An approximate mathematical model of contact melting of an unfixed material in an elliptical capsule is developed. The main characteristic scales and non-dimensional parameters which describe the principal features of the melting process are found. Choosing a special heat flux distribution on the wall of the capsule allows us to derive a closed-form evolution equation for the motion of the solid accounting for the energy convection in the liquid, expressed through the non-linearity of the temperature distribution across the molten layer. It is shown that the melting rate of the solid depends on the shape of the capsule. Generally, elliptical capsules show higher rate of melting than circular ones. Elongated capsules provide more effective melting than oblate ones, even though they have the same aspect ratios and vertical cross-sectional areas. This phenomenon is caused by the fact, that the pressure necessary to support the solid is larger for the elongated capsules than that for oblate ones, which leads to thinning of the molten layer along with the increase of the heat flux across it. The time required for complete melting can be achieved by the right choice of the shape of the capsule, which is specified by the value of the aspect ratio. The found influence of the capsule shape on the melting rate can be used for design and optimization of practical latent-heat-thermal-energy systems.