Heat Transfer Correlations — Family Index
Working reference for convective heat-transfer correlations (single-phase forced, natural, two-phase), extended-surface efficiency, compact heat exchangers, radiation, contact resistance, and heat-pipe limits. SI throughout. Each correlation lists its valid range — extrapolation outside is the most common source of design error.
1. At a glance — the pillars
The convective h-coefficient has no closed-form first-principles value; it is built from empirical correlations binned by:
- Forced convection internal — flow inside pipes, ducts, annuli, microchannels. Driver: pump/fan ΔP. Re from bulk velocity.
- Forced convection external — flow over plates, cylinders, spheres, tube banks. Driver: free-stream velocity. Boundary-layer development.
- Natural / free convection — buoyancy-driven (Gr, Ra). Vertical plates, horizontal plates up/down, cylinders, spheres, enclosures.
- Two-phase — boiling — nucleate, transition, film. Pool vs flow boiling. CHF (departure from nucleate boiling).
- Two-phase — condensation — film vs dropwise. Vertical plate, horizontal tube, in-tube, tube banks.
- Two-phase — evaporation — spray, falling film, falling-film evaporator.
- Fin / extended surfaces — straight, annular, pin, louvered. η_f and overall η_o.
- Compact heat exchangers — high area density (>700 m²/m³). j-Colburn and f vs Re. NTU-ε.
- Radiation — σT⁴, view factors, gray-diffuse network, combined convective-radiative h.
- Contact resistance — TIM thermal interface materials, bond-line, joint conductance.
- Heat pipes / vapor chambers — capillary, sonic, entrainment, boiling limits.
2. Dimensionless groups — the language
| Group | Definition | Physical meaning | Typical values |
|---|---|---|---|
| Re | ρVL/µ = VL/ν | inertial / viscous | pipe laminar <2300, turbulent >4000 |
| Pr | µcp/k = ν/α | momentum / thermal diffusivity | air ~0.71, water 6.1 at 20°C, light oil ~3000, liquid metal Na ~0.005 |
| Nu | hL/k_fluid | convective / conductive in fluid | 4–10 laminar pipes, 100–1000 turbulent |
| Gr | gβΔT·L³/ν² | buoyancy / viscous | natural-convection driver |
| Ra | Gr·Pr | buoyancy × Pr | onset of turbulence ~10⁹ vertical plate |
| Pe | Re·Pr | advection / diffusion | important for liquid metals |
| Bi | hL_c/k_body | conduction inside / convection outside | <0.1 → lumped capacitance OK |
| Fo | αt/L_c² | dimensionless time | transient conduction |
| St | Nu/(Re·Pr) = h/(ρcpV) | wall heat flux / advective enthalpy flux | drag-heat-transfer analogy |
| Ja | cp·ΔT/h_fg | sensible / latent | phase-change driver |
| Bo | q”/(G·h_fg) | boiling number | flow-boiling map |
| Co | (ρ_v/ρ_l)^0.5·((1-x)/x)^0.8 | convective number | flow-boiling map |
Properties evaluated at film temperature T_f = (T_s + T_∞)/2 unless correlation specifies otherwise. Variable-property corrections (µ_∞/µ_s)^n appear in many.
3. Internal forced convection — pipes and ducts
Laminar fully-developed circular tube (Re < 2300)
- Constant wall heat flux q”: Nu_D = 4.36
- Constant wall temperature T_s: Nu_D = 3.66
- Entry-length combined (Kays-Crawford, Hausen for T-const): Nu_D = 3.66 + 0.0668·(D/L)·Re·Pr / (1 + 0.04·[(D/L)·Re·Pr]^(2/3))
- Sieder-Tate (large ΔT, property variation): Nu_D = 1.86·(Re·Pr·D/L)^(1/3)·(µ/µ_s)^0.14, valid (Re·Pr·D/L)^(1/3)·(µ/µ_s)^0.14 ≥ 2.
Turbulent fully-developed circular tube
Dittus-Boelter (1930) — simplest, widely used: Nu_D = 0.023·Re^0.8·Pr^n, n = 0.4 heating fluid, 0.3 cooling fluid. Range: 0.7 < Pr < 160, Re > 10000, L/D > 10. Accuracy ±25%, properties at bulk T.
Gnielinski (1976) — more accurate, modest Re extended: Nu_D = (f/8)(Re - 1000)·Pr / (1 + 12.7·√(f/8)·(Pr^(2/3) - 1)) with Petukhov friction factor for smooth tubes: f = (0.790·ln(Re) - 1.64)^(-2). Range: 0.5 < Pr < 2000, 3000 < Re < 5×10⁶. Accuracy ±10%.
