- 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.
