Distribution of Retention Times in Biological Filters. I: Mathematical Model

Abstract

An assemblage of spheres having a rhombic arrangement, whose spacing among the particle centers is adjusted according to the specified porosity, is used as a model to investigate the physics of the unsaturated flow through biological filters. The model was used to develop a mutual correspondence between the flow over a single sphere, the jet dropping from it, and the extremely complex flow through filters. It has been possible to clarify the mechanism of flow, and to express the various parameters characterizing the dispersion of soluble matter through beds in terms of the properties of the media and the fluid. These parameters correspond to the characteristic points of a curve of tracer concentration versus the elapsed time. They include the mean residence time, the modal time, and the theoretical detention time. Relations among these various definitions of flow times are investigated and equations expressing flow times as a function of surface tension of liquid; grain size, porosity, and other physical quantities related to the bed; and hydraulic quantities are given.