Heat transfer formula sheet

The formulas of a first heat transfer course in one place, ready to print. Each row is a quantity and its formula, grouped by topic.

The three modes

Quantity	Formula
Conduction (Fourier's law)	$q = - k A \frac{d T}{d x}$
Convection (Newton's law of cooling)	$q = h A (T_{s} - T_{\infty})$
Radiation	$q = ε σ A (T_{s}^{4} - T_{sur}^{4})$
Stefan–Boltzmann constant	$σ = 5.67 \times 1 0^{- 8} \frac{W}{m ^{2} K ^{4}}$

Quantity	Formula
Biot number	$B i = \frac{h L _{c}}{k}, L_{c} = \frac{V}{A _{s}}$
Lumped capacitance (Bi < 0.1)	$\frac{T - T _{\infty}}{T _{i} - T _{\infty}} = exp (- \frac{h A _{s}}{ρ V c} t)$
Thermal diffusivity	$α = \frac{k}{ρ c _{p}}$

Quantity	Formula
Nusselt number	$N u = \frac{h L}{k}$
Prandtl number	$P r = \frac{ν}{α} = \frac{c _{p} μ}{k}$
Flat plate, laminar, average	$\overline{N u}_{L} = 0.664 R e_{L}^{1/2} P r^{1/3}$
Pipe, fully developed laminar	$N u = 3.66 (T_{s} const), 4.36 (q^{''} const)$
Dittus–Boelter (turbulent)	$N u = 0.023 R e^{0.8} P r^{n}, n = 0.4 heating, 0.3 cooling$

Quantity	Formula
Heat exchanger	$q = U A Δ T_{l m}$
Log mean temperature difference	$Δ T_{l m} = \frac{Δ T _{1} - Δ T _{2}}{ln ( Δ T _{1} /Δ T _{2} )}$
Effectiveness	$ε = \frac{q}{q _{m a x}}, q_{m a x} = C_{m i n} (T_{h, i} - T_{c, i})$

Radiation needs kelvin. Conduction and convection can use a Celsius difference, because a one-degree difference is the same on both scales.

For fins, θ_b = T_b − T_∞ is the temperature excess at the fin base. Dittus–Boelter applies roughly for Re > 10,000 and 0.6 < Pr < 160.

Before an exam, check your course syllabus for whether a formula sheet is allowed and which.