The ratio of actual vapor pressure to saturation pressure, expressed as a percentage.

Dew point temperature

The temperature at which air becomes saturated (φ = 100%) and moisture begins to condense, at constant pressure and ω.

Wet-bulb temperature

The temperature from evaporative cooling at a wetted thermometer. Equals dry-bulb when air is saturated; lower otherwise.

Psychrometric chart

Plots dry-bulb (x-axis) vs humidity ratio (y-axis), with lines of constant φ, wet-bulb, and enthalpy for moist-air processes.

Vapor-compression refrigeration cycle

Four steps: compressor → condenser (reject heat) → expansion valve (throttle) → evaporator (absorb heat). Refrigerant cycles between phases.

Refrigeration COP (vapor compression)

COP = q_evap/w_comp = (h₁ − h₄)/(h₂ − h₁). Evaporator cooling effect over compressor work input.

Ton of refrigeration

1 ton = 12,000 Btu/h = 3.517 kW. The cooling rate to freeze one short ton of water in 24 hours.

Evaporator and condenser roles

Evaporator absorbs heat at low pressure/temperature (cooling effect); condenser rejects heat at high pressure/temperature.

Superheat and subcooling

Superheat: vapor heated above saturation leaving the evaporator (protects compressor). Subcooling: liquid cooled below saturation leaving the condenser (more capacity).

Chiller

A machine that cools water (or brine) via a refrigeration cycle; chilled water is piped to air-handler coils for building cooling.

Cooling tower

Rejects condenser heat by evaporative cooling of water against an air stream; approach = leaving water minus ambient wet-bulb.

Air handling unit (AHU)

Conditions and distributes air using fans, heating/cooling coils, filters, and dampers within a duct system.

Fan laws

Q ∝ N, ΔP ∝ N², Power ∝ N³ at fixed diameter/density. Identical in form to pump affinity laws.

Fan power

P = QΔP/η_fan, where Q = airflow, ΔP = static pressure rise, η = fan efficiency.

Duct friction loss

Pressure drop per unit length rises with airflow and falls with duct size; sized via friction-rate charts (e.g., ~0.08 in. wg per 100 ft).

Velocity pressure (air)

VP = ρV²/2. In I-P units VP (in. wg) = (V/4005)² for standard air. Total pressure = static + velocity pressure.

Air mixing (return + outdoor)

Mixed condition is the mass-flow-weighted average: T_mix = (ṁ_oa T_oa + ṁ_ra T_ra)/ṁ_total; same form for humidity ratio.

Sensible heat ratio (SHR)

SHR = q_sensible/q_total. It sets the slope of the coil process line on the psychrometric chart.

Heating/cooling degree days

Sum of daily differences between a base temperature (e.g., 65°F) and mean outdoor temperature; estimates seasonal energy demand.

Ventilation rate (ASHRAE 62.1)

Outdoor air = R_p·P_z + R_a·A_z, where R_p is per person, P_z = occupancy, R_a per area, A_z = floor area.

Thermal comfort (ASHRAE 55)

Comfort depends on temperature, humidity, air speed, metabolic rate, and clothing; the comfort zone is roughly 68–78°F at moderate humidity.

Heat pump heating mode

A reversing valve swaps evaporator and condenser so the outdoor coil absorbs heat and the indoor coil rejects it; COP_HP = COP_ref + 1.

Evaporative cooling

Adiabatic humidification lowers dry-bulb toward the wet-bulb temperature; effective in hot, dry climates with low energy use.

Economizer

Uses cool outdoor air to provide free cooling when conditions allow, reducing mechanical refrigeration load and energy.

Refrigerant pressure-enthalpy diagram

P–h diagram maps the vapor-compression cycle: horizontal evaporator/condenser legs and the throttling line at constant enthalpy.

Bypass factor (cooling coil)

BF = fraction of air passing the coil unchanged. ADP (apparent dew point) and BF set the coil's leaving air condition.

Pump/pipe sizing for chilled water

q = ṁ c_p ΔT; flow GPM ≈ Btu/h ÷ (500·ΔT°F) for water. A larger design ΔT lowers required flow and pump energy.

Air change rate

ACH = (60·Q_cfm)/V_room. Number of times the room air volume is replaced per hour.

Static vs total vs velocity pressure

Total pressure = static pressure + velocity pressure. Static pushes outward on duct walls; velocity pressure relates to air speed.

VAV system

Variable air volume modulates supply airflow (not temperature) to match changing zone loads, saving fan energy at part load.

Refrigerant selection factors

Choose by operating pressures/temperatures, COP, safety (toxicity/flammability), and environmental impact (ODP and GWP).

Coil leaving air enthalpy

From energy balance: h_leaving = h_entering − q_coil/ṁ_air. The process line slope is set by the sensible heat ratio.

Outdoor air load

q_oa = ṁ_oa(h_oa − h_room). Conditioning ventilation air is often a large part of total HVAC load in humid climates.

