Streaming current development in a rectangular channel

Abstract

In this paper, we fast analyse the flow distribution in a metallic pipe of rectangular cross section. In the case of a laminar flow, an analytical solution has been found for the velocity distribution in the pipe cross section. Concerning the space charge density profile, in the case of weak space charge density in the whole section, the general system of equations for a rectangular channel have first been solved. Thus, the space charge density profile can be expressed in terms of the channel dimensions, the diffusion constants and the total charge in the diffuse layer, which is an indicator of the diffuse layer development. Indeed, in the case of a diffuse layer development controlled by a weak wall current ( as it is very often the case ), the space charge density profile is close to a quasi fully developed diffuse layer profile for which the total charge in the diffuse layer increases slowly in order to reach a constant value. Then, the space charge density convected in a such rectangular pipe has been computed, in terms of the geometry of the pipe and the state of development of the diffuse layer, that is to say, in terms of the total space charge in a given cross section. In the case of a turbulent flow, an analysis shows that in our experimental results, a large part of the diffuse layer is perturbed by the turbulence even for a relatively low Reynolds number. Therefore, the space charge density convected in a given cross section is nearly proportional to the mean space charge density in the whole section, or the total charge in this section. Thus, the mesurement of the space charge density convected in a given section indicates the mean space charge density in that section, and so the state of development. We present new experimental streaming current results for a rectangular channel. The experiments were conducted for Reynold's numbers varying from 2000 to 5000 and channel lengths from the entry of the pipe varying from a few centimeters to one meter. These results give the evolution of the space charge density development for different Reynolds numbers and for different lengths from the entry of pipe.