ME POST cheat sheet

Quadratic ax²+bx+c=0: roots x = [−b ± √(b²−4ac)]/(2a); discriminant b²−4ac > 0 gives two real roots, = 0 one repeated root, < 0 complex conjugate pair.

Derivative rules: d/dx(xⁿ)=nxⁿ⁻¹, d/dx(eˣ)=eˣ, d/dx(ln x)=1/x, product rule (uv)'=u'v+uv', quotient rule (u/v)'=(u'v−uv')/v², chain rule dy/dx = dy/du · du/dx.

Integration by parts: ∫u dv = uv − ∫v du; choose u by LIATE order (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential).

Laplace transforms: L{1}=1/s, L{e^(at)}=1/(s−a), L{sin ωt}=ω/(s²+ω²), L{cos ωt}=s/(s²+ω²), L{f'}=sF(s)−f(0).

First-order linear ODE y' + P(x)y = Q(x) has integrating factor μ = e^∫P dx and solution y = (1/μ)∫μQ dx.

Second-order system with characteristic roots: overdamped when discriminant positive, critically damped when zero, underdamped when negative (oscillatory).

Statistics: mean = Σx/n, sample standard deviation s = √[Σ(x−x̄)²/(n−1)], coefficient of variation = s/x̄; normal distribution has 68.3% of values within ±1σ.

Vector calculus: gradient gives direction of steepest ascent, divergence of a vector field measures source density, curl measures rotation; ∇·(∇×F)=0.

Complex numbers: Euler form z = r·e^(jθ) with r = √(a²+b²) and θ = atan2(b,a); multiplication multiplies magnitudes and adds angles.

Continuity for steady incompressible flow: A₁V₁ = A₂V₂; mass conservation in general form ṁ = ρAV = constant.

Bernoulli along a streamline (inviscid, steady, incompressible): p/ρg + V²/2g + z = constant; add head-loss h_L and pump head h_p for real systems.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below Re ≈ 2300 and turbulent above ≈ 4000, transitional between.

Darcy–Weisbach head loss h_f = f(L/D)(V²/2g); for laminar pipe flow the friction factor is f = 64/Re.

Hagen–Poiseuille laminar pipe flow: Q = πΔp D⁴/(128μL), so flow scales with the fourth power of diameter.

Pump affinity laws: Q ∝ N, H ∝ N², P ∝ N³ for geometrically similar pumps at constant impeller diameter; trimming diameter gives Q ∝ D, H ∝ D².

Specific speed N_s = N√Q/H^(3/4) classifies impellers: low N_s suits radial/centrifugal, high N_s suits axial flow; cavitation risk is tracked by NPSH available versus NPSH required.

Drag force F_D = C_D·½ρV²A; for a sphere at low Re, Stokes law gives F_D = 3πμVD.

First law for a closed system: Q − W = ΔU; for a steady-flow control volume: q − w = Δh + ΔKE + ΔPE (per unit mass).

Ideal gas relations: pV = mRT, cp − cv = R, γ = cp/cv; isentropic process obeys pV^γ = constant and T₂/T₁ = (p₂/p₁)^((γ−1)/γ).

Carnot efficiency of a heat engine η = 1 − T_L/T_H (absolute temperatures); COP of a refrigerator = T_L/(T_H − T_L) for the reversible cycle.

Entropy change for an ideal gas: Δs = cp ln(T₂/T₁) − R ln(p₂/p₁); entropy generation is zero for reversible and positive for irreversible processes.

Conduction: Fourier law q = −kA dT/dx; plane wall resistance R = L/(kA), and thermal resistances in series add.

Convection: Newton cooling q = hA(T_s − T_∞); Nusselt number Nu = hL/k relates convection to conduction.

Radiation: Stefan–Boltzmann q = εσA(T_s⁴ − T_surr⁴) with σ = 5.67×10⁻⁸ W/(m²·K⁴); gray-body exchange uses emissivity ε and view factors.

