In this study, general relationships applicable to a wide variety of microbial-mediated treatment processes are developed using a number of continuous-how stirred tank reactors-in-series. Simultaneous equations resulting from mass balance on substrate and biomass are solved numerically taking the longitudinal biomass gradient into account. A relationship between substrate and biomass concentrations is developed as a function of hydraulic residence time. Monod kinetics and mathematical models of reactors-in-series are used to represent the actual conditions resulting from varying degrees of axial dispersion and wastewater quality. Dimensionless quantities are used to reduce the number of parameters to be taken into account. Computer techniques are applied to express the results generally. By means of the numerical approach, the variation of the ratio between the hydraulic residence time of a given reactor and that of an equivalent plug-flow tank with the same inlet and outlet conditions is investigated. Results are expressed in the form of graphs to characterize the plug-flow activated sludge systems and to provide a basis of design. Experimental data reported in the literature are also evaluated to demonstrate the cases where the existing classical solution to the problem differs from the actual results obtained from the computer program and the design graphs given. (C) 2000 Elsevier Science Ltd. All rights reserved.