EQ-FOURIER-HEAT · Thermal Transport
Fourier's Law of Heat Conduction (1D)
dT_dx*k + q = 0
Derivative form
k*Derivative(T(x), x) + q = 0
Variables
variable
dT_dx
temperature gradient in the +x direction (atomic alias for ∂T/∂x)
- Object
- heat_conducting_medium
- Property
- TemperatureGradient
- Context
- near_equilibrium
variable
k
thermal conductivity of the medium
- Object
- heat_conducting_medium
- Property
- ThermalConductivity
- Context
- near_equilibrium
variable
q
heat flux (energy per area per time) in the +x direction
- Object
- heat_conducting_medium
- Property
- HeatFluxDensity
- Context
- near_equilibrium
Axioms
classical constant_coefficients deterministic differential homogeneous isotropic linear
Assumptions
- Isotropic, homogeneous medium (k constant in space and direction)
- Linear (Fourier) regime: flux ∝ gradient, no higher-order terms
- No radiative or convective transport in the control volume
- Quasi-static: the gradient is well-defined (not shock-like)
Derivation
- Empirical; Fourier, Théorie analytique de la chaleur, 1822
- Derivable from kinetic theory: q = -(1/3) n <v> λ c_v dT/dx, where c_v is the molecular heat capacity
- Structurally identical to Fick's first law (EQ-FICK-DIFFUSION) under the substitution (q,k,T) → (J,D,C) — this is the first KNOWN cross-domain identity the engine must find
References
- Incropera, Fundamentals of Heat and Mass Transfer, 7th ed., §2.2
- Carslaw & Jaeger, Conduction of Heat in Solids, §1.2