Thermal diffusion recast as energy-density transport: Irrotational velocity field, boundary conservation, and depth–time scaling

Infrared thermography (IRT) is an important technique in the field of non-destructive evaluation of aerospace composite materials. However, under non-uniform heating and noise conditions, traditional scalar diffusion models often produce errors exceeding 10–15% in defect depth estimation for heterogeneous media. This study presents a revolutionary reformulation of the thermal diffusion equation, treating it as a conservative transmission of energy density e = ρcT , driven by an equivalent velocity field uₑ = q/e = - α ∇lnT, exhibiting an invariant form ∂ₜ e + ∇·( e uₑ ) = S , which holds under the simplest assumptions (passive cooling state, S = 0; isotropic approximate constant α). The diffusion driven by uₑ is incorporated into the vector framework, and the divergence Σ = ∇· uₑ = - α ∇²ln T reveals the existence of source-sink loops across the non-uniform boundary, thus forming three verifiable principles: (i) a boundary conservation identity that links subdomain energy decay to outflow flux; (ii) the gain-invariant nonnegativity ∇· uₑ + ∂ₜln T ≥ 0; (iii) a general scaling law in which the first sign flip at the center of the sink loop follows t *_ring ∝ d²/α ( β ≈ 3). Through finite element simulation and pulsed thermal imaging experiments on flat, L -shaped, and curved aerospace composite materials with pre-fabricated voids, this method achieves good depth evaluation, which is more stable than traditional PCT. In addition, It can provide a quantitative early warning of non-ideal heat transfer such as convection and anisotropy through curl and conserved residuals, thereby improving the reliability of automated quantitative IRT non-destructive testing.