Heat Transfer — Engineering Reference

See also (Tier 3 family index): Heat-Transfer Correlations

1. At a glance

Heat transfer is the discipline that analyses energy transport driven by a temperature gradient. It is the engineering counterpart to thermodynamics: where thermodynamics tells you the equilibrium energy balance and bounds the work / heat exchange of a process, heat transfer tells you the rate at which energy crosses a boundary and the spatial temperature field that develops. Without rate information a chemical reactor cannot be sized, a transistor cannot be packaged, a building cannot be conditioned, and an engine cannot be cooled.

Three mechanisms cover every practical problem:

• Conduction — energy diffusion through a stationary medium by molecular vibration (insulators), free-electron drift (metals), or phonon transport (ceramics). The dominant mode in solids and stagnant fluids.
• Convection — energy transport by the bulk motion of a fluid, plus the conduction within that fluid. Sub-divides into forced (pumped or fan-driven), natural (buoyancy-driven), and phase-change (boiling, condensation) regimes.
• Radiation — energy emitted as electromagnetic waves (mostly infrared at engineering temperatures) by any surface above 0 K. Needs no medium; dominates at high temperature and in vacuum.

Real systems combine all three. A bare hot pipe in a room loses heat by natural convection to the surrounding air and by infrared radiation to the room surfaces; the pipe wall itself transfers heat by conduction. Engineering practice decomposes the network into mechanism-specific thermal resistances and sums them.

Where heat transfer drives design decisions:

• Heat exchangers — shell-and-tube, plate-frame, plate-fin, air-cooled bundles, cooling towers; the workhorse unit operation of refining, petrochemicals, power, HVAC, food, refrigeration, pharma.
• Electronics cooling — heatsinks, vapour chambers, cold plates, immersion cooling; θ_JA, θ_JC, T_j limits; characterised per JEDEC JESD51.
• Building HVAC — envelope loads, duct heat loss, AHU coil sizing, radiant systems; ASHRAE 90.1, 62.1, 55.
• Power generation — boiler tube heat flux, condenser sizing, gas-turbine blade cooling, nuclear fuel-rod heat transfer.
• Engine and powertrain cooling — radiator, intercooler, oil cooler, EGR cooler, battery thermal management.
• Manufacturing processes — quenching, casting solidification, friction-stir welding heat input, induction heating, injection-mould cooling.
• Spacecraft thermal control — radiative-only environment; MLI, louvres, loop heat pipes, radiator sizing.
• Fire safety and process safety — flame heat flux, jet-fire impingement, BLEVE radiation distances.

Place in the engineering stack: thermodynamics (energy bookkeeping) → fluid mechanics (flow field) → heat transfer (temperature field, rates) → thermal-system design (heat exchangers, electronics packaging, HVAC) → CFD / coupled-physics simulation.

2. First principles

2.1 The three rate laws

Fourier’s law (Fourier 1822) — conduction:

q" = −k · ∇T               [W/m²]

q" is the heat flux (rate per unit area, vector), k the thermal conductivity [W/m·K], and ∇T the temperature gradient. The minus sign enforces the second law: heat flows down the gradient.

Newton’s law of cooling (Newton 1701) — convection:

q" = h · (T_s − T_∞)        [W/m²]

h is the convective heat-transfer coefficient [W/m²·K], T_s the surface temperature, T_∞ the free-stream (or bulk) fluid temperature. h is not a material property — it depends on flow regime, geometry, fluid, and temperature; finding h is the central problem of convection.

Stefan–Boltzmann law (Stefan 1879, Boltzmann 1884) — radiation:

q"_emit = ε · σ · T_s⁴       [W/m²]
σ = 5.670 374 419 × 10⁻⁸  W/m²·K⁴

ε is the surface emissivity (0 ≤ ε ≤ 1). Net radiative exchange between a small grey surface and a large surrounding enclosure at T_∞:

q"_net = ε · σ · (T_s⁴ − T_∞⁴)

All temperatures must be absolute (K, never °C, never °F-converted-incorrectly). A 5 °C error in absolute temperature at room temperature is < 2 % in T⁴; a 5 °C error treated as a delta is meaningless.

2.2 Thermal-resistance analogy

For 1-D steady-state problems, each mechanism is rewritten as a resistance with the heat-flow / temperature-difference relation q = ΔT / R:

R_cond,plane    = L / (k · A)              [K/W]
R_cond,cyl      = ln(r₂/r₁) / (2π · L · k)
R_cond,sph      = (1/r₁ − 1/r₂) / (4π · k)
R_conv          = 1 / (h · A)
R_rad           = 1 / (h_rad · A)          with h_rad = ε σ (T_s + T_∞)(T_s² + T_∞²)

Series and parallel rules carry over directly from electrical-circuit analogy: stack R for layers in series, 1/R_total = Σ 1/R_i for parallel paths. R_total and Δ T are the only two numbers an engineer typically needs for a steady-state load calculation; everything else is computing those resistances.

The linearised radiation coefficient h_rad lets you fold radiation into a single overall h_total = h_conv + h_rad for ambient-temperature problems, but the linearisation is only valid for small (T_s − T_∞) — re-evaluate at the design point.

2.3 Conservation of energy

Volumetric form for a stationary medium (no advection):

ρ · c_p · ∂T/∂t = ∇·(k ∇T) + q̇        [W/m³]

with q̇ the volumetric heat generation (Joule heating, nuclear, chemical reaction, viscous dissipation). Constant k reduces this to the heat equation:

∂T/∂t = α · ∇²T + q̇ / (ρ c_p)
α = k / (ρ c_p)        [m²/s]   thermal diffusivity

Steady state with no generation collapses to Laplace’s equation ∇²T = 0. With generation, Poisson’s equation ∇²T = −q̇/k. These are the same partial differential equations that govern electrostatic potential and incompressible-flow stream functions; a solution method from one transfers to the others.

