The physics of dispersion in fixed beds of randomly-placed particles are considered against the background of experimental and theoretical work published during the past several years. The general equations for the reactor are shown as developments of the Central Limit Theorem, that may be presented as a second order equation defined by two-point boundary conditions. When dispersion is convective it is shown that the boundary conditions may be transformed into initial conditions and a numerical solution, if required, may be obtained without iteration. Illustrations of the method are given for non-linear processes including single and non-isothermal reactions. The application of the method to multiple reactions is described. (C) 2004 Elsevier Ltd. All rights reserved.