Heat exchanger effectiveness ε = actual heat transfer / maximum possible; NTU = UA/C_min, with ε rising monotonically as NTU increases.

LMTD for a heat exchanger ΔT_lm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂), with counterflow giving the highest effectiveness for equal areas.

Axial stress σ = F/A and axial strain ε = ΔL/L; Hooke's law σ = Eε applies within the proportional limit.

Poisson's ratio ν = −ε_lateral/ε_axial; the shear modulus relates as G = E/[2(1+ν)].

Bending stress σ = My/I with maximum at the outer fiber; deflection of a simply supported beam under central point load P is PL³/(48EI).

Torsion of a solid circular shaft: τ_max = Tc/J with J = πd⁴/32; angle of twist θ = TL/(GJ).

Principal stresses for plane stress: σ₁,₂ = (σx+σy)/2 ± √[((σx−σy)/2)² + τxy²]; maximum shear τ_max = (σ₁−σ₂)/2.

Von Mises equivalent stress σ' = √(σ₁² − σ₁σ₂ + σ₂²) for plane stress; Tresca criterion uses maximum shear stress.

Stress concentration factor K_t = σ_max/σ_nominal; fatigue life is estimated with the S–N curve and endurance limit, often near 0.5 S_ut for steels.

Shaft design under combined bending and torsion uses equivalent torque T_e = √(M² + T²) or √(M² + 0.75T²) depending on the failure theory.

Belt drives: ratio = D₂/D₁; for a flat belt the limiting tension ratio is T₁/T₂ = e^(μθ) with θ in radians.

Cutting speed V = πDN/1000 (m/min with D in mm); feed, depth of cut, and speed jointly set material removal rate MRR ≈ V·f·d.

Taylor tool life equation VTⁿ = C; increasing cutting speed shortens tool life sharply, with n typically 0.1–0.4 for HSS and higher for carbide.

Casting: Chvorinov's rule solidification time t = B(V/A)², so a larger volume-to-surface ratio cools more slowly and feeds shrinkage.

Welding: heat input per unit length = VI/v (J/mm); preheat and interpass control reduce cracking in high-carbon and alloy steels.

Carbon steel classification: AISI 1045 is 0.45% C plain carbon; increasing carbon raises hardness and strength but lowers ductility and weldability.

Iron–carbon diagram: eutectoid at 0.76% C and 727 °C forms pearlite; austenite above the A₃ line transforms to ferrite plus pearlite on slow cooling.

Heat treatment: annealing softens and relieves stress, normalizing refines grain, quenching forms martensite, tempering restores toughness.

Rolling and forging refine grain and align inclusions, producing directional properties; cold working raises strength and lowers ductility (strain hardening).

Static sensitivity is the slope of the output–input curve; hysteresis is the maximum difference between upscale and downscale readings at the same input.

Accuracy describes closeness to the true value; precision describes repeatability. Systematic (bias) errors shift all readings, random errors scatter them.

Uncertainty propagation: for y = f(x₁,x₂), u_y = √[Σ(∂f/∂x_i · u_i)²] when inputs are independent.

A strain gauge has gauge factor GF = (ΔR/R)/(ΔL/L), about 2 for metal foil gauges; a Wheatstone bridge converts ΔR to voltage.

Thermocouples produce a Seebeck emf proportional to the junction temperature difference; cold-junction compensation is required for accurate readings.

A Pitot-static tube measures velocity from the pressure difference: V = √(2Δp/ρ), assuming incompressible flow.

First-order instrument response to a step input: output reaches 63.2% of final value in one time constant τ; settling to 98% takes about 4τ.

A manometer reading h of a liquid of density ρ gives gauge pressure Δp = ρgh; the reading must be corrected for the connecting-fluid density.

Transfer function G(s) = output/input in the Laplace domain with zero initial conditions; poles are roots of the denominator and set the response modes.

First-order system G(s) = K/(τs+1); second-order G(s) = Kω_n²/(s² + 2ζω_n s + ω_n²) with damping ratio ζ and natural frequency ω_n.