2.4 Dimensionless groups

Convection collapses an unmanageable parameter list (ρ, µ, c_p, k, V, L, g·β·ΔT) onto a few dimensionless groups. The seven that engineers actually use:

Symbol	Name	Definition	Physical meaning
Re	Reynolds	ρ V L / µ	inertia / viscous (forced flow regime)
Pr	Prandtl	µ c_p / k = ν / α	momentum diffusivity / thermal diffusivity (fluid only)
Nu	Nusselt	h L / k_fluid	convective / conductive heat transfer at the wall — the answer to the convection problem
Gr	Grashof	g β ΔT L³ / ν²	buoyancy / viscous (natural convection)
Ra	Rayleigh	Gr · Pr	buoyancy × thermal diffusion (natural-convection regime)
Bi	Biot	h L_c / k_solid	external convection / internal conduction (lumped-capacitance test)
Fo	Fourier	α t / L²	dimensionless time for transient conduction

Correlations report Nu = f(Re, Pr) for forced convection or Nu = f(Ra, Pr) for natural convection, evaluated at a defined characteristic length and reference (film or bulk) temperature. Mis-defining the characteristic length is the single most common student error; the geometry of each correlation must be matched to the original derivation.

3. Practical math / design equations

3.1 Conduction — canonical 1-D geometries

Plane wall, thickness L, area A, surface temperatures T₁ and T₂:

q   = k A (T₁ − T₂) / L                       [W]
R   = L / (k A)                               [K/W]
T(x) = T₁ − (T₁ − T₂) x / L                    (linear profile, no generation)

Cylindrical wall (pipe insulation), inner r₁, outer r₂, length L:

q   = 2π L k (T₁ − T₂) / ln(r₂ / r₁)
R   = ln(r₂ / r₁) / (2π L k)

The critical insulation radius for a small bare pipe in still air: adding insulation can increase heat loss until the outside-surface area gains beat the added conduction resistance. The crossover is:

r_cr = k_insulation / h_outside     (cylinder)
r_cr = 2 k_insulation / h_outside   (sphere)

For a typical pipe in still air (h ≈ 10 W/m²·K) with fiberglass (k = 0.04 W/m·K): r_cr ≈ 4 mm. Most steam-piping is far above this — insulation always reduces loss. Small electrical-wire insulation can sit below it.

Spherical wall (tank insulation), inner r₁, outer r₂:

q = 4π k (T₁ − T₂) / (1/r₁ − 1/r₂)
R = (1/r₁ − 1/r₂) / (4π k)

Composite wall (n layers in series, e.g. wall + sheathing + insulation + drywall + interior air film):

1 / U·A = Σ Rᵢ = 1/(h_o·A) + Σ Lᵢ/(kᵢ·Aᵢ) + 1/(h_i·A)

The result U [W/m²·K] is the overall heat-transfer coefficient — the single number that goes into a building-load calculation or HX rating.

3.2 Conduction — multi-dimensional and transient

Shape factors S [m] turn 2-D conduction problems into a 1-D form q = S · k · (T₁ − T₂). Carslaw & Jaeger and the appendix of Incropera tabulate S for buried pipes, edges of slabs, corner conduction, isothermal disc on semi-infinite medium, parallel cylinders, etc.

Lumped-capacitance (transient cooling of a small body in a fluid):

Bi = h L_c / k_solid       L_c = V / A_s

If Bi < 0.1, internal conduction resistance is negligible and the solid behaves as a single thermal node:

T(t) − T_∞                 t                          ρ V c
─────────── = exp ( − ─── ) ,        τ = ─────────
T_i − T_∞                  τ                          h A_s

For Bi > 0.1, use Heisler charts (or the analytical series for plane wall, infinite cylinder, sphere) — the dimensionless centreline temperature θ_c*/θ_i* is plotted against Fo for a family of Bi values.

Semi-infinite solid with step surface temperature T_s applied at t = 0:

(T − T_i) / (T_s − T_i) = erfc( x / (2 √(α t)) )
q"_s(t) = k (T_s − T_i) / √(π α t)              (decays as 1/√t)

Used for thermal penetration depth in quenching, fire-protection of structural steel, ground heat exchange, transistor-pulse self-heating.

3.3 Forced convection — internal flow (pipes, ducts)

Flow regime by Reynolds number based on hydraulic diameter D_h = 4 A_c / P_wetted:

• Re_D < 2 300 — laminar
• 2 300 < Re_D < 4 000 — transition (avoid for HX design — heat transfer is unpredictable)
• Re_D > 4 000 — turbulent
• Re_D > 10 000 — fully turbulent (Dittus–Boelter valid)

Laminar, fully developed, circular pipe:

• Constant wall temperature: Nu_D = 3.66
• Constant wall heat flux: Nu_D = 4.36

Turbulent, fully developed, smooth pipe — Dittus–Boelter (Dittus & Boelter 1930):

Nu_D = 0.023 · Re_D^0.8 · Pr^n
n = 0.4 (heating: wall hotter than fluid)
n = 0.3 (cooling: wall cooler than fluid)
valid for 0.7 < Pr < 160, Re > 10 000, L/D > 10, T_wall − T_bulk moderate

Gnielinski (Gnielinski 1976) — better accuracy across the transitional and high-Pr ranges:

Nu_D = ( (f/8) · (Re_D − 1000) · Pr ) / ( 1 + 12.7 (f/8)^0.5 (Pr^(2/3) − 1) )
f = (0.790 ln Re_D − 1.64)⁻²        (Petukhov, smooth pipe)
valid 3 000 < Re < 5×10⁶, 0.5 < Pr < 2000

Then h = Nu_D · k_fluid / D. Properties are evaluated at the bulk mean temperature for internal flow.

