Approximate residence time distribution of fully develop laminar flow in a straight rectangular channel

Abstract

For microfluidic applications the residence time distribution (RTD) of laminar flow in rectangular channels is of interest. The exact velocity profile for this type of flow consists of an infinite series and does not allow analytical evaluation of the RTD curve. In this paper we adopt a simpler binomial product profile which was proposed in literature and serves as good approximation. This allows us to determine in an analytical manner approximate expressions for the diffusion-free RTD of fully developed laminar flow in a straight rectangular channel of arbitrary aspect ratio. Since the evaluation of this RTD is computationally elaborate because it involves the Gauss hypergeometric function, we fit it by an empirical model which is suitable for engineering applications. We find that for a Newtonian fluid there is a narrowing of the RTD as the aspect ratio decreases from unity (square channel) to zero (parallel plates). We investigate the range of applicability of the diffusion-free RTD and show that it is a good estimation for liquids in a certain range of Reynolds numbers. The actual limits of this range depend on the Schmidt number and on the aspect ratio and length-to-hydraulic-diameter ratio of the channel.