📑 Table of Contents (Tap to view sections)
Overview
Heat always flows from a region of higher temperature to one of lower temperature. Three distinct physical mechanisms can transport this energy — conduction, convection, and radiation. Most real engineering systems involve all three simultaneously; understanding each mode and how they combine is the starting point for any heat-transfer analysis.
Mode 1: Conduction
Fourier's Law
Conduction is heat transfer through a stationary medium (solid or stagnant fluid) by molecular interaction — vibrating atoms pass kinetic energy to their neighbours. Fourier's Law states:
$$ \dot{q} = -k \nabla T \quad \Rightarrow \quad Q = -kA\frac{dT}{dx} $$where Q [W] is heat flow rate, k [W/m·K] is thermal conductivity, A [m²] is cross-sectional area perpendicular to flow, and dT/dx [K/m] is the temperature gradient. The minus sign confirms heat flows down the gradient.
Thermal Conductivity of Engineering Materials
| Material | k (W/m·K) | Notes |
|---|---|---|
| Diamond | 2 000 | Highest of any bulk material |
| Copper | 401 | Heat exchangers, bus bars |
| Aluminium | 237 | Air-cooled fins, plate exchangers |
| Carbon steel | 50 | Standard pipe, vessel walls |
| 316 Stainless steel | 16 | Corrosive service — 3× lower than C-steel |
| Water (25°C) | 0.613 | High vs most liquids |
| Engine oil | 0.145 | Low — convection dominates in oil coolers |
| Air (25°C) | 0.026 | Poor conductor — excellent insulator when trapped |
| Mineral wool insulation | 0.035–0.045 | Mostly trapped air with fiber matrix |
| Polyurethane foam | 0.022–0.028 | Excellent for cold insulation |
Steady-State Conduction Solutions
For common geometries at steady state (dT/dt = 0), Fourier's Law reduces to simple analytical solutions. The flat wall gives:
$$ Q = k A \frac{T_1 - T_2}{L} $$The hollow cylinder (pipe wall) with inner radius ri and outer radius ro:
$$ Q = \frac{2\pi k L (T_i - T_o)}{\ln(r_o/r_i)} $$The hollow sphere:
$$ Q = \frac{4\pi k r_i r_o (T_i - T_o)}{r_o - r_i} $$Thermal Diffusivity & Transient Conduction
In transient problems (temperature varying with time), conduction is governed by the heat equation:
$$ \frac{\partial T}{\partial t} = \alpha \nabla^2 T $$where α [m²/s] is the thermal diffusivity:
$$ \alpha = \frac{k}{\rho c_p} $$High α means the material equilibrates rapidly. Metals (α ≈ 10⁻⁵ m²/s) respond ~100× faster than soils or concrete (α ≈ 10⁻⁷ m²/s), which is why metal components reach thermal equilibrium quickly after a process upset.
Mode 2: Convection
Newton's Law of Cooling
Convection transfers heat between a solid surface and an adjacent moving fluid. The mechanism involves the combined effect of fluid conduction and bulk fluid motion (advection):
$$ Q = h A (T_s - T_\infty) $$h [W/m²·K] is the convective heat-transfer coefficient — the single most important parameter in process heat-exchanger design. Unlike k, h is not a material property; it depends on geometry, flow velocity, fluid properties, and whether the flow is laminar or turbulent.
Natural vs. Forced Convection
Natural (free) convection is driven by buoyancy: heated fluid becomes less dense and rises, cooler fluid descends. The Grashof number (Gr) characterises natural convection:
$$ Gr = \frac{g \beta (T_s - T_\infty) L^3}{\nu^2} $$where β is the volumetric thermal expansion coefficient [1/K], g = 9.81 m/s², and ν is kinematic viscosity [m²/s]. Natural convection dominates when Gr/Re² ≫ 1.
Forced convection uses an external driver (pump, fan, compressor) to move fluid past the surface. Forced convection always achieves much higher h values than natural convection for the same fluid. The Nusselt number (Nu = hL/k) correlates with Reynolds and Prandtl numbers:
$$ Nu = C \cdot Re^m \cdot Pr^n $$The Dittus-Boelter equation for turbulent pipe flow (Re > 10 000, 0.7 < Pr < 160):
$$ Nu = 0.023 \cdot Re^{0.8} \cdot Pr^n, \quad n = 0.4 \text{ (heating)}, \ n = 0.3 \text{ (cooling)} $$Typical Convective Heat-Transfer Coefficients
| Service | h (W/m²·K) |
|---|---|
| Natural convection, air | 2 – 25 |
| Forced convection, air | 25 – 250 |
| Forced convection, light oil | 50 – 1 500 |
| Forced convection, water | 1 000 – 15 000 |
| Boiling water (pool) | 3 000 – 60 000 |
| Condensing steam | 5 000 – 100 000 |
The 5 000× range from natural air convection to condensing steam is why a bare-metal surface cooled by air and a steam-heated surface are so dramatically different in size requirements.
Boundary Layer Theory
Convective h arises because of the thin thermal boundary layer — a region near the surface where the temperature transitions from Ts to T∞. Heat must conduct across this layer, and the layer thickness δT determines h:
$$ h \approx \frac{k_{fluid}}{\delta_T} $$Turbulence disrupts the boundary layer, reduces δT, and dramatically increases h. This is the fundamental reason turbulent flow dramatically outperforms laminar flow in heat exchangers, and why tube-side velocity is critical in heat-exchanger design.
