Variable-order fractional model for enhancing solar still thermal performance with hybrid nanofluid and phase change material: 6E analysis

Abstract This study introduces a variable-order fractional model (VOGL) to assess the performance of four solar still desalination with four different configurations: conventional solar still (CSS), solar still with hybrid nanofluid (SS + HNF), solar still with phase change material (SS + PCM), and a combined system with both enhancements (SS + HNF + PCM). The VOGL model is compared to the classical integer-order model under varying climatic conditions, showing significantly improved prediction accuracy. The methodology involves formulating energy balance equations for the solar still layers, incorporating a variable fractional order. MATLAB-implemented numerical simulations are verified against summer and winter experimental data. The mean absolute percentage error for water and glass temperatures was reduced to 2.254% and 2.509%, respectively, compared to 9.071% and 11.757% in the classical model. Beyond modeling accuracy, the study evaluates the impact of Ag/Fe 2 O 3 nanofluids and PCM on system performance. The SS + HNF + PCM configuration yielded the highest productivity at 1.4871 kg m −2 , with energy and exergy efficiencies of 59.15% and 7.58%, respectively. It also achieved the lowest cost per liter ($0.00746/m 2 ) and the highest annual enviroeconomic ($777.88) and exergoenviroeconomic ($77.782) savings. Enhanced systems further demonstrated higher CO 2 mitigation and improved sustainability.