Sieder-Tate (turbulent variable-property): Nu_D = 0.027·Re^0.8·Pr^(1/3)·(µ/µ_s)^0.14 Used for oils with large viscosity variation.
Non-circular ducts
Hydraulic diameter D_h = 4A_c/P (P wetted perimeter). Turbulent: use D_h in Dittus-Boelter / Gnielinski directly. Laminar: Nu varies with cross-section shape.
| Cross-section | Nu_D,T (T const) | Nu_D,H (q” const) |
|---|---|---|
| Circular | 3.66 | 4.36 |
| Square (a×a) | 2.98 | 3.61 |
| 2:1 rectangle | 3.39 | 4.12 |
| 4:1 rectangle | 4.44 | 5.33 |
| 8:1 rectangle | 5.60 | 6.49 |
| Parallel plates (b→∞) | 7.54 | 8.23 |
| Triangular equilateral | 2.47 | 3.11 |
Curved pipes / coils
Dean number De = Re·(D/D_coil)^0.5. Secondary flow enhances h. Schmidt: Nu_curved/Nu_straight = 1 + 3.6·(1 - D/D_coil)·(D/D_coil)^0.8 in turbulent.
Rough tubes
Friction factor from Moody (Colebrook) replaces Petukhov-smooth in Gnielinski; the (f/8) numerator increases but the friction term in the denominator also rises — net Nu increase modest (factor ~2 typical for sand-grain ε/D = 0.05).
4. External forced convection
Flat plate (parallel flow)
Laminar (Re_x < 5×10⁵):
- Local: Nu_x = 0.332·Re_x^0.5·Pr^(1/3), Pr ≥ 0.6
- Average 0→L: Nu_L = 0.664·Re_L^0.5·Pr^(1/3)
- Low Pr (liquid metals, Pr < 0.05): Nu_x = 0.565·(Re_x·Pr)^0.5
Turbulent (5×10⁵ < Re_x < 10⁸):
- Local: Nu_x = 0.0296·Re_x^0.8·Pr^(1/3), 0.6 < Pr < 60
- Average mixed boundary (transition at Re_c = 5×10⁵): Nu_L = (0.037·Re_L^0.8 - 871)·Pr^(1/3)
Cylinder cross-flow
Churchill-Bernstein (single correlation, all Re·Pr > 0.2): Nu_D = 0.3 + (0.62·Re^0.5·Pr^(1/3)) / (1 + (0.4/Pr)^(2/3))^(1/4) · (1 + (Re_D/282000)^(5/8))^(4/5) Valid all Re_D; properties at film T. Accuracy ±20%.
Hilpert (older, Re-banded): Nu_D = C·Re_D^m·Pr^(1/3), C and m vary with Re band (e.g. 40<Re<4000 → C=0.683, m=0.466).
Sphere
Whitaker: Nu_D = 2 + (0.4·Re_D^0.5 + 0.06·Re_D^(2/3))·Pr^0.4·(µ_∞/µ_s)^(1/4) Range: 3.5 < Re_D < 7.6×10⁴, 0.71 < Pr < 380, 1.0 < µ_∞/µ_s < 3.2. Limit Re→0: Nu_D = 2 (pure conduction sphere into infinite fluid).
Ranz-Marshall (droplets, gas): Nu_D = 2 + 0.6·Re^0.5·Pr^(1/3).
Tube banks (cross-flow)
Zukauskas (1972): Nu_D = C₁·C₂·Re_D,max^m·Pr^0.36·(Pr/Pr_s)^0.25 Re_D,max uses V_max at minimum free-area gap. C₁ depends on layout (aligned vs staggered) and S_T/S_L. C₂ corrects for number of rows N_L < 16. m bins by Re_D,max (e.g. 1000–2×10⁵ → m ≈ 0.60 staggered, 0.63 aligned). Tables in Incropera Ch. 7.
Impinging jets
Martin correlation for single round jet: Nu_D = 2·Re^0.5·(1 + 0.005·Re^0.55)^0.5·Pr^0.42·[1 - 1.1·(D/r)] / [1 + 0.1·(H/D - 6)·(D/r)] where H/D is nozzle-to-plate spacing, r is radial position. Local h peaks at r/D ≈ 0.5 stagnation, secondary at r/D ≈ 6–8.
5. Natural / free convection
Driver: gβΔT, where β = thermal expansion coefficient (= 1/T_film for ideal gas).
