Heat transfer in incompressible magnetohydrodynamic pipe flows: An experiment-based modelling methodology

Convective heat transfer in incompressible magnetohydrodynamic (MHD) pipe flows is central to the system-level design of electrically conducting fluid loops, such as the breeding blanket components of fusion reactors, yet it remains markedly less characterised than its hydrodynamic counterpart—especially under mixed convection. This work develops a data-driven modelling framework that first characterises the phenomenon across the available evidence and then condenses it into compact heat transfer correlations. An experimental database of 353 points is assembled from the liquid-metal MHD campaigns retrieved from the open literature, spanning three working fluids, uniform and differential heating, and horizontal, upward and downward flow orientations; numerical studies inform the physical interpretation but do not enter the regression database. Every entry is referred to a single target quantity — the Nusselt number built on the perimeter-averaged wall temperature — reconstructed source by source under a documented procedure. A combined qualitative and quantitative analysis then identifies the governing dimensionless groups and the functional dependencies that control the heat transfer process. On this basis, symbolic regression with the PySR library is applied under a fixed derivation protocol — multi-run campaigns with declared seeds, structural recurrence analysis, physical-admissibility screening, and frozen-structure cross-validation at the level of facilities and campaigns — which alone governs model selection. Three main correlations are retained, each with an explicitly bounded validity domain, and are offered as a toolkit rather than as a single recommended law: the widest-coverage form spans all regimes at modest accuracy, whereas the most accurate ones are restricted to negligible-buoyancy conditions, one of which remains continuous between the hydrodynamic and MHD limits and saturates under strong fields. A prototypical operating configuration, held out from every stage of the derivation, is predicted within ± 20 % for 90 – 100 % of its points, with a systematic underprediction that favours the conservative side for cooling-oriented design.