We study experimentally semi-dilute solutions of xanthan, a semi-rigid polymer, flowing through an axisymmetric orifice. The pressure drop through the orifice and the size of the secondary vortex upstream of the orifice are measured simultaneously as a function of the flow rate. The results indicate that xanthan solutions behave like suspensions of rigid rods in a Newtonian solvent. Quantitatively, this analogy is supported by a theoretical analysis which combines a macroscopic flow model and the use of a microscopic variable reflecting the contribution of the rods to the bulk stress. With this simple model, it is possible to obtain the elongation viscosity of semi-rigid polymer solutions and estimate unknown macromolecular parameters.