Vertical plate
Churchill-Chu (all Ra): Nu_L = {0.825 + 0.387·Ra_L^(1/6) / [1 + (0.492/Pr)^(9/16)]^(8/27)}² Laminar-only form (better for Ra < 10⁹): Nu_L = 0.68 + 0.670·Ra_L^(1/4) / [1 + (0.492/Pr)^(9/16)]^(4/9)
Horizontal plate
Hot surface up / cold surface down:
- Nu_L = 0.54·Ra_L^(1/4) for 10⁴ ≤ Ra_L ≤ 10⁷
- Nu_L = 0.15·Ra_L^(1/3) for 10⁷ ≤ Ra_L ≤ 10¹¹
Hot surface down / cold surface up (stable):
- Nu_L = 0.27·Ra_L^(1/4) for 10⁵ ≤ Ra_L ≤ 10¹⁰
Characteristic length L_c = A_s/P (area / perimeter).
Horizontal cylinder (Churchill-Chu)
Nu_D = {0.60 + 0.387·Ra_D^(1/6) / [1 + (0.559/Pr)^(9/16)]^(8/27)}² Valid Ra_D < 10¹².
Sphere (Churchill)
Nu_D = 2 + 0.589·Ra_D^(1/4) / [1 + (0.469/Pr)^(9/16)]^(4/9) Range: Ra_D < 10¹¹, Pr ≥ 0.7.
Vertical rectangular enclosure (aspect H/L)
Berkovsky-Polevikov correlations (cold wall T_c, hot wall T_h, L horizontal gap):
- For 2 < H/L < 10, Pr < 10⁵, Ra·Pr/(0.2+Pr) > 10³: Nu_L = 0.22·(Ra·Pr/(0.2+Pr))^0.28·(H/L)^(-1/4)
- For 1 < H/L < 2, 10⁻³ < Pr < 10⁵, Ra·Pr/(0.2+Pr) > 10³: Nu_L = 0.18·(Ra·Pr/(0.2+Pr))^0.29
Horizontal enclosure (Rayleigh-Bénard)
- Conduction only if Ra < 1708.
- Turbulent (Globe-Dropkin): Nu_L = 0.069·Ra_L^(1/3)·Pr^0.074, 3×10⁵ < Ra_L < 7×10⁹.
Concentric cylinders / spheres
Raithby-Hollands effective-conductivity formulation k_eff/k = 0.386·(Pr/(0.861+Pr))^(1/4)·Ra*^(1/4) with a modified Ra*.
6. Boiling — pool and flow
Pool-boiling curve (Nukiyama)
ΔT_e = T_s - T_sat. Four regimes:
- Free convection ΔT_e < ~5 K (water at 1 atm): natural-convection correlations.
- Nucleate boiling 5 < ΔT_e < ~30 K: bubble nucleation at cavities; q” rises ~ΔT_e³.
- Transition 30 < ΔT_e < ~120 K: unstable film patches; q” decreases.
- Film boiling ΔT_e > ~120 K: continuous vapor blanket; radiation contributes.
Nucleate pool boiling — Rohsenow (1952)
q”_s = µ_l·h_fg · [g(ρ_l - ρ_v)/σ]^0.5 · [cp_l·ΔT_e / (C_sf·h_fg·Pr_l^n)]³
| Surface / fluid combo | C_sf | n |
|---|---|---|
| Water-copper polished | 0.0130 | 1 |
| Water-copper scored | 0.0068 | 1 |
| Water-stainless polished | 0.0130 | 1 |
| Water-brass | 0.0060 | 1 |
| Benzene-chromium | 0.0101 | 1.7 |
| n-Pentane-copper polished | 0.0154 | 1.7 |
| R-134a-copper | 0.0049 | 1.7 |
Note: n = 1 for water, 1.7 for other liquids. C_sf is empirical; surface preparation matters enormously.
Critical heat flux (CHF / boiling crisis) — Zuber
q”_max = C_cr·h_fg·ρ_v^0.5·[σ·g·(ρ_l - ρ_v)]^(1/4) C_cr = π/24 ≈ 0.131 (Zuber). Lienhard: 0.149 for large flat heaters. Kutateladze: 0.131–0.18 typical.
Water at 1 atm: q”_max ≈ 1.26 MW/m². Industrial designs derate to ~25–50%.
Film boiling — Bromley horizontal cylinder
Nu_D = C·[g·ρ_v·(ρ_l - ρ_v)·h’_fg·D³ / (ν_v·k_v·(T_s - T_sat))]^(1/4) C = 0.62 horizontal cylinder, 0.67 sphere. h’_fg = h_fg + 0.8·cp_v·ΔT_e includes vapor superheat. Add radiation: h_total = h_conv + (3/4)·h_rad (Bromley correction when h_rad < h_conv).
