Some solutions of the differential equations, which describe an elementary model of the disposition of material carried by a fluid flowing in a pipe, are derived and investigated. The solutions pertain to a variety of assumptions made regarding the initial concentrations of the material in the fluid within the pipe and on the pipe interior surface, and the concentration of the material in the fluid entering the pipe. The means of obtaining some of the model parameters using the derived solutions and measured values of the concentrations of matter on the pipe surface and in the fluid are illustrated.