Modeling plasmas in fluid codes is essentially limited to the region of plasma quasineutrality since, due to their high thermodynamic equilibrium, fluid models near the plasma boundaries fail. Recently a concept of the polytropic coefficient function, which is a local quantity (rather than a constant, as usually assumed in plasma physics), has been introduced by Kuhn et al. [Phys. Plasmas 13, 013503 (2006)] . This concept has been already applied to the Tonks-Langmuir discharges in the case of ions created in plasma from a cold ion source, yet, due to the non-reliability of the existing models never to the important case with finite ion sources. Recently a highly reliable solution of the plasma equation with finite temperature in the limit e ≡ lD/ℓ = 0 (where lD is the Debye length and ℓ is a proper characteristic length of the discharge) has been reported by Kos et al. [Phys. Plasmas 16, 093503 (2009)]. Unlike previous Bissell-Johnson models [Phys. Fluids 30, 779 (1987)], the validity of which was limited to the rather narrow ranges of ion source temperatures, with the model by Kos et al. this range is unlimited and solutions are obtained with a high reliability and in a high resolution. Here we employ this model to find relevant plasma parameters at the sheath edge. Special attention is given to the fluid Bohm criterion, which with the ion polytropic coefficient function turns out to be exact. It shows that a kinetic generalization of this criterion might be disregarded for practical purposes.