Flow boiling — Chen (1966) superposition
h_tp = h_nb·S + h_cv·F
- h_cv from Dittus-Boelter with liquid-only Re_l = G(1-x)D/µ_l, then multiplied by enhancement F = f(X_tt) (Martinelli parameter).
- h_nb from Forster-Zuber, multiplied by suppression S = f(Re_tp).
Kandlikar (1990) maps two regimes (convective-dominated CBD vs nucleate-dominated NBD) using Co and Bo.
Gungor-Winterton, Liu-Winterton are common refinements. Stiel-Thodos for new refrigerants.
Subcooled boiling
Bergles-Rohsenow: q”_total² = q”_conv² + q”_boil² (interpolation between FC and saturated NB).
7. Condensation
Hierarchy: dropwise (h 5–10× higher) → film. Dropwise is hard to sustain industrially without promoter coatings; design assumes film.
Nusselt 1916 film theory (laminar)
Vertical plate, isothermal wall: h̄_L = 0.943·[ρ_l·(ρ_l - ρ_v)·g·h’_fg·k_l³ / (µ_l·(T_sat - T_s)·L)]^(1/4) h’_fg = h_fg + 0.68·cp_l·(T_sat - T_s) (Rohsenow correction for liquid subcooling). Valid Re_δ = 4Γ/µ_l < 30 laminar smooth film, where Γ = ṁ_cond per unit width.
Wavy laminar (30 < Re_δ < 1800): Kutateladze h̄_L = Re_δ·k_l / (1.08·Re_δ^1.22 - 5.2)·(ν_l²/g)^(-1/3). Turbulent film (Re_δ > 1800): Labuntsov h̄_L = Re_δ·k_l / (8750 + 58·Pr_l^(-0.5)·(Re_δ^0.75 - 253))·(ν_l²/g)^(-1/3).
Horizontal tube outside (Nusselt)
h̄_D = 0.729·[ρ_l·(ρ_l - ρ_v)·g·h’_fg·k_l³ / (µ_l·(T_sat - T_s)·D)]^(1/4)
Horizontal tube bank (N tubes vertically stacked)
h̄_N,D = h̄_1,D · N^(-1/4) (Nusselt — film thickening from above). Kern correction h̄_N,D = h̄_1,D · N^(-1/6) for inundation in real bundles.
Sphere
h̄_D = 0.815·[same bracket as horizontal tube with D]^(1/4).
Inside horizontal tubes (in-tube condensation)
Two regimes by vapor velocity:
- Stratified (low G): Chato — h ≈ 0.555·[same Nusselt bracket].
- Annular (high G): Cavallini-Zecchin or Shah (1979): Nu = 0.023·Re_l^0.8·Pr_l^0.4 · [(1-x)^0.8 + 3.8·x^0.76·(1-x)^0.04 / p_r^0.38] where x is vapor quality, p_r = p/p_crit.
- Recent: Cavallini 2006 (refrigerants), Thome flow-regime-based.
Dropwise promoters
Hydrophobic coatings (silane SAM, PTFE, parylene-C, lubricant-infused). Heat flux 5–10× film at same ΔT but long-term durability is the issue — most coatings degrade in hundreds of hours.
8. Fin / extended-surface efficiency
Straight rectangular fin (uniform cross-section)
Fin parameter m = √(h·P / (k·A_c)). For rectangular fin width w, thickness t: m = √(2h·(w+t) / (k·w·t)) ≈ √(2h/(k·t)) for w ≫ t.
Adiabatic tip: η_f = tanh(mL)/(mL). Corrected length L_c = L + t/2 absorbs tip-convection: η_f = tanh(mL_c)/(mL_c).
Pin fin (cylindrical, diameter D)
m = √(4h/(k·D)). Same η_f form with L_c = L + D/4.
Annular (radial) fin
Closed-form involves modified Bessel functions K and I; commonly use Schmidt approximation: η_f = tanh(m·r_2c·φ)/(m·r_2c·φ), φ = (r_2c/r_1 - 1)·[1 + 0.35·ln(r_2c/r_1)] r_2c = r_2 + t/2.
Overall surface efficiency
η_o = 1 - (A_f/A_total)·(1 - η_f) where A_f is total fin surface area, A_total = A_f + A_base. Used to apply to a finned-side h.
Fin selection rules of thumb
- Fins help when (k·t/h)^0.5 > ~1 (fin Biot small). High-k base, modest h, modest t.
