Latent heat thermal energy storage applications are increasing in modern days for various purposes, such as solar thermal applications, air conditioning, textile industries, pharmaceutical industries, etc. In the present work melting of Laurie acid (PCM) in a square enclosure with rectangular as well as T-shaped fin in the vertical and horizontal direction is studied numerically. It is observed that among T-shaped fins and rectangular fins, a vertical T-shaped fin has the least melting time. A vertical T-shaped fin reduces the melting time by up to 70.3% in comparison with a no-fin case in the bottom wall. But up to 80% melting process, a horizontal rectangular fin placed at a lower height gives the highest melting rate. Vertical fins always perform better than horizontal fins, for having a symmetrical convection current on both sides of the fin. Further horizontal T-shaped fin placed in the middle of the vertical wall performs better than the horizontal rectangular fin placed in the same position. But, the rectangular fin placed below the centre of the vertical wall performs melting much faster than the horizontal T-shaped fin, because of intensified convection current. Rectangular fin placed below the centre of the vertical wall reduces the melting time by 57.2% in comparison with no fin casein the side wall.