3.4 Forced convection — external flow

Flat plate, parallel flow:

• Laminar (Re_x < 5×10⁵): Nu_x = 0.332 · Re_x^0.5 · Pr^(1/3) (Pohlhausen); average over plate Nu_L = 0.664 · Re_L^0.5 · Pr^(1/3).
• Turbulent: Nu_x = 0.0296 · Re_x^0.8 · Pr^(1/3) (local); Nu_L = 0.037 · Re_L^0.8 · Pr^(1/3) (average, fully turbulent boundary layer).

Cylinder in cross-flow — Churchill–Bernstein (Churchill & Bernstein 1977):

Nu_D = 0.3 + (0.62 · Re_D^0.5 · Pr^(1/3)) / [1 + (0.4/Pr)^(2/3)]^(1/4) · [1 + (Re_D / 282 000)^(5/8)]^(4/5)
valid Re_D · Pr > 0.2

Sphere — Whitaker (Whitaker 1972):

Nu_D = 2 + (0.4 · Re_D^0.5 + 0.06 · Re_D^(2/3)) · Pr^0.4 · (µ_∞ / µ_s)^(1/4)
valid 3.5 < Re_D < 7.6×10⁴, 0.71 < Pr < 380

The ”+ 2” is the limiting Nu for a sphere in stagnant fluid — pure conduction in an infinite medium.

Banks of tubes — Žukauskas (Žukauskas 1972) — staggered and aligned arrangements, with a correction factor C₂ for fewer than 20 rows. Workhorse for finned-tube banks and shell-side flow over baffled tube bundles.

3.5 Natural convection

Characteristic group is Rayleigh number Ra_L = g β ΔT L³ / (ν α). For an ideal gas, β = 1/T_f at film temperature T_f = (T_s + T_∞)/2.

Vertical plate — Churchill–Chu (Churchill & Chu 1975), valid all Ra:

Nu_L = { 0.825 + (0.387 · Ra_L^(1/6)) / [1 + (0.492/Pr)^(9/16)]^(8/27) }²

Horizontal plate, hot surface facing up (or cold surface facing down):

• 10⁴ < Ra_L < 10⁷: Nu_L = 0.54 · Ra_L^(1/4)
• 10⁷ < Ra_L < 10¹¹: Nu_L = 0.15 · Ra_L^(1/3)

L = A / P (characteristic length for horizontal plates).

Horizontal plate, hot surface facing down (suppressed natural convection):

• 10⁵ < Ra_L < 10¹⁰: Nu_L = 0.27 · Ra_L^(1/4)

Long horizontal cylinder — Churchill–Chu (1975), valid Ra_D < 10¹²:

Nu_D = { 0.60 + (0.387 · Ra_D^(1/6)) / [1 + (0.559/Pr)^(9/16)]^(8/27) }²

Mixed convection is significant when Gr / Re² ≈ 1; below 0.1 forced dominates, above 10 natural dominates. Always check both and use whichever predicts higher h, or use a combined correlation Nu³ = Nu_forced³ ± Nu_nat³ (sign depends on aiding or opposing).

3.6 Boiling and condensation

Pool boiling follows the Nukiyama curve (Nukiyama 1934): natural convection at low ΔT_excess = T_s − T_sat, then nucleate boiling, peak heat flux (CHF) at the Zuber limit (Zuber 1959):

q"_max = 0.131 · h_fg · ρ_v^0.5 · [ σ_st · g · (ρ_l − ρ_v) ]^(1/4)

then transition boiling (unstable), then film boiling. Rohsenow correlation (Rohsenow 1952) covers nucleate pool boiling for water with a surface-pair coefficient C_sf.

Flow boiling — Chen correlation (Chen 1966) — superposition of nucleate + convective contributions; still the basis of most evaporator-tube design.

Film condensation on a vertical plate — Nusselt (Nusselt 1916), laminar wave-free:

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 c_p,l (T_sat − T_s)             (Rohsenow correction for subcooling)

Film condensation inside horizontal tubes — Chato (Chato 1962) for low vapour velocity; Shah (Shah 1979) for general inside-tube condensation with finite shear.

3.7 Heat-exchanger design — LMTD and ε-NTU

LMTD method — preferred when both inlet and outlet temperatures are known:

q = U · A · F · ΔT_LM
ΔT_LM = (ΔT₁ − ΔT₂) / ln(ΔT₁ / ΔT₂)

ΔT₁, ΔT₂ are the end-temperature differences; for counter-flow, ΔT₁ = T_h,in − T_c,out and ΔT₂ = T_h,out − T_c,in. F is the correction factor for non-counter-flow (1-shell 2-tube, cross-flow); F ≥ 0.75 required for a viable design (chart from Bowman, Mueller & Nagle 1940, reproduced in TEMA).

ε-NTU method — preferred when only inlet temperatures are known (rating an existing HX):

NTU = U · A / C_min
C_min = min(ṁ c_p)_hot, (ṁ c_p)_cold
C_r = C_min / C_max
ε = q / q_max = q / [ C_min · (T_h,in − T_c,in) ]
ε = f(NTU, C_r, arrangement)              (closed-form expressions tabulated)

For counterflow: ε = (1 − exp[−NTU(1 − C_r)]) / (1 − C_r · exp[−NTU(1 − C_r)]), reducing to NTU/(1+NTU) when C_r = 1.

