transient-conduction

name	transient-conduction
description	Transient heat conduction — lumped capacitance, Biot number, Heisler charts, semi-infinite solid, 1D Fourier series solution, product solution for multi-D problems.
metadata	{"priority":7,"promptSignals":{"phrases":["transient conduction","transient heat","lumped capacitance","Heisler","Biot number","semi-infinite","thermal time constant"],"minScore":3}}

Transient Heat Conduction — Complete Skill

Lumped Capacitance Method

Validity: Biot Number Check

Bi = h L_c / k_solid (L_c = V/A_s = characteristic length) Lumped valid when Bi < 0.1 (temperature uniform within solid)

Solution

(T - T_∞) / (T_i - T_∞) = e^(-t/τ) = e^(-Bi × Fo) τ = ρ c_p V / (h A_s) = thermal time constant [s] Fo = αt/L_c² = Fourier number (dimensionless time) α = k/(ρ c_p) = thermal diffusivity [m²/s]

Time to reach fraction θ: t = τ × ln(1/θ) = (ρ c_p V / h A_s) × ln((T_i - T_∞)/(T - T_∞))

Total heat transferred: Q = m c_p (T_i - T_∞)(1 - e^(-t/τ))

One-Dimensional Heisler Charts (Bi ≥ 0.1)

Plane Wall (thickness 2L, centered at x=0)

At centerline (x=0): θ₀* = (T_0 - T_∞)/(T_i - T_∞) = C₁ exp(-ζ₁² Fo) ζ₁ cot(ζ₁) = Bi = hL/k (first transcendental root) C₁ = 4 sin(ζ₁)/(2ζ₁ + sin(2ζ₁))

At interior position x: θ* = θ₀* × cos(ζ₁ x/L)

For Fo > 0.2: one-term approximation valid (error < 2%)

Heisler chart reading: X-axis: Fo = αt/L²; Y-axis: θ₀*; Parameter: Bi = hL/k Separate chart for position correction: θ*/θ₀* = f(x/L, Bi)

Cumulative heat: Q/Q₀ = 1 - θ₀* × sin(ζ₁)/ζ₁ (Q₀ = ρ c_p V (T_i - T_∞))

Cylinder (radius r_0)

Same Fo, but: Bi = h r_0/k; ζ₁ J₀(ζ₁) = Bi J₁(ζ₁) (Bessel functions) θ₀* = C₁ exp(-ζ₁² Fo) (centerline) Position correction: J₀(ζ₁ r/r_0) / J₀(ζ₁)

Sphere (radius r_0)

Bi = h r_0/k; 1-ζ₁ cot(ζ₁) = Bi θ₀* = C₁ exp(-ζ₁² Fo) (center) Position: sin(ζ₁ r/r_0)/(ζ₁ r/r_0 sin(ζ₁))

Semi-Infinite Solid

Applies when: solid extends to infinity (short times, thick bodies) Boundary condition: constant surface temperature T_s

Interior temperature: (T - T_i)/(T_s - T_i) = erfc(x / (2√(αt)))

Surface heat flux: q_s = k(T_s - T_i)/√(παt) [decreases as 1/√t]

For convective BC (h, T_∞): (T - T_i)/(T_∞ - T_i) = erfc(x/(2√(αt))) - exp(hx/k + h²αt/k²) × erfc(x/(2√(αt)) + h√(αt)/k)

Contact temperature (two semi-infinite bodies touching): T_contact = (k₁ρ₁c₁)^0.5 T₁ + (k₂ρ₂c₂)^0.5 T₂ / [(k₁ρ₁c₁)^0.5 + (k₂ρ₂c₂)^0.5] Contact feels hot if k₁ρ₁c₁ (thermal effusivity) is high (metals feel cold to touch)

Product Solution (Multi-Dimensional)

θ*(x,y,z,t) = θ₁*(x,t) × θ₂*(y,t) × θ₃*(z,t)

Short cylinder: θ* = θ_plane(x,t) × θ_cylinder(r,t) Corner of semi-infinite bodies: θ* = θ_1D(x,t) × θ_1D(y,t) × θ_1D(z,t)

Valid when each dimension can be analyzed independently (constant properties, uniform initial T)

Neumann (Melting/Solidification) Problem

Moving phase boundary in semi-infinite domain Stefan condition: ρ L_f dS/dt = k_s ∂T_s/∂x - k_l ∂T_l/∂x at interface S(t) = 2λ√(αt) where λ satisfies transcendental equation

Output

Provide: Biot number Bi (lumped validity), θ(x,t) = (T-T∞)/(Ti-T∞) at specified location and time, t to reach T_target [s], cumulative Q [J], thermal time constant τ [s], appropriate method selection (lumped/Heisler/semi-infinite).