Recently, phase change materials (PCMs) have gained great attention from engineers and researchers due to their exceptional properties for thermal energy storing, which would effectively aid in reducing carbon footprint and support the global transition of using renewable energy. The current research attempts to enhance the thermal performance of a shell-and-tube heat exchanger by means of using PCM and a modified tube design. The enthalpy–porosity method is employed for modelling the phase change. Paraffin wax is treated as PCM and poured within the annulus; the annulus comprises a circular shell and a fined wavy (trefoil-shaped) tube. In addition, copper nanoparticles are incorporated with the base PCM to enhance the thermal conductivity and melting rate. Effects of many factors, including nanoparticle concentration, the orientation of the interior wavy tube, and the fin length, were examined. Results obtained from the current model imply that Cu nanoparticles added to PCM materials improve thermal and melting properties while reducing entropy formation. The highest results (27% decrease in melting time) are obtained when a concentration of nanoparticles of 8% is used. Additionally, the fins’ location is critical because fins with 45° inclination could achieve a 50% expedition in the melting process.