Thermal energy storage (TES) with phase change materials (PCM) can improve energy efficiency by reducing energy availability and demand mismatch. However, despite the potential of this technology, the low thermal conductivity of PCM slows down the storage cycle and limits its commercialisation. The literature has proposed many solutions to improve heat transfer, such as fins, encapsulation, or metal foam inserts, which are too expensive and complicated. This study focuses on an inexpensive and practical solution, the eccentricity of shell and tube heat exchanger (STHE). Eccentricity's impact has been well established, but its effects on multi-tube cases have not yet been examined. This study investigates the eccentricities of four horizontal STHE with 1, 2, 3, and 4 tubes. The paper presents the ideal combinations of tube number and eccentricity for maximising TES efficiency and explores the underlying heat transfer and fluid mechanics phenomena. The enthalpy double porosities model accurately predicts the PCM melting and solidifying characteristics with varied eccentricities. Results reveal that the eccentric charge performance is better than the concentric tube(s) arrangement. The optimal eccentricity of the charging cycle depends on the tube number. The optimal eccentricities of the single and multi-tube systems are 6 and 10. The total melting time of optimal eccentricity for the four-tube, three-tube, two-tube, and single-tube is optimised by 63 %, 63 %, 60 % and 54 % compared to the concentric cases, respectively. The optimal eccentricity of the single-tube case has the shortest melting time, which increases linearly with the Rayleigh number. The concentric cases consistently exhibit the best performance and shortest solidification time as the Rayleigh number increases. The total solidification time of the single-tube case is reduced by 56 %, 52 % and 49 % compared to the four-tube, three-tube and two-tube, respectively. The multiphysical mechanisms of the charge and discharge cycles are discussed.