Specific volume of moist air

v = R_a T/(P − P_v) per kg dry air. Used to convert volumetric airflow to mass flow on the psychrometric chart.

Rankine cycle

Pump → boiler (heat in) → turbine (work out) → condenser (heat out). The standard vapor power cycle using water/steam.

Rankine thermal efficiency

η = w_net/q_in = (w_turbine − w_pump)/q_boiler. Pump work is small relative to turbine work in liquid pumping.

Pump work (Rankine)

w_pump = v_f(P₂ − P₁), using saturated-liquid specific volume. Liquids are nearly incompressible, so pump work is small.

Reheat Rankine cycle

Steam is expanded, reheated, then expanded again. It raises average heat-addition temperature and keeps turbine exit quality high.

Regenerative Rankine (feedwater heating)

Bleed steam preheats feedwater in heaters, raising cycle efficiency by reducing boiler heat addition at low temperature.

Brayton cycle

Compressor → combustor (heat in) → turbine → exhaust. The ideal gas-turbine cycle, with all components at steady flow.

Brayton cycle efficiency

η = 1 − 1/[r_p raised to (k−1)/k], where r_p = P₂/P₁ pressure ratio and k = c_p/c_v. Efficiency rises with pressure ratio.

Regeneration in Brayton

A recuperator uses hot turbine exhaust to preheat compressed air before combustion, raising efficiency at low pressure ratios.

Combined cycle

A Brayton (gas-turbine) topping cycle exhausts to a Rankine (steam) bottoming cycle, reaching combined efficiencies over 60%.

Energy balance (control volume)

Q̇ − Ẇ = Σ_out ṁ(h + V²/2 + gz) − Σ_in ṁ(h + V²/2 + gz). The steady-flow first law for multiple streams.

Mass balance (steady state)

Σṁ_in = Σṁ_out. Mass is conserved across a control volume with no accumulation at steady state.

Isentropic stagnation properties

T_0 = T(1 + (k−1)/2·Ma²). Stagnation (total) temperature is reached when flow is brought to rest isentropically.

Choked flow

Flow reaches Ma = 1 at the throat of a converging nozzle; mass flow then becomes maximum and independent of further downstream pressure drop.

Converging-diverging nozzle

Subsonic flow accelerates in the converging section; once sonic at the throat, the diverging section accelerates it supersonically.

Normal shock wave

An abrupt supersonic-to-subsonic transition with rising pressure, temperature, and entropy, but falling stagnation pressure.

Compressible isentropic relations

T_0/T = 1 + (k−1)/2·Ma²; P_0/P = (T_0/T) raised to k/(k−1). Link static and stagnation states for an ideal gas.

Stoichiometric combustion

The exact air-fuel ratio for complete combustion with no excess oxygen, producing CO₂ and H₂O. For methane, AFR ≈ 17.2 by mass.

Excess air

Air supplied beyond stoichiometric to ensure complete combustion; too much wastes energy heating extra nitrogen.

Heating value (HHV vs LHV)

HHV includes the latent heat of water vapor condensation; LHV does not. HHV > LHV by the fuel's water vapor latent heat.

Air-fuel ratio

AFR = m_air/m_fuel. Equivalence ratio φ = AFR_stoich/AFR_actual; φ > 1 is fuel-rich, φ < 1 is fuel-lean.

Adiabatic flame temperature

The maximum combustion temperature with no heat loss; reached at stoichiometric conditions and lowered by excess air or incomplete combustion.

Boiler efficiency

η = useful heat to steam/fuel energy input = ṁ_steam(h_out − h_in)/(ṁ_fuel·HV). Stack losses are the main inefficiency.

Pump/turbine specific speed

N_s = N√Q / H raised to (3/4). A dimensionless-style index that guides selection between centrifugal, mixed, and axial machine types.

Centrifugal vs positive-displacement pump

Centrifugal: flow varies with system head (uses a pump curve). Positive-displacement: near-constant flow regardless of pressure.

System curve vs pump curve

Operating point is where the pump head-flow curve intersects the system resistance curve (static + friction).

Turbine work (steady flow)

w_t = h_in − h_out (adiabatic, negligible KE/PE). Actual work = isentropic work times turbine efficiency.

Nozzle exit velocity

V_exit = √(2(h_in − h_out)) for an adiabatic nozzle with small inlet velocity. Enthalpy drop converts to kinetic energy.

Heat rate (power plant)

Heat rate = fuel energy input per unit electrical output (Btu/kWh). Lower heat rate means higher overall efficiency.

Cogeneration (CHP)

Combined heat and power uses turbine exhaust or extraction steam for process heating, raising total fuel utilization well above electric-only efficiency.

Engineering economics: present worth

P = F/(1 + i)ⁿ. The present value of a future amount F discounted n periods at interest rate i.

Future worth factor

F = P(1 + i)ⁿ. A present amount grows to F after n compounding periods at rate i.

Uniform series (annuity) capital recovery

A = P·[i(1 + i)ⁿ]/[(1 + i)ⁿ − 1]. Converts a present amount into n equal periodic payments (A/P factor).

