We show that dispersion effects due to local velocity gradients and transverse molecular diffusion in chromatographs and chemical reactors can be better described in terms of averaged models that are hyperbolic in the longitudinal coordinate and time, and with an effective local time or length scale in place of the traditional axial dispersion coefficient. This description not only eliminates the use of artificial exit boundary conditions but also inconsistencies such as upstream propagation and infinite speed of signals associated with the traditional parabolic averaged models. We also show that the hyperbolic models can describe dispersion effects accurately and have a much larger region of validity in the physical parameter space compared to the traditional parabolic models. Our method of obtaining averaged models from the governing partial differential equations is based on the Lyapunov-Schmidt technique of classical bifurcation theory and is rigorous. We illustrate our approach using three well known chemical engineering problems.