Your FREE PE Mechanical Flashcards 2026 – 300+ Cards

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

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.

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).

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

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

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

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.

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

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

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

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).

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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: 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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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).

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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).

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.

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.

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.

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

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

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.

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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).

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

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.

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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