Sinking fund factor

A = F·i/[(1 + i)ⁿ − 1]. The uniform deposit needed each period to accumulate a future sum F (A/F factor).

Effective vs nominal interest

i_eff = (1 + r/m) raised to m, − 1, where r = nominal annual rate and m = compounding periods per year. Effective exceeds nominal when m > 1.

Rate of return / payback

ROR is the interest rate making net present worth zero. Payback period = time for cumulative cash inflows to recover the initial investment.

Straight-line depreciation

D = (cost − salvage)/n. Equal depreciation each year over the asset's useful life n.

Benefit-cost ratio

B/C = present worth of benefits / present worth of costs. A project is economically justified when B/C ≥ 1.

SI base units

Length (m), mass (kg), time (s), temperature (K), current (A), amount (mol), luminous intensity (cd).

Force unit consistency

1 N = 1 kg·m/s². In US units, 1 lbf = 32.174 lbm·ft/s², requiring the g_c conversion constant.

Common pressure conversions

1 atm ≈ 101.325 kPa ≈ 14.7 psi ≈ 760 mmHg ≈ 33.9 ft of water. 1 bar = 100 kPa.

Energy and power units

1 hp = 0.746 kW = 550 ft·lbf/s; 1 Btu = 1055 J; 1 kWh = 3.6 MJ; 1 ton refrigeration = 3.517 kW.

Temperature conversions

K = °C + 273.15; °R = °F + 459.67; °F = 1.8·°C + 32. ΔT: 1 K = 1°C, 1°R = 1°F.

ASME Boiler & Pressure Vessel Code

Governs design, fabrication, and inspection of boilers and pressure vessels for safe operation under pressure.

Codes vs standards

A code states what must be done and is enforceable when adopted into law; a standard gives how-to methods and may be referenced by a code.

Feedback control loop

A sensor measures output, a controller compares it to setpoint, and an actuator corrects the error to drive deviation toward zero.

PID controller

Output = K_p·e + K_i∫e dt + K_d(de/dt). Proportional acts on present error, integral on accumulated error, derivative on rate of change.

Open vs closed loop control

Open loop has no feedback (acts blindly on input). Closed loop measures output and corrects error, improving accuracy and disturbance rejection.

Accuracy vs precision

Accuracy = closeness to the true value. Precision = repeatability of measurements. A device can be precise but inaccurate (biased).

Thermocouple

Generates a voltage from a temperature difference between two dissimilar metal junctions (Seebeck effect); used for wide-range temperature sensing.

RTD vs thermistor

RTD: metal (platinum) resistance rises nearly linearly with temperature, accurate and stable. Thermistor: ceramic, more sensitive but nonlinear.

Strain gauge

Measures strain via resistance change: gauge factor GF = (ΔR/R)/ε. Typically wired in a Wheatstone bridge for sensitivity.

Factor of safety in design codes

Codes mandate minimum safety factors based on load type, material reliability, and failure consequence to protect against uncertainty.

OSHA / workplace safety role

Sets and enforces occupational safety standards (e.g., machine guarding, lockout-tagout, pressure systems) to protect workers.

Lockout-tagout (LOTO)

A procedure isolating and de-energizing equipment during service so it cannot start unexpectedly, preventing injury.

Significant figures in results

Report answers consistent with the least precise input data; carrying excess digits implies false precision in engineering calculations.

Pump/system efficiency chain

Overall efficiency = motor × drive × pump efficiencies. Multiply component efficiencies to get the wire-to-water efficiency.

Pressure relief / safety valve

A spring-loaded device that opens to limit system pressure to a safe set point, protecting vessels and piping from overpressure.

Gradient series factor

Converts a uniformly increasing cash flow (arithmetic gradient G per period) into an equivalent uniform series or present worth (A/G, P/G factors).

P&ID diagram

A piping and instrumentation diagram showing equipment, piping, valves, and instruments with their control connections for a process system.

Mollier (h-s) diagram

Plots enthalpy vs entropy for steam; used to find turbine work and exit quality by reading the enthalpy drop along an isentropic line.

Triple point and critical point

Triple point: solid, liquid, and vapor coexist (water 0.01°C, 0.6113 kPa). Critical point: liquid and vapor become indistinguishable (water 374°C, 22.1 MPa).

Energy grade line vs hydraulic grade line

EGL = P/γ + V²/2g + z (total head). HGL = EGL minus velocity head. EGL drops along flow due to friction.

Heat exchanger NTU max

Q_max = C_min(T_hot,in − T_cold,in). The thermodynamic limit if the C_min stream reached the opposite inlet temperature.

Moment of inertia (rectangle)

I = bh³/12 about the centroidal axis. b = width, h = height; the cube on height makes depth dominate bending stiffness.

Coriolis acceleration

a_cor = 2ω × v_rel. Appears when a point moves radially on a rotating reference frame; magnitude 2ωv_rel.