This paper reports the two-dimensional (2D) transient numerical simulation of a phase change material (PCM) based finned heat sink to investigate the heat transfer performance for passive cooling of electronic devices. The finned heat sinks of 2 mm and 3 mm fin thickness are employed with a constant fin volume fraction of 9%, acting as thermal conductivity enhancer (TCE). The n-eicosane is employed as a PCM inside the heat sink to store the heat generated from the electronic device applied at the heat sink base. Transient numerical simulations are performed using finite-volume-method and conjugate heat transfer and melting/solidification phenomenon are investigated by applying various power levels. The numerical results show that the employed PCM with low temperature keeps the heat sink base temperature in lower limits and uniform melting is observed inside the finned heat sink. With the increase of heating power level, the PCM melting time is decreased for fin thickness heat sinks. By increasing the power level from 4 to 6 W, for the case of 3 mm fin thickness, the melting time increases by 6.63%,3.59% and 1.90% by 3 mm fin thickness heat sink, compared to the 2 mm fin thickness heat sink. The developed equations of liquid fraction and modified Nusselt number are obtained as function of modified Fourier number, Stefan number, and Rayleigh number which provide guidelines for generalizing the thermal performance of PCM based finned heat sinks.