Heat Conduction Model Based on the Explicit Euler Method for Non-Stationary Cases

This article presents an optimization of the explicit Euler method for a heat conduction model. The starting point of the paper was the analysis of the limitations of the explicit Euler scheme and the classical CFL condition in the transient domain, which pointed to the oscillation occurring in the intermediate states. To eliminate this phenomenon, we introduced the No-Sway Threshold given for the Fourier number (K), stricter than the CFL, which guarantees the monotonic approximation of the temperature-time evolution. Thereafter, by means of the identical inequalities derived based on the Method of Equating Coefficients, we determined the optimal values of Δt and Δx. Finally, for the construction of the variable grid spacing (M2), we applied the equation expressing the R of the identical inequality system and accordingly specified the thickness of the material elements (Δξ). As a proof-of-concept, we demonstrate the procedure on an application case with major simplifications: during an emergency shutdown of the Flexblue® SMR, the temperature of the air inside the tank instantly becomes 200 °C, while the initial temperatures of the water and the steel are 24 °C. For a 50.003 mm × 50.003 mm surface patch of the tank, we keep the leftmost and rightmost material elements of the uniform-grid (M1) and variable-grid (M2) single-line models at constant temperature; we scale the results up to the total external surface (6714.39 m²). In the M2 case, a larger portion of the heat power taken up from the air is expended on heating the metal, while the rise in the heat power delivered to the seawater is more moderate. At the 3000th min, the steel-wall temperature in M1 falls between 26.229 °C and 25.835 °C, whereas in M2 the temperature gradient varies between 34.648 °C and 30.041 °C, which confirms the advantage of the combination of variable grid spacing and the No-Sway Threshold.