Exact analytical solutions for transient local thermal non-equilibrium heat conduction in a fluid-saturated porous slab

Exact analytical solutions are developed for one-dimensional transient heat conduction in a fluid-saturated porous slab under local thermal non-equilibrium (LTNE) conditions. The coupled two-energy equations for the fluid and solid phases are solved by an eigenfunction expansion method. Four representative transient problems are considered: sudden heating from one sidewall, sudden cooling from both sidewalls, the sudden onset of uniform volumetric heat generation within the solid phase, and sudden cooling by convection at the outer surface. The convective-cooling solution is shown to be exact only under the restriction k s e / k f e = 1 − ε / ε . The convergence of the series solution is examined, showing that five to six eigenfunctions give excellent accuracy for weakly non-equilibrium combinations (e.g., air‑aluminum), whereas strongly non-equilibrium combinations (e.g., water‑aluminum) require substantially more terms to suppress Gibbs-type oscillations near the thermal front at early times. Results demonstrate that boundary-driven heating and cooling cause the fluid and solid temperatures to approach local thermal equilibrium, whereas solid-phase heat generation produces the opposite trend, with the degree of local thermal non-equilibrium increasing as the transient progresses. The analytical solutions provide benchmark results for validating numerical methods and offer physical insight into transient LTNE heat transfer in porous media.