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Parabolic trough solar collectors (PTSCs) are commonly used in solar thermal implementations to achieve high-temperatures. The current investigation looks at entropy formation and the effect of nano solid particles on a parabolic trough surface collector (PTSC) mounted aboard a solar-powered ship (SPS). The non-Newtonian first grade viscoelastic type, as well as a porous medium and Darcy-Forchheimer effects, were utilised in the current study. The flowing of PTSC was created by a non-linear stretching sheet, and the changing thermal conductivity, heat source, and viscous dissipation effects were used to calculate the heat flux in the thermal boundary layer. To convert partial differential equations (PDEs) into solvable ordinary differential equations (ODEs) with boundary-constraints, a similarity transformation strategy was used. The boundary-constraints and PDEs have been reduced to a set of non-linear ODEs (ordinary differential equations). To reach the approximated solution of ODEs, the Galerkin finite element method (G-FEM) is used. As working fluids, copper-sodium alginate (Cu-SA) and molybdenum disulfide-copper/sodium alginate (MoS2-Cu/SA) hybrid nanofluids were used. According to the findings, the permeability factor diminished the Nusselt number whilst boosting the skin friction factor. Furthermore, overall entropy variance throughout the domain was increased for flow speeds using the Reynolds number, and viscosity changes were tracked using the Brinkman number. When compared to MoS2-Cu/SA, using Cu-SA nanofluid boosted thermal efficiency by 1.3–18.8%.