- Aluminum fins on Cu base for cost; pure-Cu fins for very high heat flux.
- Pin fins shed boundary-layer better in cross-flow but pressure drop higher.
- Louvered and offset-strip fins increase j by repeated boundary-layer restart at penalty of f.
9. Compact heat exchangers
Area density α > 700 m²/m³ (gas) or > 400 m²/m³ (liquid). Built up of fins between flat plates (plate-fin), tubes with fins (tube-fin), or chemically etched plates (PCHE).
Kays-London j-Colburn factors
For each surface (continuous-flat, louvered, wavy, offset-strip, perforated, pin) the curves j = St·Pr^(2/3) vs Re_Dh and Fanning f vs Re_Dh are tabulated.
| Surface | j relative | f relative | Notes |
|---|---|---|---|
| Plain flat | 1.0 | 1.0 | baseline |
| Wavy | 1.5–2.0 | 1.5–2.0 | low-cost enhancement |
| Louvered | 2–3 | 2–3 | auto radiators/condensers, R-134a/R-1234yf |
| Offset-strip (OSF) | 3–4 | 3–4 | aerospace, cryogenic |
| Perforated | 1.5 | 1.5 | obsolete-ish |
| Pin fin | 3+ | 4+ | very high f penalty |
NTU-ε effectiveness method
Definitions:
- C_min = (ṁ·cp)_min, C_max = (ṁ·cp)_max, C_r = C_min/C_max.
- NTU = UA/C_min.
- ε = Q_actual / Q_max, Q_max = C_min·(T_h,i - T_c,i).
Closed-form ε for common flows:
- Counterflow: ε = [1 - exp(-NTU·(1-C_r))] / [1 - C_r·exp(-NTU·(1-C_r))]; C_r = 1 → ε = NTU/(1+NTU).
- Parallel: ε = [1 - exp(-NTU·(1+C_r))] / (1+C_r).
- Cross-flow both unmixed: ε ≈ 1 - exp[(NTU^0.22/C_r)·(exp(-C_r·NTU^0.78) - 1)] (Eckert).
- Cross-flow one mixed: ε = (1/C_r)·{1 - exp[-C_r·(1 - exp(-NTU))]}.
- Shell-and-tube (1 shell, 2N tube passes): TEMA formula with E = √(1+C_r²).
- Boiler/condenser (C_r = 0): ε = 1 - exp(-NTU).
Multipass and TEMA designations
TEMA shell types: E (single-pass, most common), F (two-pass longitudinal baffle), J (divided flow), K (kettle reboiler), X (cross-flow). Heads: A (channel + cover), B (bonnet), C (channel integral), N (channel integral fixed tubesheet). Rear: L (fixed), M (fixed bonnet), N (fixed), P (outside-packed floating), S (split-ring floating), T (pull-through), U (U-tube), W (packed floating).
10. Heat-exchanger types — selection map
| Type | Vendors / standards | Use | Pressure | Temp range |
|---|---|---|---|---|
| Shell-and-tube TEMA | Alfa Laval, Kelvion, Koch, Smithco | refining, petrochem, steam | <300 bar | -200 to 800°C |
| Plate-and-frame gasketed | Alfa Laval (M-series), GEA Sondex, SWEP | HVAC, food, dairy | <25 bar | -40 to 200°C (gasket) |
| Brazed plate (BPHE) | SWEP, DanFoss, Kelvion, Alfa Laval CB | small refrig, HP, EV battery | <50 bar | -195 to 225°C |
| Welded plate (semi-welded) | Alfa Laval AlfaRex, GEA Bloc, Compabloc | aggressive media | <40 bar | -50 to 350°C |
| Printed-circuit (PCHE) | Heatric (Meggitt), Vacuum Process, Alfa Laval DC | sCO₂, LNG, hydrogen | <600 bar | -200 to 900°C |
| Plate-fin (PFHE, brazed Al) | Linde, Chart Industries, Kobe Steel, FIVES | cryogenic (LNG, ASU) | <120 bar | -270 to 65°C (Al) |
| Spiral | Alfa Laval, Kelvion | fouling slurries, sludges | <25 bar | up to 400°C |
| Double-pipe (hairpin) | Brown Fintube, Koch | small duties, viscous | <100 bar | -100 to 600°C |
| Finned-tube air cooler (ACHE) | Hudson (SPX), Harsco, Wabash | refinery cooling, HVAC outdoor | <300 bar (tubes) | up to 400°C |
| Coil-wound (CWHE) | Linde, Air Products, IHI | base-load LNG | <100 bar | -200°C |
| Bayonet | custom | reformers, low-ΔT | high | high |
| Falling-film evaporator | GEA Wiegand, SPX Anhydro, Alfa Laval AlfaVap | food, dairy, juice | low | <120°C |
PCHE for sCO₂ Brayton (10–25 MPa, 500–700°C): Heatric Inconel 617 / 800H diffusion-bonded.