Overall U from the resistance stack (tube-side and shell-side, with fouling):

1/(U_o A_o) = 1/(h_i A_i) + R_f,i / A_i + ln(r_o/r_i)/(2π k_t L) + R_f,o / A_o + 1/(h_o A_o)

3.8 Fin performance

Pin or plate fins extend a surface; fin efficiency η_f relates real to ideal (isothermal fin) heat transfer:

m = √(h · P / (k · A_c))           [1/m]
η_f = tanh(m · L_c) / (m · L_c)    (straight fin with insulated tip; L_c = L + t/2 for adiabatic-tip correction)

Surface efficiency for a finned surface with N_f fins:

η_o = 1 − (N_f A_f / A_t) · (1 − η_f)

then h is applied to A_t · η_o instead of bare area. Targets: η_f > 0.7 for a well-designed fin; below 0.5 means the fin is too long or too thin for the chosen material.

4. Reference data

4.1 Thermal conductivity (room temperature, W/m·K)

Material	k (W/m·K)	Notes
Diamond (type IIa)	2 000 – 2 300	highest known
Silver	429
Copper (pure)	401
Gold	317
Aluminum 6061-T6	167
Aluminum 1100 (pure)	222
Brass (C36000)	116
Steel, mild (A36)	51
Stainless 304	14.9	low for a metal
Stainless 316L	14.0
Inconel 625	9.8
Titanium Ti-6Al-4V	6.7
Aluminum nitride (AlN)	170–230	electronics substrate
Silicon carbide (sintered)	110–150
Beryllium oxide (BeO)	270	toxic, restricted
Alumina 96 % (Al₂O₃)	24
Silicon (single crystal)	130	wafer / die
Glass (soda-lime)	1.0
Concrete (normal weight)	1.4
Brick (common)	0.7
Water (liquid, 20 °C)	0.60
Wood (Douglas fir, ⊥ grain)	0.12
Polycarbonate	0.20
PEEK	0.25
Fiberglass batt (R-13)	0.045
Polyurethane foam	0.024–0.030	closed-cell
Aerogel (silica)	0.012–0.018	best non-vacuum insulator
Vacuum-insulated panel	0.004–0.008
Air (still, 20 °C)	0.026	conductive only

Graphite and carbon-composite are anisotropic: in-plane k can exceed 1 000 W/m·K (pyrolytic graphite, k-Core, Momentive TPG); through-plane is 5–15 W/m·K. Specify direction when quoting.

4.2 Typical convective h ranges (W/m²·K)

Mode	h (W/m²·K)
Natural convection, gases	2 – 25
Natural convection, liquids	50 – 1 000
Forced convection, gases	25 – 250
Forced convection, liquids	100 – 20 000
Boiling water	2 500 – 100 000
Condensing steam (film)	5 000 – 100 000
Condensing steam (dropwise)	30 000 – 300 000
Liquid metals, forced	5 000 – 250 000

These are order-of-magnitude only; the actual number comes from a correlation evaluated at the design point.

4.3 Typical overall U for heat exchangers (W/m²·K, after Kakaç and TEMA)

Service	U (W/m²·K)
Gas–gas, low pressure	5 – 35
Gas–gas, high pressure	150 – 500
Gas–liquid (air cooler)	25 – 60 (referred to fin-side area)
Water–water	800 – 2 500
Light hydrocarbon–water	350 – 900
Heavy oil–water	60 – 300
Condensing steam–water	1 500 – 5 000
Condensing refrigerant–water	700 – 1 100
Evaporating refrigerant (DX)	300 – 1 200
Plate-frame water–water	3 000 – 7 000

4.4 Emissivity of common surfaces (total hemispherical, 300 K unless noted)

Surface	ε
Polished aluminum	0.04
Anodised aluminum (black)	0.82
Mill-finish stainless 304	0.30
Stainless oxidised at 800 K	0.85
Copper, polished	0.03
Copper, oxidised	0.78
Steel, polished	0.07
Steel, heavily oxidised	0.81
Cast iron, oxidised	0.78
Concrete	0.85
Brick (red)	0.93
Glass, plate	0.92 (long-wave IR)
Water	0.96
Human skin	0.97
White paint (matte)	0.90 in IR, 0.20 in solar
Black paint (matte)	0.95
Anodised black	0.88
Gold (vapour deposited)	0.02 (IR)

Note: solar absorptivity α_s is decoupled from emissivity ε in the IR for many engineered surfaces. Spacecraft radiators want low α_s / high ε (white paint, OSR mirrors); solar collectors want the opposite (selective coatings, α_s/ε > 6).

4.5 Fouling resistances (TEMA Table RGP-T-2.4, m²·K/W)

Stream	R_f (m²·K/W)
Treated boiler feedwater	0.000 09
Cooling-tower water (treated)	0.000 18
Cooling-tower water (untreated)	0.000 53
Seawater (below 50 °C)	0.000 09
Brackish water	0.000 35
Fuel oil	0.000 88
Light hydrocarbon vapours	0.000 18
Refrigerant liquids	0.000 18
Steam (clean, oil-free)	0.000 09

Adding R_f shifts U downward (R_f = 0.000 5 m²·K/W on water–water service typically drops U by 30–50 %). Design U is always based on fouled condition; clean U is for commissioning.

