A finite-element method for predicting flow through ducts of arbitrary and varying cross sections is presented. The governing differential equations are discretized using a control-volume approach. For this purpose, specially constructed three-dimensional control volumes are employed. The method is of the marching type and is based an a fully implicit formulation. Here, in part I of the paper, only the basic methodology is described; in the accompanying pan II, results of application of the method to some test problems are presented.