11. Contact resistance — TIMs
Bond-line thermal resistance R”_bond = t_BL/k_TIM + R”_c1 + R”_c2 (top + bottom contact). Typical chip-to-heatsink stack: 0.05–0.5 K·cm²/W.
| TIM class | Examples | k (W/m·K) | Form | Notes |
|---|---|---|---|---|
| Greases | Arctic Silver 5, Thermal Grizzly Kryonaut, ShinEtsu G-751 | 3–12 | paste | pump-out over thermal cycles |
| Phase-change pads (PCM) | Honeywell PTM7950, Henkel Bergquist Hi-Flow 650P | 5–8 | solid → soft | LCD/laptop, very stable |
| Silicone gel | Dow TC-3022, Henkel SS-1700 | 1–3 | cured | structural, no pump-out |
| Silicone pad | Bergquist Sil-Pad, Laird Tflex | 1–6 | sheet | easy assembly, thicker bond-line |
| Graphite sheet | Panasonic PGS Pyrolytic, Henkel BRTS-1, GrafTech eGraf SS | 600–1900 (in-plane!) | sheet | anisotropic — high in-plane only |
| Indium foil / metallized | Indium 5.7, Hi-Tech HTP-100 | 86 | soldered | space, defense, high-flux |
| Liquid metal | Galinstan, Thermal Grizzly Conductonaut | 70–80 | liquid | Al-corrosive; do not use on Al lid |
| Solder TIM | indium-based, SnBi | 50–90 | reflowed | hardest to assemble, lowest R |
| Carbon nanotube pads | Carbice, Cambridge Nanomaterials | 30+ | dry adhesive | space/defense |
Roughness-cleared joint conductance: h_c = function of P_contact / H_micro (microhardness), surface roughness σ, and λ (mean-plane slope). Yovanovich correlations.
12. Radiation
Stefan-Boltzmann
Black surface emission: E_b = σ·T⁴, σ = 5.670374×10⁻⁸ W/(m²·K⁴). Gray surface: q = ε·σ·(T_s⁴ - T_surr⁴), 0 < ε < 1.
View factors F_{ij}
F_{ij} = fraction of radiation leaving surface i striking surface j directly. Reciprocity: A_i·F_{ij} = A_j·F_{ji}. Summation: Σ_j F_{ij} = 1 for enclosure. Tabulated formulas for parallel rectangles, aligned squares, coaxial disks, perpendicular rectangles with a common edge.
Radiosity network
For a gray-diffuse N-surface enclosure: solve N equations (E_bi - J_i)/((1-ε_i)/(ε_i·A_i)) = Σ_j (J_i - J_j)/(1/(A_i·F_{ij})) for radiosity J_i, then q_i = (E_bi - J_i)/((1-ε_i)/(ε_i·A_i)).
Combined h (parallel convection + radiation)
For surface losing to surroundings at near-ambient: define radiation h_rad such that q_rad = h_rad·(T_s - T_surr), h_rad = ε·σ·(T_s + T_surr)·(T_s² + T_surr²) Total h_total = h_conv + h_rad. Useful in natural-convection electronics enclosures where h_rad often equals or exceeds h_conv.
Gas radiation (participating media)
CO₂ and H₂O in combustion products absorb/emit on bands. Hottel charts give ε_g(p_g·L, T_g) with overlap correction Δε for mixtures. WSGGM (Weighted Sum of Gray Gases) for CFD. Soot dominates in flames.
Selective surfaces
Solar absorber: high α_solar (0.3–2.5 µm), low ε_thermal (>2.5 µm). Black chrome, TiNOX, sputtered cermet. ε/α ratio drives Concentrating Solar plant efficiency.
13. Transient lumped capacitance
Valid when Bi = h·L_c/k_body < 0.1 (L_c = V/A — internal gradient negligible).
T(t) = T_∞ + (T_i - T_∞)·exp(-t/τ), τ = ρ·V·cp / (h·A) = (Bi·Fo)⁻¹·(L_c/α)… rewriting: T(t) - T_∞ / (T_i - T_∞) = exp(-Bi·Fo)
For Bi > 0.1: solve 1D Heisler charts (slab, cylinder, sphere). Series solution θ*= Σ C_n·exp(-ζ_n²·Fo)·f_n(r*) with eigenvalues ζ_n from transcendental equation. First-term approximation valid for Fo > 0.2.
