Modeling of solidification via Galerkin method within storage unit utilizing nano-powders incorporating unsteady source term

The employed numerical method, which incorporates nanomaterial and fins, represents a noteworthy advancement in simulating enhanced freezing processes. This technique leverages an adaptive grid approach, providing a sophisticated framework for estimating the characteristics of the nanomaterial. A key innovation lies in considering the final material as a homogeneous mixture. Notably, the exclusion of gravity's influence on the freezing process, leading to the omission of velocity terms, adds to the precision of the model. This approach holds substantial promise for a range of applications requiring controlled and efficient freezing. The study's significance lies in its potential to revolutionize freezing technologies, offering improved outcomes across various industrial contexts. The analysis's physical outputs, demonstrating notable reductions in completion time, further underscore the method's efficacy in enhancing freezing processes. The Finite Element Method (FEM) has been employed to obtain a comprehensive solution, allowing for an assessment of the effect of the shape factor “m” and nanoparticle concentration (ϕ) on the freezing process. Notably, the introduction of nanoparticles leads to a significant acceleration of the process, resulting in a remarkable 26.77 % reduction in completion time. This finding underscores the crucial role that nanoparticle dispersion plays in enhancing the efficiency of the solidification process. This insight holds substantial promise for applications where controlled and rapid freezing is paramount. The choice of particles with higher “m” values leads to an enhancement in the conduction mode. Consequently, this results in a nearly 7 % reduction in completion time, demonstrating the efficiency of this selection. The fastest process achieves solidification in a mere 538.98 s, while in the absence of nanomaterial, the process with water alone takes significantly longer at 975.69 s. This showcases the substantial improvement that can be achieved with the incorporation of nanoparticles. This information is vital for industries where rapid, controlled solidification is critical.