In this paper a monitoring tool is designed for a class of plug flow reactors whose mathematical model is described by a set of first-order partial differential equations with different coefficients in the convective terms. The infinite dimensional structure of such a tool is derived according to the methodology established in the design of the well-known asymptotic observer. As a consequence, it preserves the robustness of the aforementioned observer against the lack of information of the nonlinear terms involved in the model. The original structure of the estimator is then represented as a couple of integral equations by means of the method of characteristics and its behaviour is analyzed through simulation experiments. These simulations show that the mean square observation error is 0.58 when the proposed observer is implemented in a solid-waste anaerobic digestion process to estimate the evolution of biomass concentration.