Steady-state error depends on system type: step input needs type 0, ramp input needs type 1, and parabolic input needs type 2 for finite error.

Routh–Hurwitz criterion counts sign changes in the first column of the array; any sign change indicates a right-half-plane pole and instability.

PID controller: u = K_p e + K_i ∫e dt + K_d de/dt; integral action removes steady-state offset, derivative action adds damping and anticipates error.

Bode plot: a pole adds −20 dB/decade slope and −90° phase; a zero adds +20 dB/decade and +90°. Gain margin and phase margin indicate relative stability.

Closed-loop transfer function T = G/(1+GH); the system is stable when all roots of 1+GH=0 lie in the left half-plane.

A sensor with a time constant much larger than the process dynamics acts as a lag and can destabilize the loop by adding phase delay.

Rankine cycle efficiency rises with higher boiler pressure and temperature and with lower condenser pressure; reheat and regeneration further improve it.

Rankine cycle efficiency η = (W_turbine − W_pump)/Q_in; pump work is usually small compared with turbine work for large steam plants.

Specific steam consumption and heat rate (kJ/kWh) are plant performance indicators; lower heat rate means higher efficiency.

Boiler efficiency by indirect method accounts for losses: dry flue gas, moisture, unburned carbon, radiation, and blowdown.

Condenser vacuum improves turbine output but is limited by cooling-water temperature; back pressure is set by the achievable vacuum.

Gas turbine Brayton cycle efficiency η = 1 − 1/r_p^((γ−1)/γ); regeneration, intercooling, and reheat raise performance.

Combined cycle plants reach about 55–62% efficiency by using gas turbine exhaust in a heat recovery steam generator for a steam bottoming cycle.

Nuclear plant thermal efficiency is limited by low steam temperature and pressure; most of the heat is rejected to the condenser.

Sensible heat for air: Q = ṁ cp ΔT, with cp ≈ 1.005 kJ/(kg·K); latent heat uses the moisture content difference and latent heat of vaporization.

Psychrometrics: relative humidity is the ratio of actual vapor pressure to saturation pressure at the same dry-bulb temperature.

Cooling coil apparatus dew point and bypass factor determine the leaving air condition; the room sensible heat ratio fixes the supply air temperature difference.

Air conditioning load components include conduction through envelope, solar gain, internal people, lighting, equipment, and ventilation/infiltration latent load.

Refrigeration COP = cooling effect / compressor work; for a vapor-compression cycle it is typically 3–5 and falls as condenser temperature rises.

Chiller efficiency is expressed as kW/ton or COP; 1 ton of refrigeration equals 3.517 kW of cooling capacity.

Duct sizing by equal friction uses a constant pressure drop per unit length; velocity limits control noise and pressure loss.

Heat pump COP for heating = COP_cooling + 1 for the same cycle; performance drops as outdoor temperature falls.

Time value of money: F = P(1+i)ⁿ and P = F/(1+i)ⁿ; these compound-interest factors convert present and future worth.

Uniform series: P = A[(1+i)ⁿ − 1]/[i(1+i)ⁿ] and A = P[i(1+i)ⁿ]/[(1+i)ⁿ − 1] for annual payments.

Net present value discounts all cash flows to time zero at the minimum attractive rate of return; accept projects with NPV > 0.

Internal rate of return is the discount rate making NPV zero; accept if IRR exceeds the MARR, but IRR can be ambiguous with multiple sign changes in cash flow.

Payback period is the time to recover the initial investment from net cash flows; it ignores cash flows after payback and the time value of money unless discounted.

Straight-line depreciation D = (cost − salvage)/n; declining balance uses a fixed rate applied to the book value each year.

Benefit–cost ratio = present worth of benefits / present worth of costs; a ratio above 1 indicates an economically acceptable project.

Break-even quantity = fixed cost/(unit price − unit variable cost); above this volume the operation is profitable.

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