Mode 3: Radiation
Stefan-Boltzmann Law
Radiation transfers heat by electromagnetic waves (primarily infrared) — uniquely, it requires no intervening medium and can cross a vacuum. The maximum possible radiative flux from a blackbody surface at absolute temperature T [K]:
$$ E_b = \sigma T^4, \quad \sigma = 5.670 \times 10^{-8} \ \text{W/m}^2\text{K}^4 $$Real surfaces emit less than a blackbody; their emissive power is:
$$ E = \varepsilon \sigma T^4 $$where ε ∈ [0, 1] is the surface emissivity. Net radiation exchange between two large parallel plates (or a small surface in a large enclosure):
$$ Q_{rad} = \varepsilon \sigma A (T_s^4 - T_{surr}^4) $$Emissivity of Engineering Surfaces
| Surface | ε |
|---|---|
| Polished aluminium / copper | 0.02 – 0.05 |
| Oxidised steel (mill scale) | 0.60 – 0.80 |
| Black paint / carbon | 0.95 – 0.98 |
| Fired heater refractory | 0.85 – 0.95 |
| Human skin | 0.95 |
| Polished stainless steel | 0.10 – 0.20 |
View Factors
For radiation between two surfaces that do not entirely "see" each other, the view factor F12 accounts for geometry. Net heat exchange from surface 1 to surface 2:
$$ Q_{12} = A_1 F_{12} \varepsilon_{12} \sigma (T_1^4 - T_2^4) $$View factors satisfy the summation rule (F11 + F12 + … = 1) and the reciprocity rule (A1F12 = A2F21). Enclosure analysis with multiple gray surfaces requires solving a system of radiosity equations.
Combined Heat Transfer: Worked Example
A bare 6-inch steam pipe (surface temperature 170°C) runs horizontally in ambient air at 25°C. Pipe outer diameter = 168.3 mm. Estimate total heat loss per metre.
Radiation component (ε = 0.75 for oxidised steel):
$$ Q_{rad} = 0.75 \times 5.67 \times 10^{-8} \times (\pi \times 0.1683 \times 1) \times [(170+273)^4 - (25+273)^4] $$ $$ = 0.75 \times 5.67 \times 10^{-8} \times 0.5285 \times [(443)^4 - (298)^4] $$ $$ = 2.246 \times 10^{-8} \times [3.85 \times 10^{10} - 7.90 \times 10^9] = 2.246 \times 10^{-8} \times 3.06 \times 10^{10} = 687 \ \text{W/m} $$Natural convection component (horizontal cylinder, Churchill-Chu correlation):
$$ h_{nc} \approx 8 \ \text{W/m}^2\text{K} \quad \text{(for air at 100°C film temperature, 6-inch pipe)} $$ $$ Q_{conv} = 8 \times (\pi \times 0.1683) \times (170 - 25) = 8 \times 0.5285 \times 145 = 614 \ \text{W/m} $$Total bare pipe heat loss:
$$ Q_{total} = Q_{rad} + Q_{conv} = 687 + 614 = 1301 \ \text{W/m} $$With 75 mm mineral wool insulation, the same pipe loses approximately 60–80 W/m — a reduction of ~95%. This example demonstrates why radiation must be included in bare-surface insulation economics calculations.
Relative Importance by Temperature Level
The dominant heat-transfer mode depends strongly on the absolute temperature level:
- Below 200°C — Conduction and forced convection dominate in most process equipment. Radiation is secondary (<15% of total on bare surfaces, negligible through insulation).
- 200–700°C — Radiation grows rapidly (Qrad ∝ T⁴). At 500°C, bare surface radiation and convection are comparable.
- Above 700°C — Radiation is dominant. Fired heaters, furnaces, and kilns transfer 60–90% of duty by radiation.
Process Industry Applications
| Equipment | Dominant Mode(s) | Notes |
|---|---|---|
| Shell-and-tube heat exchanger | Forced convection (both sides) | Conduction through tube wall (small resistance) |
| Air-cooled heat exchanger | Forced convection (tube-side); forced convection + fins (air-side) | Air-side h is the controlling resistance |
| Fired heater / furnace | Radiation (radiant section) + convection (convection section) | Radiant section: 60–75% of total duty |
| Steam condenser | Condensation (shell-side) + forced convection (tube-side) | Shell-side h: 5 000–50 000 W/m²·K |
| Reboiler / vaporiser | Nucleate boiling + convection | Critical heat flux limits design flux |
| Pipe / vessel insulation | Conduction (through insulation) | Natural convection + radiation at outer surface |
Key Dimensionless Numbers Summary
| Number | Definition | Physical Meaning |
|---|---|---|
| Nusselt (Nu) | hL / kf | Dimensionless convective h; ratio of convective to conductive heat transfer |
| Prandtl (Pr) | cpμ / k = ν / α | Ratio of momentum to thermal diffusivity; Pr < 1 for metals, Pr ≈ 7 for water, Pr ≫ 1 for oils |
| Reynolds (Re) | ρvL / μ | Inertia vs. viscous forces; Re > 10 000 → turbulent pipe flow |
| Grashof (Gr) | gβΔTL³/ν² | Buoyancy vs. viscous forces; governs natural convection |
| Biot (Bi) | hL / ksolid | Surface convection vs. internal conduction; Bi ≪ 1 → lumped capacitance valid |
| Fourier (Fo) | αt / L² | Dimensionless time for transient conduction |
Try the Calculator
Use the Heat Transfer Coefficient Calculator on this page to combine all three modes in a thermal resistance network — enter film coefficients (convection), wall conductance (conduction), and fouling factors, and compute the overall heat duty instantly.
Heat Exchanger Rating
Apply this methodology directly in the ChemProCal calculator.
Open Calculator →Live Overall Heat Transfer Coefficient (U-Value) Estimator
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