4.6 Correlation summary by regime

Geometry / regime	Correlation	Range
Internal turbulent, smooth	Dittus–Boelter (1930)	Re > 10⁴, 0.7 < Pr < 160
Internal turbulent, accurate	Gnielinski (1976)	3 000 < Re < 5×10⁶
Internal laminar	constant Nu = 3.66 or 4.36	fully developed
Flat plate laminar	Pohlhausen	Re_x < 5×10⁵
Cylinder cross-flow	Churchill–Bernstein (1977)	Re·Pr > 0.2
Sphere	Whitaker (1972)	3.5 < Re < 7.6×10⁴
Tube banks	Žukauskas (1972)	tabulated by arrangement
Vertical plate, natural	Churchill–Chu (1975)	all Ra
Horizontal cylinder, natural	Churchill–Chu (1975)	Ra < 10¹²
Pool boiling, nucleate	Rohsenow (1952)	water + tabulated C_sf
Pool boiling, CHF	Zuber (1959)	clean horizontal surface
Flow boiling	Chen (1966)	inside vertical tubes
Film condensation, vertical	Nusselt (1916)	laminar, smooth film
In-tube condensation	Chato (1962), Shah (1979)	low and general shear

5p. Theory

The three rate laws of §2 are special cases of a single transport principle: flux is proportional to the gradient of an intensive property, weighted by a transport coefficient that depends on the medium and the regime. Fick’s law (mass diffusion), Newton’s law of viscosity (momentum diffusion), and Fourier’s law (heat diffusion) share this structure; the Reynolds analogy and Chilton–Colburn analogy (j_H = j_M = f/2) exploit it to predict mass-transfer coefficients from a heat-transfer measurement on the same geometry.

Conduction at the molecular level: in metals, free electrons carry energy (Wiedemann–Franz law k/σ_e · T = L_0 = 2.44 × 10⁻⁸ W·Ω/K² ties thermal conductivity to electrical conductivity); in non-metals, phonons (lattice vibrations) dominate, with k limited by phonon scattering at boundaries, defects, isotopes, and other phonons (Umklapp). At cryogenic temperatures or in nanoscale devices, ballistic conduction replaces diffusive — Fourier’s law breaks down below ~100 nm in silicon.

Convection is conduction in a moving frame: the energy equation in convective form ρ c_p DT/Dt = ∇·(k∇T) + q̇ + Φ (Φ = viscous dissipation) carries the substantial derivative D/Dt = ∂/∂t + **v**·∇. The wall heat flux is always computed from q"_wall = −k_fluid (∂T/∂y)_{y=0} — i.e. Fourier in the fluid evaluated at the wall. Newton’s h is a definition that absorbs the wall-gradient detail into a single number. CFD does not need h; it solves the field directly.

Radiation is a quantum-statistical phenomenon: Planck’s spectral distribution E_bλ(λ,T) = 2π h c² / [λ⁵ (exp(hc/λkT) − 1)] integrates over wavelength to Stefan–Boltzmann and peaks at Wien’s displacement wavelength λ_max T = 2 897.8 µm·K. At 300 K, λ_max ≈ 9.7 µm — far-IR. At 6 000 K (the sun), λ_max ≈ 0.48 µm — visible green. This wavelength shift is why “low-emissivity” glass on a building can be highly transparent to solar (short-wave) while reflecting interior IR (long-wave) — selective spectral behaviour, not a violation of Kirchhoff.

Kirchhoff’s law (Kirchhoff 1859) states α_λ = ε_λ at the same wavelength and direction for a body in thermal equilibrium. The misuse to claim α_total = ε_total is wrong whenever the source and the surface are at very different temperatures (sun on roof). Engineering surfaces are characterised by two numbers: solar absorptivity α_s (weighted over the 5 800 K solar spectrum) and IR emissivity ε (weighted near the surface’s own temperature).

6p. Application — worked examples

Example A — IC package θ_JA estimation

A QFN-32 package, 5 mm × 5 mm × 0.9 mm, sits on a 9 mm × 9 mm copper pad with a 4 × 4 thermal-via array (0.3 mm dia, 1.6 mm FR-4). Dissipation = 1 W. Ambient = 25 °C. Estimate junction temperature.

Resistance network (junction → ambient):

1. Die → exposed pad (Si conduction): die 3 × 3 × 0.3 mm, k_Si = 130 W/m·K. R₁ = L / (k A) = 0.0003 / (130 × 9×10⁻⁶) = 0.26 K/W.

2. Pad → PCB top copper through solder: 50 µm SAC305, k = 58 W/m·K, A = 25 mm². R₂ = 50×10⁻⁶ / (58 · 25×10⁻⁶) = 0.034 K/W.

3. PCB through-via array: 16 vias, copper plated 25 µm. Each via R = L / (k A_Cu) = 0.0016 / (401 · π·(150²−125²)×10⁻¹²) ≈ 0.0016 / (401 · 2.16×10⁻⁸) ≈ 185 K/W per via. 16 in parallel: 11.6 K/W.

4. PCB plane spreading: assume 30 mm × 30 mm × 35 µm copper plane. Spreading resistance from disc-on-half-space: R₄ ≈ 8 K/W (Lee 1995 spreading correlation).

5. PCB bottom → ambient by combined convection + radiation: still air, h_conv = 6 W/m²·K, h_rad ≈ 5 W/m²·K (ε_PCB ≈ 0.9), A = 900 mm² = 9×10⁻⁴ m². R₅ = 1 / [(6+5) · 9×10⁻⁴] = 101 K/W.

θ_JA ≈ Σ R = 0.26 + 0.034 + 11.6 + 8 + 101 ≈ 121 K/W.

At 1 W, ΔT_JA = 121 °C; T_j = 25 + 121 = 146 °C — above the 125 °C limit for industrial-grade silicon.

Mitigations the math suggests: (a) the via array (11.6 K/W) and the still-air convection (101 K/W) dominate; reducing either by ½ saves real budget. Forced airflow (1 m/s) raises h to ~25 W/m²·K, dropping R₅ to ~40 K/W and θ_JA to ~60 K/W. (b) A vapour chamber or copper coin slot is needed if still-air convection has to be retained.

Caveat: this is a back-of-envelope physical model. JEDEC JESD51-2 reports θ_JA on a standard 2s2p test board that is not the customer’s PCB; real θ_JA scales with the actual copper area the part is soldered to.

