Heat Transfer Notes

Last reviewed 16 Sept 2026 · 8 min read

Modes of heat transfer

Mode	Mechanism	Needs medium?
Conduction	Energy transfer by molecular interaction/free electrons within a body or between bodies in contact	Yes (solids, stationary fluids)
Convection	Conduction combined with bulk fluid motion	Yes (fluids)
Radiation	Electromagnetic waves emitted by all bodies above absolute zero	No — can occur in vacuum

Conduction

FormulaFourier's law

Q = - k A \frac{d T}{d x}

$k$ = thermal conductivity (W/m·K); the negative sign indicates heat flows from higher to lower temperature.

Thermal conductivity (typical order of values)

Material	k (W/m·K, approx.)
Copper	~ 385–400
Aluminium	~ 200–240
Steel	~ 45–50
Concrete	~ 1–1.8
Brick	~ 0.6–0.8
Glass	~ 0.8–1.0
Water	~ 0.6
Wood	~ 0.1–0.2
Insulation (glass wool, EPS, PUF)	~ 0.02–0.05
Air (still)	~ 0.026

Metals — high k (free electrons); gases — low k; insulators trap still air.
k of most solids decreases with moisture absorption increasing for porous insulating materials (wet insulation performs poorly).
Thermal diffusivity $α = \frac{k}{ρ c}$ — how fast temperature changes propagate.

Plane walls and thermal resistance

FormulaThermal resistance

Plane wall: $Q = \frac{T _{1} - T _{2}}{R}$ , $R_{co n d} = \frac{L}{k A}$

Convection resistance: $R_{co n v} = \frac{1}{h A}$

Composite wall (layers in series): $R_{t o t a l} = \frac{1}{h _{i} A} + \sum \frac{L _{j}}{k _{j} A} + \frac{1}{h _{o} A}$

Overall heat transfer coefficient (U-value):

Q = U A Δ T, \frac{1}{U} = \frac{1}{h _{i}} + \sum \frac{L _{j}}{k _{j}} + \frac{1}{h _{o}}

Parallel paths (e.g. wall with window): add conductances $U A$ .

Cylinders (pipes)

R_{cy l} = \frac{ln ( r _{2} / r _{1} )}{2 π k L}

Critical radius of insulation

Adding insulation to a small pipe or wire can increase heat loss initially (outer surface area increases faster than conduction resistance).

r_{cr i t i c a l} = \frac{k _{in s}}{h} (cylinder), r_{cr i t i c a l} = \frac{2 k _{in s}}{h} (sphere)

If outer radius < $r_{c}$ : insulation increases heat loss (useful for electric cables to dissipate heat).
For effective insulation, outer radius should exceed $r_{c}$ .

Convection

FormulaNewton's law of cooling

Q = h A (T_{s} - T_{\infty})

$h$ = convective heat transfer coefficient (W/m²·K)

Type	Description	Typical h (W/m²·K, order)
Free (natural) convection	Fluid motion due to buoyancy (density differences)	Air: ~ 2–25
Forced convection	Fluid moved by fans, pumps, wind	Air: ~ 25–250; water: ~ 100–15 000
Boiling and condensation	Phase change	Very high (thousands)

Dimensionless numbers

Number	Definition	Significance
Nusselt (Nu)	$\frac{h L}{k}$	Ratio of convective to conductive heat transfer
Reynolds (Re)	$\frac{ρ V L}{μ}$	Inertia/viscous forces — forced convection, laminar or turbulent
Prandtl (Pr)	$\frac{μ c _{p}}{k} = \frac{ν}{α}$	Momentum/thermal diffusivity (air ≈ 0.7)
Grashof (Gr)	$\frac{g β Δ T L ^{3}}{ν ^{2}}$	Buoyancy/viscous — natural convection
Biot (Bi)	$\frac{h L _{c}}{k _{so l i d}}$	Internal/surface resistance — lumped analysis valid if Bi < 0.1

Forced convection: $N u = f (R e, P r)$ (e.g. Dittus–Boelter $N u = 0.023 R e^{0.8} P r^{n}$ for turbulent pipe flow); natural convection: $N u = f (G r, P r)$ .

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Modes of heat transfer

Conduction

Thermal conductivity (typical order of values)

Plane walls and thermal resistance

Cylinders (pipes)

Critical radius of insulation

Convection

Dimensionless numbers

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