Semi-infinite solid (sudden surface T_s): θ/θ_i = erf(x/(2√(αt))) for constant surface T.
14. Heat pipes and vapor chambers
Closed evacuated tube with wick. Working fluid evaporates at hot end, vapor flows to cold end, condenses, capillary returns through wick. Effective k_eff = 10⁴–5×10⁴ W/m·K — orders of magnitude above solid Cu.
Operating limits (whichever is smallest sets q_max)
- Capillary limit — wick can’t return liquid fast enough; 2σ/r_c ≥ ΔP_l + ΔP_v + ΔP_g.
- Boiling limit — nucleate boiling inside wick disrupts capillary action.
- Sonic limit — vapor flow chokes at exit of evaporator (low-pressure startup, alkali metal).
- Entrainment limit — high vapor velocity shears liquid off wick surface (Weber number criterion).
- Viscous limit — at very low T, vapor viscous ΔP dominates (cryogenic).
Working fluid by temperature range
| Range | Fluid | Application |
|---|---|---|
| 4–77 K | He, H₂ | cryogenic |
| 60–200 K | N₂, Ne, O₂ | cryogenic |
| 230–400 K | NH₃, methanol, acetone, R-134a | electronics, satellites |
| 280–500 K | water | CPU, GPU, server |
| 400–650 K | toluene, naphthalene, Dowtherm | industrial |
| 550–1100 K | mercury, cesium, potassium | nuclear, space power |
| 1100–2300 K | sodium, lithium, silver | reactor, leading edge |
Wick types
Sintered powder (best capillarity, isotropic), groove (axial, high q but low gravity tolerance), mesh screen (cheap, planar), composite (powder + groove), arteries (high-power axial).
Vendors
Aavid Genie/Boyd, ACT Advanced Cooling Technologies, Cooler Master, Wakefield-Vette, Furukawa Electric, Fujikura, Sunon. Vapor chambers for CPU/GPU spreaders: AMECTHERM, Coolerage, Asia Vital Components.
15. Useful property reference (engineering quick-look)
Thermal conductivity k (W/m·K) at ~25°C
| Material | k |
|---|---|
| Diamond | 2200 |
| Silver | 429 |
| Copper (OFHC) | 401 |
| Gold | 317 |
| Aluminum 6061 | 167 |
| Aluminum 1100 | 222 |
| Brass C26000 | 120 |
| Carbon steel 1018 | 51 |
| Cast iron, gray | 52 |
| Stainless 304 | 16 |
| Stainless 316 | 14 |
| Inconel 600 | 15 |
| Ti-6Al-4V | 6.7 |
| Alumina 96% | 24 |
| AlN | 170 |
| BeO | 250 |
| Si single crystal | 148 |
| Macor machinable ceramic | 1.5 |
| Pyrex | 1.4 |
| FR-4 (in-plane) | 0.30 |
| FR-4 (through) | 0.29 |
| PTFE | 0.25 |
| PEEK | 0.25 |
| Polyimide (Kapton) | 0.12 |
| Air (1 atm) | 0.026 |
| Water (liquid) | 0.61 |
| Engine oil SAE 50 | 0.145 |
| Ethylene glycol | 0.252 |
| 50/50 EG-water | 0.40 |
| R-134a liquid | 0.083 |
| R-410A liquid | 0.090 |
| Glass wool insulation | 0.045 |
| PU foam closed-cell | 0.025 |
| Aerogel | 0.014 |
Prandtl number Pr at 25°C
| Fluid | Pr |
|---|---|
| Mercury | 0.025 |
| Sodium (400°C) | 0.005 |
| Helium | 0.69 |
| Air | 0.71 |
| Water | 6.13 |
| Ammonia | 1.6 |
| R-134a | 3.5 |
| Ethylene glycol | 150 |
| 50/50 EG-water | 30 |
| Engine oil SAE 50 | 8500 |
| Glycerin | 12500 |
Coefficient of thermal expansion β (1/K) at 25°C
- Ideal gas: β = 1/T(K) ≈ 0.00335 at 25°C.
- Water: 0.000257.
- Ethylene glycol: 0.00065.
- Engine oil: 0.0007.