Example B — Counterflow shell-and-tube oil cooler

Duty. Cool 5.0 kg/s hot oil from 150 → 80 °C using 25 °C cooling water heated to ≤ 60 °C. Oil c_p = 2.1 kJ/kg·K, water c_p = 4.18 kJ/kg·K. Counter-flow 1-1 shell-and-tube, plain 19 mm OD × 1.65 mm wall steel tubes, 6 m straight length.

Heat duty (oil side). q = ṁ_oil · c_p,oil · ΔT_oil = 5.0 · 2 100 · (150 − 80) = 735 kW (2.51 MMBtu/h).

Water flow. ṁ_w = q / (c_p,w · ΔT_w) = 735 000 / (4 180 · 35) = 5.03 kg/s (≈ 80 gpm).

LMTD (counter-flow). ΔT₁ = T_h,in − T_c,out = 150 − 60 = 90 °C ΔT₂ = T_h,out − T_c,in = 80 − 25 = 55 °C ΔT_LM = (90 − 55) / ln(90/55) = 35 / 0.4925 = 71.1 °C F = 1.0 (true counterflow).

Assumed overall U. Light-oil to water with TEMA fouling: take U_design = 500 W/m²·K.

Area. A = q / (U · F · ΔT_LM) = 735 000 / (500 · 1.0 · 71.1) = 20.7 m² (referred to outside surface).

Tube count. Each tube A_o = π · 0.019 · 6 = 0.358 m²/tube. N_t = 20.7 / 0.358 = 58 tubes. Round to 60 (typical TEMA tube count for 8-inch shell, triangular pitch).

Tube-side check. Cross-section per tube A_c = π/4 · (0.0157)² = 1.94×10⁻⁴ m² (16 mm ID). 1 pass over 60 tubes: total A_c = 0.0116 m². Water velocity v = ṁ / (ρ · A_c) = 5.03 / (995 · 0.0116) = 0.44 m/s — too low for good heat transfer and fouling control; switch to 2-pass tube (velocity doubles to 0.88 m/s, Re ≈ 14 000, turbulent). F drops to ~0.95 for 1-shell 2-tube; re-iterate A.

This is the canonical hand-calc that precedes any Aspen EDR or HTRI run. The detailed mechanical design (TEMA AES vs BEM, baffle pitch, tie-rod count, ASME thickness) is the next stage.

Example C — Natural convection from a vertical heated panel

A vertical panel, 1.0 m tall × 0.5 m wide, surface T_s = 80 °C, room air T_∞ = 20 °C, ε = 0.9 to surroundings at 20 °C.

Air properties at film T_f = 50 °C (323 K): k = 0.0282 W/m·K, ν = 1.80×10⁻⁵ m²/s, α = 2.55×10⁻⁵ m²/s, Pr = 0.71, β = 1/323 = 3.10×10⁻³ K⁻¹.

Rayleigh. Ra_L = g β ΔT L³ / (ν α) = 9.81 · 3.10×10⁻³ · 60 · 1.0³ / (1.80×10⁻⁵ · 2.55×10⁻⁵) = 1.824 / 4.59×10⁻¹⁰ = 3.97×10⁹ — turbulent natural convection.

Churchill–Chu. Term: (0.492/Pr)^(9/16) = (0.693)^(9/16) ≈ 0.815; [1 + 0.815]^(8/27) = 1.815^0.296 ≈ 1.193. 0.387 · Ra^(1/6) = 0.387 · (3.97×10⁹)^(1/6) = 0.387 · 40.0 = 15.5. Nu_L = [0.825 + 15.5 / 1.193]² = [0.825 + 13.0]² = 13.82² = 191.

Convective h. h_conv = Nu · k / L = 191 · 0.0282 / 1.0 = 5.39 W/m²·K. q”_conv = 5.39 · 60 = 323 W/m².

Radiation. q”rad = ε σ (T_s⁴ − T∞⁴) = 0.9 · 5.67×10⁻⁸ · (353⁴ − 293⁴) = 0.9 · 5.67×10⁻⁸ · (1.553×10¹⁰ − 0.737×10¹⁰) = 0.9 · 5.67×10⁻⁸ · 8.16×10⁹ = 416 W/m².

Total heat loss. A = 0.5 m². q = (323 + 416) · 0.5 = 370 W total; radiation contributes 56 % of the loss at this geometry and ΔT. The “natural-convection heat loss” engineers often quote without checking radiation is, here, a ~half-truth.

7p. Edge cases & assumptions

1. Property variation with temperature. All correlations specify where to evaluate fluid properties (film T for external flow; bulk mean T for internal flow; sometimes a Sieder–Tate viscosity correction (µ/µ_s)^0.14 for large wall-bulk ΔT). Using room-temperature properties for an oil cooler operating at 150 °C is a common 20–40 % error.

2. LMTD assumes constant U and linear T profile. Boilers and condensers with phase change have piecewise-constant T on one side; the HX must be zoned (sub-cooled, two-phase, superheated) and each zone designed separately. Petroleum-fractionator condensers with composition changes need rigorous step-by-step methods (HTRI Xphlex, Aspen EDR).

3. Fouling growth is time-dependent. R_f represents the design fouling — TEMA values are equilibrium / cleaning-cycle averages. Real systems start at clean-U and degrade; mid-cycle U is what determines the schedule for cleaning (chemical, mechanical, online ball-cleaning systems like Taprogge HPS).

4. Flow-induced vibration. Shell-and-tube failure mode #1 outside fouling. TEMA Section RGP-RCB-4.6 specifies the critical-velocity check (Connors equation for fluid-elastic instability, vortex-shedding lock-in, acoustic resonance). Exceeding it cracks tubes at the baffle support in weeks.

