Conduction

Conduction is heat flow through a solid (or still fluid) by molecular contact. Learn Fourier's law, thermal conductivity, plane walls, cylinders (pipes and insulation), and transient heating.

Hold one end of a metal spoon in hot tea and the other end soon feels warm. Heat travels through the metal from hot to cold by conduction: faster-vibrating atoms and free electrons pass energy to their neighbours.

Conduction controls heat loss through walls and pipes, how quickly a hot part cools, and how well a heat sink spreads heat from a chip.

An analogy

A bucket chain of people passing buckets: each passes energy to the next. Good conductors (metals) have a fast, efficient chain; insulators (foam) have a slow, sluggish one.

Thermal conductivity

k measures how well a material conducts heat. Copper ≈ 400, aluminium ≈ 205, steel ≈ 50, stainless ≈ 16, glass ≈ 1, brick ≈ 0.7, water ≈ 0.6, wood ≈ 0.15, mineral wool ≈ 0.04, still air ≈ 0.026 W/m·K. Insulation works by trapping still air.

Steady conduction in walls and pipes

In a plane wall the temperature falls linearly; in a pipe wall it falls logarithmically with radius. Insulating pipes and walls cuts heat loss dramatically — but adding insulation to very small wires can increase loss (critical radius r_c = k/h).

Transient conduction

When temperatures change with time (quenching, cooking, start-up), thermal diffusivity α governs how fast. If Bi < 0.1 the body is almost uniform in temperature (lumped model): T − T∞ = (T₀ − T∞)e^(−t/τ) with τ = ρcV/(hA).

Multidimensional conduction and numerics

The heat equation ∂T/∂t = α∇²T + q̇/ρc is solved analytically for simple shapes (Heisler charts, Fourier series) and numerically (finite difference/element) for real parts. Contact resistance between surfaces, anisotropic k (composites) and temperature-dependent k matter in electronics and aerospace.

Formulas

• Fourier's law: q″ = −k · dT/dx
• Plane wall: Q̇ = k A (T₁ − T₂) / L
• Cylindrical wall (pipe): Q̇ = 2π k L (T₁ − T₂) / ln(r₂/r₁)
• Biot number (transient): Bi = h L_c / k (lumped if Bi < 0.1)

Worked example: Heat loss through a wall

Given: Brick k = 0.7 W/m·K, 0.2 m thick; Area 10 m², ΔT = 20 K

Find: Heat flow.

• Q̇ = 0.7 × 10 × 20/0.2 = 700 W

Answer: 700 W

Worked example: Insulated pipe

Given: Pipe OD 50 mm, insulation to 150 mm OD; k = 0.04, inner surface 150 °C, outer 30 °C

Find: Loss per metre.

• Q̇/L = 2π × 0.04 × 120/ln(75/25)
• = 30.16/1.0986 = 27.5 W/m

Answer: ≈ 27.5 W/m

Common mistakes

• Wrong: Using the plane-wall formula for thick pipe insulation. — Right: Use the logarithmic cylinder formula when r₂/r₁ is significant.
• Wrong: Forgetting to convert mm thickness to m. — Right: k is per metre — use metres.

Applications

• Building insulation: U-values of walls and roofs are set by building codes to limit heating demand.
• Electronics: Thermal interface materials and copper spreaders conduct heat from chips to heat sinks.
• Heat treatment: Transient conduction decides quench rates and hardness through the thickness of steel parts.

Key takeaways

• q″ = −k dT/dx.
• Wall: kAΔT/L; pipe: 2πkLΔT/ln(r₂/r₁).
• Insulators trap still air (k ≈ 0.03–0.04).
• Lumped transient model if Bi < 0.1.

Going further

• Fins: Extended surfaces add area; fin efficiency η_f = tanh(mL)/(mL) with m = √(hP/kA_c).
• Contact resistance: Microscopic gaps between clamped surfaces add resistance; thermal pastes and pads fill them.
• Phonons and electrons: In metals electrons carry most heat (Wiedemann–Franz law); in ceramics and diamond, lattice vibrations (phonons) dominate.