PE Mechanical Thermal Fluids cheat sheet

Steady-flow energy: q - w = Δh + ΔKE + ΔPE per unit mass; for adiabatic nozzles/diffusers w = 0 and q = 0, so h1 + V1²/2 = h2 + V2²/2.

Continuity: ṁ = ρAV = AV/v = constant for steady one-inlet one-exit flow; incompressible area-velocity relation A1V1 = A2V2.

Bernoulli (incompressible, frictionless, steady, along a streamline): P/ρ + V²/2 + gz = constant; add head-loss hL and pump head when losses or work are present.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below Re ≈ 2300, transitional to 4000, turbulent above; external flow over a flat plate transitions near Re_x ≈ 5×10⁵.

Darcy-Weisbach: h_f = f (L/D)(V²/2g); for laminar pipe flow f = 64/Re regardless of roughness; turbulent f from Moody chart or Colebrook equation using ε/D.

Minor losses h_m = K V²/2g; K depends on fitting geometry and Reynolds number; exit loss K = 1, entrance loss K ≈ 0.5 for square-edged inlet.

Conduction: Fourier law q = -kA dT/dx; plane wall R = L/(kA); cylindrical shell R = ln(r2/r1)/(2πkL); convection R = 1/(hA).

Convection: Nu = hL/k; Dittus-Boelter Nu = 0.023 Re^0.8 Pr^n with n = 0.4 heating and 0.3 cooling of the fluid, valid for 0.6 ≤ Pr ≤ 160 and Re ≥ 10⁴ in fully developed turbulent pipe flow.

Heat exchanger LMTD: ΔT_lm = (ΔT1 - ΔT2)/ln(ΔT1/ΔT2) for counterflow; multiply by F ≤ 1 for multipass/crossflow; q = U A F ΔT_lm.

Carnot efficiency η = 1 - T_L/T_H with absolute temperatures; no real cycle between the same reservoirs exceeds it; COP_HP = COP_R + 1.

Rankine cycle: η = (w_t - w_p)/q_in; boiler and condenser are constant-pressure; pump work ≈ v_f (P2 - P1); reheat and regeneration raise mean temperature of heat addition.

Brayton cycle: η = 1 - 1/r_p^((k-1)/k) for air-standard ideal; back work ratio = w_c/w_t; regeneration uses exhaust to preheat compressor discharge.

Otto cycle: η = 1 - 1/r^(k-1); Diesel cycle: η = 1 - (1/r^(k-1))·(r_c^k - 1)/(k(r_c - 1)); both increase with compression ratio.

Vapor-compression refrigeration: COP_R = q_L/w_c = (h1 - h4)/(h2 - h1); actual COP = Carnot COP × isentropic and component efficiencies.

Psychrometrics: humidity ratio W = 0.622 p_v/(P - p_v); enthalpy h ≈ 1.0T + W(1061 + 0.444T) in Btu/lbm dry air with T in °F; dew point is where p_v equals saturation pressure.

Combustion: stoichiometric air-fuel ratio from atom balance; excess air = (actual AF - stoich AF)/stoich AF; adiabatic flame temperature from energy balance with no heat loss.

Conduction shape/transient: lumped capacitance valid when Bi = hL_c/k < 0.1; Fourier number Fo = αt/L² governs transient temperature response.

Radiation: q = εσA(T_s⁴ - T_surr⁴); σ = 5.67×10⁻⁸ W/(m²·K⁴); view factors satisfy reciprocity A_i F_ij = A_j F_ji and summation ΣF_ij = 1.

Pump/fan laws (affinity): Q ∝ N, H ∝ N², P ∝ N³ for same impeller; Q ∝ D, H ∝ D², P ∝ D³ for geometrically similar impellers.

NPSH_a = (P_atm - P_v)/(ρg) + z - h_L on the suction side; cavitation is avoided when NPSH_a > NPSH_r with margin.

System curve: H_sys = H_static + KQ²; operating point is the intersection of pump curve and system curve; throttling moves the operating point along the pump curve.

Specific speed N_s = N√Q/H^(3/4) (US units, rpm, gpm, ft) classifies impellers; low N_s is radial, high N_s is axial; best efficiency near design specific speed.

Compressors: polytropic work w = n/(n-1)·P1v1[(P2/P1)^((n-1)/n) - 1]; intercooling between stages reduces work and approaches isothermal; equal pressure ratio per stage minimizes work.

Duct flow: friction factor from Moody using hydraulic diameter D_h = 4A/P; equivalent length method converts fittings to added duct length; velocity pressure = ρV²/2.

Fan laws and duct design: total pressure = static + velocity pressure; system effect arises from poor inlet/outlet connections and reduces delivered performance.

Two-phase flow: quality x = m_vapor/m_total; homogeneous void fraction and pressure drop depend on flow pattern; flow regime maps use mass flux and quality.

Valves and flow control: choked flow occurs when downstream pressure falls below critical ratio; for ideal gas critical ratio P*/P0 = (2/(k+1))^(k/(k-1)), about 0.528 for air.

Overall heat transfer coefficient: 1/UA = R_total = 1/(h_i A_i) + R_fouling,i/A_i + ln(r_o/r_i)/(2πkL) + R_fouling,o/A_o + 1/(h_o A_o); fouling adds resistance and lowers U.

Effectiveness-NTU: ε = q/q_max, q_max = C_min(T_h,in - T_c,in); NTU = UA/C_min; counterflow effectiveness ε = (1 - exp[-NTU(1 - C_r)])/(1 - C_r exp[-NTU(1 - C_r)]) with C_r = C_min/C_max.

Heat exchanger energy balance: q = ṁ_h c_p,h (T_h,in - T_h,out) = ṁ_c c_p,c (T_c,out - T_c,in); use c_p at mean bulk temperature.

Extended surfaces: fin efficiency η_f = tanh(mL)/(mL) for adiabatic-tip rectangular fin, m = √(hP/(kA_c)); overall surface efficiency η_o = 1 - (A_f/A_t)(1 - η_f).

Boiling and condensation: nucleate boiling flux rises steeply with ΔT_excess until critical heat flux (CHF), then film boiling; dropwise condensation coefficients exceed filmwise by roughly an order of magnitude.

Thermal resistance network analogy: treat each layer as a resistor; for parallel paths sum conductances; contact resistance between layers can dominate when surfaces are rough or pressures low.

Log-mean temperature difference correction: use F from charts for shell-and-tube with multiple passes; F < 1 and approaches 1 for pure counterflow; q = U A F ΔT_lm.

Insulation economics: critical insulation radius r_cr = k/h for a cylinder; adding insulation increases heat transfer when outer radius is below r_cr and decreases it above.

Solar and radiation exchange: net radiation between gray surfaces uses q = σ(T1⁴ - T2⁴)/((1-ε1)/(ε1A1) + 1/(A1F12) + (1-ε2)/(ε2A2)).