5. Two-phase pressure drop. Single-phase Darcy–Weisbach does not predict pressure drop in evaporators and condensers. Use Lockhart–Martinelli (1949), Friedel (1979), or homogeneous-flow models; check that pressure-drop budget doesn’t starve flow distribution between parallel tubes (mal-distribution kills heat transfer).

6. Latent heat dominates phase change. A 1 °C ΔT across an R-410A evaporator at 5 bar (h_fg ≈ 200 kJ/kg) at h = 3 000 W/m²·K transfers 3 kW/m². Compare 1 °C ΔT in single-phase liquid R-410A: q ≈ 3 kW/m² also — but the saturation temperature is clamped, so the whole evaporator runs at near-uniform T. That clamping is what makes refrigeration cycles work.

7. Thermal contact resistance. Bolted, screwed, and pressed joints have non-zero contact resistance even at high pressure. Typical R_contact for bare metal-metal at 1 MPa: 1×10⁻⁴ m²·K/W. With a thermal-interface material (TIM — Bergquist Gap Pad, Honeywell PTM7950, Laird Tflex, Indium Heat-Spring): 0.2 – 1×10⁻⁵ m²·K/W. For a CPU heatsink, the TIM resistance is often comparable to the rest of the package; specify TIM as carefully as fin geometry.

8. Edge / bypass / leakage flows in shell-side flow around segmental baffles can reduce effective shell-side heat transfer by 30–60 %. The Bell–Delaware method (Bell 1963) accounts for the five stream types (B, C, A, E, F) and is the basis of modern shell-and-tube rating software.

9. Compressibility and recovery temperature. In high-Mach flow, T_recovery = T_∞ + r · V²/(2 c_p) with r ≈ Pr^(1/2) (laminar) or Pr^(1/3) (turbulent). The adiabatic wall sits at T_recovery, not T_∞ — relevant to engine blade cooling, hypersonic vehicle skin, supersonic wind-tunnel design.

10. Anisotropic conductivity. Pyrolytic graphite, k-Core (Momentive), TGON (Laird) have k_in-plane = 1 000–1 700 W/m·K but k_through-plane ≈ 10 W/m·K. Useful as heat spreader; not as a bulk conductor. Carbon-fibre composites and laminated sheet stock are similarly anisotropic.

11. Radiation in semi-transparent media. Glass, fused silica, alumina, water below 1 m optical depth all transmit IR partially. At high T (kilns, glass tanks) radiation penetrates the bulk; surface-only emission models under-predict. Use band-modelled radiation or discrete-ordinates in CFD.

12. Mixed-convection regime. Reynolds and Grashof in similar magnitude — neither pure natural nor pure forced correlations apply. Re-check after sizing forced-convection equipment for low-velocity operation (off-design, turndown).

8p. Tools & software

Hand calculation + tables is correct for any canonical problem (steady 1-D conduction, single-correlation convection, simple HX). Workhorse property references: Incropera 8th ed appendix tables, ASHRAE Handbook Fundamentals, NIST REFPROP, CoolProp (open-source, Bell 2014).

Heat-exchanger rating and design:

• Aspen EDR (Exchanger Design and Rating) — shell-and-tube design + rating; air coolers; plate-frame; bundles into Aspen Plus / HYSYS for full-process integration.
• HTRI Xchanger Suite — Xist (S&T), Xfist (fired heaters), Xphe (plate-frame), Xphlex (multicomponent), Xtlo (cooling towers). The refinery / petrochem industry standard, owned by an industry-research consortium.
• TEMA Excel sheets — mechanical-design templates that interlock with rating software.
• AHED (Alfa Laval) for plate-frame; Kelvion KConfig, GEA OmegaSelect, SWEP SSP, Tranter PHEx — vendor selection tools (free, intended to up-sell).

Multiphysics CFD / FEA:

• ANSYS Fluent + Mechanical Thermal; ANSYS Icepak (electronics, JEDEC test-board templates built in).
• Siemens Star-CCM+ — single-environment CFD with strong CHT (conjugate heat transfer); used heavily in automotive thermal management.
• COMSOL Heat Transfer Module — strong on multiphysics (Joule + thermal, thermal + structural).
• Abaqus Heat Transfer — coupled with mechanical solver; popular in aerospace.
• OpenFOAM — chtMultiRegionFoam, buoyantSimpleFoam, reactingFoam; open-source CHT and combustion.
• Mentor / Siemens FloTHERM, 6SigmaET — board- and system-level electronics thermal.

Electronics-specific:

• Mentor FloTHERM PCB, ANSYS SIwave–Icepak coupling, Cadence Celsius — die-package-board thermal co-design.
• JEDEC JESD51-2 / -5 / -7 test boards modelled directly.

Building HVAC / energy:

• EnergyPlus (DOE, open-source), TRNSYS, IES VE, Carrier HAP, Trane TRACE 3D Plus — building energy simulation; embed ASHRAE-vetted correlations.
• Bluebeam + Excel templates for HVAC load takeoffs per ASHRAE 62.1 / 90.1.

Process / refrigeration:

• HYSYS, Aspen Plus, ChemCAD, ProMax, UniSim — process simulators with embedded HX models.
• CoolProp + Python / Jupyter — programmable property and correlation library; the modern hand-calc replacement.

Property libraries:

• NIST REFPROP (paid, authoritative).
• CoolProp (open-source, covers most refrigerants + 122 fluids).
• Cantera (open-source, combustion gases).

