| 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).