16. Selection heuristics — match the application
| Application | Recommended thermal solution |
|---|---|
| Handheld phone / wearable SoC | embedded heat pipe + graphite spreader (PGS) + chassis as heat sink |
| Laptop CPU/GPU | heat pipes + vapor chamber + axial fan + perforated fin stack |
| Data-center CPU cold-plate | Cu cold plate (skived/microchannel) + EG-water + plate-and-frame CDU + dry cooler outside |
| EV battery pack (NMC pouch) | brazed Al cold plate beneath modules + glycol-water 50/50 |
| EV power electronics (SiC inverter) | pin-fin baseplate over EG-water; some Tier-1 going to direct two-phase |
| Server PSU / DC-DC | extruded Al fin + axial fan; or AlSiC IMS PCB if dense |
| LED high-bay | extruded Al fin + natural convection; high-end uses heat pipes |
| HVAC residential outdoor coil | louvered-fin tube-and-fin condenser, Al fins on Cu tubes, axial fan |
| HVAC residential indoor coil | Cu hairpin + Al wavy/lanced fins, evaporator |
| HVAC commercial chiller condenser | shell-and-tube smooth or low-fin Cu, water-cooled |
| Refrigeration commercial evap (R-744 CO₂) | microchannel Al, brazed |
| Automotive radiator | Al brazed louvered, single-row tube |
| Automotive AC condenser (R-1234yf) | multi-pass microchannel parallel-flow Al |
| Automotive AC evaporator | laminated Al plate (B-type) + louvered fin |
| Cryogenic LNG main-cryogenic exchanger | coil-wound (Linde, AP) or plate-fin brazed Al (Chart, Linde) |
| LNG vaporizer (regas) | shell-and-tube SS or open-rack seawater |
| Air separation cold box | plate-fin brazed Al Linde/Chart |
| sCO₂ Brayton recuperator | PCHE Heatric Inconel 617 / 800H |
| Nuclear steam generator (PWR) | inverted U-tube shell-and-tube Inconel 690 |
| Concentrating solar receiver | tube-and-fin or PCHE, NaK or molten salt or sCO₂ |
| Refinery preheat train | shell-and-tube TEMA AES/AET, large duties |
| Food/dairy pasteurizer | plate-and-frame gasketed (clean-in-place) |
| Pharma evaporator | falling-film evaporator, GEA Wiegand |
| Turbine blade internal cooling | impingement + rib-roughened + pin-fin trailing edge + film holes on leading edge |
| Combustor liner | effusion cooling + transpiration + thermal-barrier coating |
17. Cross-references
- heat-transfer — undergrad-level fundamentals
- hvac-fundamentals — psychrometrics, refrigeration cycles, coils
- refrigerants — fluid properties + GWP/safety
- pumps-taxonomy — coolant circulation
- pipe-fittings — fluid distribution
- cfd-deep — CFD methods for HT problems beyond correlation reach
- aluminum-alloys — fin/plate brazing alloys (3003, 4045, 6951)
- copper-alloys — tube alloys C12200, C19400, C70600
- electric-motor-taxonomy — motor cooling architectures
18. Citations
- Incropera, DeWitt, Bergman, Lavine — Fundamentals of Heat and Mass Transfer, 8th ed (Wiley 2019). Standard university reference; correlation tables in Ch. 7–10.
- Kays & Crawford — Convective Heat and Mass Transfer, 4th ed (McGraw-Hill 2005). Authoritative on internal flow and variable-property effects.
- Kays & London — Compact Heat Exchangers, 3rd ed (Krieger 1984). The j and f data for compact surfaces.
- Çengel & Ghajar — Heat and Mass Transfer: Fundamentals and Applications, 6th ed (McGraw-Hill 2020). Practical engineering presentation.
- Shah & Sekulic — Fundamentals of Heat Exchanger Design (Wiley 2003). Industry HX design methodology.
- Carey — Liquid-Vapor Phase-Change Phenomena, 2nd ed (Taylor & Francis 2008). Definitive boiling/condensation theory.
- Thome — Engineering Data Book III (Wolverine 2010). In-tube boiling/condensation correlations.
- Hewitt et al. — Process Heat Transfer (CRC 1994). Industrial perspective, foulling and TEMA.
- Faghri — Heat Pipe Science and Technology (Taylor & Francis 1995). Heat-pipe design.
- ASHRAE Handbook — Fundamentals (2021), HVAC Systems and Equipment (2024). HVAC HX selection.
- ESDU correlation library — proprietary but the industrial gold standard.
- Thermopedia — online correlation reference (Begell House). https://www.thermopedia.com/
- ASME PTC 12.5 — single-phase HX performance test code.
Note: All correlations have ranges of validity. Out-of-range extrapolation is the dominant source of design error. When in doubt, run a CFD check (RANS k-ω SST for single-phase, VOF or Eulerian for two-phase) and validate against the nearest in-range correlation.