11. Cross-references

• thermodynamics — first/second-law foundations; cycle efficiency feeds Q̇ that heat transfer then has to deliver.
• fluid-mechanics — Re, boundary layers, pressure drop; convection is fluid mechanics with the energy equation switched on.
• materials-ceramics — refractory and insulating ceramics; k tables and use cases.
• materials-aluminum — heatsink alloys (1050, 6063), k and forming.
• electromagnetics-engineering — induction heating, dielectric heating, Joule self-heating from current density.
• pcb-design — copper pours, thermal vias, θ_JA test boards.
• op-amps — IC self-heating, thermal-noise budgets, drift specifications.
• semiconductor-devices — junction-temperature limits, T_j max, derating curves.
• power-electronics — IGBT / MOSFET thermal management, transient thermal impedance Z_θ(t).
• pumps-turbomachinery — companion fluid-machinery reference.
• hvac-fundamentals — building-load application of overall-U methods.
• scientific — CFD mesh-file formats and conjugate heat-transfer setup.

12. Citations

1. Bergman, T. L.; Lavine, A. S.; Incropera, F. P.; DeWitt, D. P. Fundamentals of Heat and Mass Transfer, 8th ed. Wiley, 2017. ISBN 978-1119353881. The canonical English-language undergraduate text; standard property tables and problem set.
2. Çengel, Y. A.; Ghajar, A. J. Heat and Mass Transfer: Fundamentals and Applications, 6th ed. McGraw-Hill, 2020. ISBN 978-0073398181. Alternative undergraduate text; strong on examples.
3. Mills, A. F.; Coimbra, C. F. M. Basic Heat and Mass Transfer, 2nd ed. Pearson, 2014. ISBN 978-0996305303. Concise, engineering-oriented.
4. Kreith, F.; Manglik, R. M.; Bohn, M. S. Principles of Heat Transfer, 8th ed. Cengage, 2017. ISBN 978-1305387102.
5. Kakaç, S.; Liu, H.; Pramuanjaroenkij, A. Heat Exchangers: Selection, Rating, and Thermal Design, 4th ed. CRC Press, 2020. ISBN 978-0367186944. The HX-design reference.
6. Shah, R. K.; Sekulić, D. P. Fundamentals of Heat Exchanger Design. Wiley, 2003. ISBN 978-0471321712. Deep treatment of ε-NTU, plate-fin, and compact HX.
7. Modest, M. F. Radiative Heat Transfer, 4th ed. Academic Press, 2021. ISBN 978-0128181430. The radiation authority.
8. Howell, J. R.; Mengüç, M. P.; Daun, K.; Siegel, R. Thermal Radiation Heat Transfer, 7th ed. CRC Press, 2020. ISBN 978-0367347079.
9. Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, 2nd ed. Oxford, 1959. ISBN 978-0198533689. The classical transient-conduction reference; shape-factor tables.
10. Rohsenow, W. M.; Hartnett, J. P.; Cho, Y. I. (eds.) Handbook of Heat Transfer, 3rd ed. McGraw-Hill, 1998. ISBN 978-0070535558.
11. TEMA Standards, 10th ed., Tubular Exchanger Manufacturers Association, 2019. The mechanical-design rules for shell-and-tube HX; fouling tables.
12. API 660 (9th ed., 2015) — Shell-and-Tube Heat Exchangers. The petroleum-industry overlay on TEMA.
13. API 661 (8th ed., 2022) — Air-Cooled Heat Exchangers for General Refinery Service.
14. ASME BPVC Section VIII Div. 1 (2023) — pressure-vessel rules; covers HX shells and channels.
15. ASHRAE Handbook — Fundamentals, 2025. ASHRAE, 2025. Property tables, comfort criteria, HVAC load calculation procedures.
16. GPSA Engineering Data Book, 15th ed., 2024. Gas Processors Suppliers Association. Practical HX, cooling-tower, and refrigeration design data.
17. JEDEC JESD51 series (JESD51-1 through JESD51-14) — semiconductor-package thermal characterisation methods. The basis for vendor θ_JA / θ_JC / Ψ_JT specs.
18. Dittus, F. W.; Boelter, L. M. K. “Heat Transfer in Automobile Radiators of the Tubular Type.” Univ. Calif. Pub. Eng. 2 (1930): 443–461.
19. Gnielinski, V. “New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow.” Int. Chem. Eng. 16 (1976): 359–368.
20. Churchill, S. W.; Bernstein, M. “A Correlating Equation for Forced Convection from Gases and Liquids to a Circular Cylinder in Crossflow.” J. Heat Transfer 99 (1977): 300–306.
21. Whitaker, S. “Forced Convection Heat Transfer Correlations for Flow in Pipes, Past Flat Plates, Single Cylinders, Single Spheres, and for Flow in Packed Beds and Tube Bundles.” AIChE J. 18 (1972): 361–371.
22. Žukauskas, A. “Heat Transfer from Tubes in Crossflow.” Adv. Heat Transfer 8 (1972): 93–160.
23. Churchill, S. W.; Chu, H. H. S. “Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate.” Int. J. Heat Mass Transfer 18 (1975): 1323–1329.
24. Chen, J. C. “Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow.” I&EC Process Design and Development 5 (1966): 322–329.
25. Chato, J. C. “Laminar Condensation Inside Horizontal and Inclined Tubes.” ASHRAE J. 4 (1962): 52–60.
26. Nusselt, W. “Die Oberflächenkondensation des Wasserdampfes.” VDI-Zeitschrift 60 (1916): 541–546, 569–575.
27. Zuber, N. “Hydrodynamic Aspects of Boiling Heat Transfer.” AEC Report AECU-4439, 1959.
28. Bell, K. J. Final Report of the Cooperative Research Program on Shell-and-Tube Heat Exchangers. University of Delaware Engineering Experiment Station Bulletin 5, 1963. (Bell–Delaware method.)
29. Bell, I. H.; Wronski, J.; Quoilin, S.; Lemort, V. “Pure and Pseudo-Pure Fluid Thermophysical Property Evaluation and the Open-Source Thermophysical Property Library CoolProp.” Ind. Eng. Chem. Res. 53 (2014): 2498–2508.