A modified fractal Langmuir non-isothermal adsorption-swelling model for CO2/CH4 in coal: Coupled effects of coal rank and temperature

The conventional isothermal Langmuir equation assumes an idealised homogeneous pore surface, making it insufficient to describe the structure-dependent adsorption of different coal ranks under varying temperatures. LF-NMR was applied to characterise the pore structures of lignite, long flame coal, bituminous coal, and anthracite. A unified dynamic structural parameter ( δ T ) was constructed by integrating the pore size distribution fractal dimensions (2.849, 2.836, 2.819, 2.829), surface tortuosities (1.256, 1.209, 1.159, 1.183), and the pore size range. By incorporating δ T into the conventional Langmuir model, a modified fractal Langmuir equation was derived. Isothermal experiments validated this model ( R 2 ≈ 0 . 900 ), revealing that the maximum adsorption capacities for CO 2 (21.318–69.530 cm 3 /g) and CH 4 (18.202–32.048 cm 3 /g), alongside their corresponding maximum volumetric swelling strains for CO 2 (1.729%–3.681%) and CH 4 (0.629%–1.296%), increased with δ T . To incorporate temperature influence, this model was extended to non-isothermal conditions using a fractional-order correction. Subsequent non-isothermal experiments on Huainan bituminous coal validated this coupled model ( R 2 > 0 . 968 ), demonstrating that while rising temperatures decreased the adsorption capacities of CO 2 (64.122–69.530 cm 3 /g) and CH 4 (26.986–32.048 cm 3 /g), the maximum volumetric swelling strains for CO 2 (3.534%–3.681%) and CH 4 (1.265%–1.296%) exhibited a decrease-then-increase trend due to the coupled effects of desorption-induced shrinkage and thermal expansion. Its applicability was further assessed using datasets from Poland and China to describe gas adsorption and swelling responses in CO 2 -ECBM engineering.