A numerical investigation of the Graetz problem in straight pipes of circular and non-circular cross-sections is carried out to analyze the influence of the theological parameters on the heat transfer enhancement with negligible axial heat conduction and viscous dissipation for a class of non-linear viscoelastic fluids constitutively represented by the simplified Phan-Thien-Tanner (SPPT) model. The analysis considers both constant wall heat flux and constant wall temperature thermal boundary conditions and concludes that the combined elastic and shear-thinning effects represented by the parameter epsilon We(2) lead to heat transfer enhancement for low values of the parameter of order O(1) whereas values of the parameter epsilon We(2) > O(10) lead to a decrease in the heat transfer rate in the case of constant wall heat flux. Nusselt number distributions in the entrance region of tubes of equilateral triangular, square and rectangular cross-sections as well as Nusselt numbers Nu = f(epsilon We(2)) for the fully developed flow in these non-circular tubes are reported for the first time for non-linear viscoelastic fluids of the SPPT type. It is concluded that for small values of elasticity (We), the computations based on the methods included in the Polyflow software are in full agreement with analytical results when available and that discrepancies exist for high values of We. Such limitations may not exist with pseudo-spectral methods. (c) 2012 Elsevier Masson SAS. All rights reserved.