Explicit solution of a generalized mathematical model for the solar collector/photovoltaic applications using nanoparticles

Abstract

This paper analyzes the system describing the absorption solar collector as an application of nanoparticles for the storage of solar energy. The system involves two fractional partial differential equations (FPDEs) utilizing the Caputo fractional definition (CFD). Explicit solutions are determined for the temperature and velocity in terms of the Wright function (WrFn) via Laplace transform (LT). In addition, the solutions are expressed in terms of some well–known special functions at selected values of the fractional–order. Moreover, the behaviors of the temperature and velocity are investigated graphically using the thermo–physical data of the Cu/Al2O3–nanoparticles. Furthermore, some numerical results are conducted about the performance of the Cu and Al2O3. The results reveal the higher efficiency/performance of the Cu–nanoparticles over the Al2O3 and thus copper enjoys higher capability than alumina for the purpose of solar energy storage. In addition, it is found that 1.41% in the enhancement of heat transfer is achieved by adding 1% of the Cu–nanoparticles while the corresponding enhancement rate of the other three nanoparticles was less. Besides, the enhancement rate reaches 7.1% by increasing the volume fraction of the Cu–nanoparticles to 5%. Furthermore, the enhancement rates for the TiO2, Ag, and Al2O3 are 5.41%, 6.47%, and 6.53%, respectively when using 5% of these nanoparticles. Finally, the advantages/effectiveness of the current approach over a previous work in the literature are discussed in detail.

Keywords

• Fractional PDE
• Wright function
• Solar collector
• Photovoltaic
• Laplace transform
• Boundary value problem