Within the framework of density functional theory, a study of approximations to the enhancement factor of the non-interacting kinetic energy functional Ts[ρ] has been presented. For this purpose, the model of Liu and Parr (Liu and Parr, Phys Rev A 1997, 55, 1792) based on a series expansion of Ts[ρ] involving powers of the density was employed. Application to 34 atoms, at the Hartree–Fock level has shown that the enhancement factors present peaks that are in excellent agreement with those of the exact ones and give an accurate description of the shell structure of these atoms. The application of Z-dependent expansions to represent some of the terms of these approximations for neutral atoms and for positive and negative ions, which allows Ts[ρ] to be cast in a very simple form, is also explored. Indications are given as to how these functionals may be applied to molecules and clusters.