| name | conduction |
| description | Heat conduction — 1D steady/transient, composite walls, cylinders, spheres, fins/extended surfaces, thermal resistance networks, 2D conduction, shape factors. |
| metadata | {"priority":7,"promptSignals":{"phrases":["conduction","thermal resistance","heat conduction","Fourier","fin","thermal conductivity","composite wall","transient heat","Biot number"],"minScore":4}} |
Heat Conduction — Complete Skill
Fourier's Law
q = -kA·dT/dx [W] (heat flux = thermal conductivity × area × temperature gradient)
q'' = -k·dT/dx [W/m²]
Thermal conductivities k [W/m·K]:
| Material | k |
|---|
| Copper | 401 |
| Aluminum 6061 | 167 |
| Steel (carbon) | 50 |
| Stainless 304 | 14.4 |
| Cast iron | 52 |
| Glass | 1.05 |
| Concrete | 1.0-1.8 |
| Brick | 0.69 |
| Insulation (glass wool) | 0.038-0.046 |
| Air (20°C) | 0.026 |
| Water (20°C) | 0.6 |
1D Steady Conduction
Plane wall:
q = kA(T₁-T₂)/L, R_cond = L/(kA)
Cylindrical wall (pipe):
q = 2πkL(T₁-T₂)/ln(r₂/r₁), R_cyl = ln(r₂/r₁)/(2πkL)
Spherical shell:
q = 4πk·r₁r₂/(r₂-r₁)·(T₁-T₂), R_sph = (r₂-r₁)/(4πk·r₁r₂)
Thermal Resistance Networks
q = ΔT_overall/R_total (analogous to Ohm's law)
Convection resistance: R_conv = 1/(hA)
Contact resistance: R_contact = R''_tc/A (R''_tc = contact conductance, 10⁻⁴-10⁻³ m²K/W)
Radiation resistance: R_rad = 1/(h_rad·A), h_rad = εσ(T_s+T_sur)(T_s²+T_sur²)
Series: R_total = R₁+R₂+... (same q)
Parallel: 1/R_total = 1/R₁+1/R₂+... (same ΔT)
Composite wall with convection:
R_total = 1/(h₁A) + L₁/(k₁A) + L₂/(k₂A) + 1/(h₂A)
q = (T_∞1 - T_∞2)/R_total
Overall heat transfer coefficient U:
q = U·A·ΔT, U = 1/R_total·A
Fins / Extended Surfaces
Governing equation: d²θ/dx² - m²θ = 0, θ = T-T_∞, m = √(hP/kA_c)
P = fin perimeter, A_c = fin cross-section area
Solutions by boundary condition:
Insulated tip (adiabatic): θ/θ_b = cosh(m(L-x))/cosh(mL)
q_fin = M·tanh(mL) where M = √(hPkA_c)·θ_b
Fin efficiency: η_f = q_fin/q_fin,max = tanh(mL)/(mL)
Corrected length for convecting tip:
L_c = L + A_c/P (rectangular: L_c = L + t/2, cylindrical: L_c = L + D/4)
Use L_c in place of L for adiabatic tip formulas
Fin effectiveness: ε_f = q_f/(hA_c·θ_b) > 2 (fin is beneficial)
Overall surface efficiency:
η_o = 1 - N·A_f/A·(1-η_f)
q_total = η_o·h·A·θ_b
Optimal fin dimensions (maximum heat transfer per unit volume):
Rectangular fin: L_opt ≈ 1.4194·(k·A_c/(h·P))^(1/2)
Optimal profile: parabolic (triangular is near-optimal, easier to manufacture)
2D Steady Conduction
Shape factors: q = kS·ΔT₁₂
| Configuration | Shape Factor S |
|---|
| Isothermal sphere in semi-infinite medium | 2πD/(1-D/4z) |
| Isothermal cylinder buried (L>>D) | 2πL/ln(4z/D) |
| Two parallel cylinders | 2πL/cosh⁻¹((4s²-D₁²-D₂²)/(2D₁D₂)) |
| Disk on surface | 2D |
| Square channel (inner a, outer b) | 2πL/(0.785·ln(b/a)) |
Transient Conduction — Lumped System (Biot ≤ 0.1)
Bi = hL_c/k ≤ 0.1 → temperature uniform in body (L_c = V/A_s)
(T-T_∞)/(T_i-T_∞) = exp(-t/τ_t) where τ_t = ρcV/(hA_s) [time constant]
t_1/2 = τ_t·ln(2) [half-time]
Transient — 1D Semi-infinite Solid
Bi > 0.1 → Heisler charts or analytical
Constant surface temperature:
(T-T_s)/(T_i-T_s) = erf(x/(2√(αt)))
Heat flux: q''_s = k(T_s-T_i)/√(παt)
Constant heat flux:
T_s - T_i = 2q''√(αt/π)/k
Thermal diffusivity: α = k/(ρc_p) [m²/s]
| Material | α (m²/s) |
|---|
| Copper | 117×10⁻⁶ |
| Aluminum | 97×10⁻⁶ |
| Steel | 12×10⁻⁶ |
| Glass | 0.34×10⁻⁶ |
| Concrete | 0.75×10⁻⁶ |
Transient — Finite Geometry (Heisler Charts)
For Fo > 0.2 (Fourier number Fo = αt/L²):
One-term approximation:
θ* = (T-T_∞)/(T_i-T_∞) = C₁·exp(-ζ₁²·Fo)·cos(ζ₁·x/L) [plane wall]
Plane wall: C₁ = 4sinζ₁/(2ζ₁+sin2ζ₁), ζ₁ from Bi·tan(ζ₁) = ζ₁ (lookup)
Output
Provide: R_thermal [K/W], q [W], temperature distribution, Biot number (lumped check), τ_t or Fourier number for transient.
