Taylor dispersion convection velocity

The use of the Coanda effect is based on the desire to have a second passive momentum to speed up mixing in addition to diffusion [55, 163], The second momentum is based on so-called transverse dispersion produced by passive structures, which is in analogy with the Taylor convective radial dispersion ( Taylor dispersion ) (see Figure 1.180 and [163] for further details). It was further desired to have a flat ( in-plane ) structure and not a 3-D structure, since only the first type can be easily integrated into a pTAS system, typically also being flat A further design criterion was to have a micro mixer with improved dispersion and velocity profiles. 

Currently, analytical approaches are still the most preferred tools for model reduction in microfluidic research community. While it is impossible to enumerate all of them in this chapter, we will discuss one particular technique - the Method of Moments, which has been systematically investigated for species dispersion modeling [9, 10]. The Method of Moments was originally proposed to study Taylor dispersion in a circular tube under hydrodynamic flow. Later it was successfully applied to investigate the analyte band dispersion in microfluidic chips (in particular electrophoresis chip). Essentially, the Method of Moments is employed to reduce the transient convection-diffusion equation that contains non-uniform transverse species velocity into a system of simple PDEs governing the spatial moments of the species concentration. Such moments are capable of describing typical characteristics of the species band (such as transverse mass distribution, skew, and variance). 

Taylor dispersion is a special case of convection, where the dispersion is caused by a mean velocity gradient. It is most often referred to in the case of laminar pipe flow, where axial dispersion arises due to the parabolic velocity gradient in the pipe. 

In the present problem (illustrated in Figure 6.2.2(a)), we will also find that the solute pulse introduced at z = 0 win show up (on a radially averaged basis) as a concentration peak with a broadened base as z becomes large. However, this broadening of the solute profile in the z-direction is not due to the molecular diffusion coefficient Djs of species i in the solvent. Rather, it arises primarily due to the radially nonuniform axial velocity profile (6.1.1b, c) of flow in a tube. It is identified as an axial dispersion or convective dispersion. This phenomenon was first studied by Taylor (1953, 1954), and is often described also as Taylor dispersion. 

In macroscopic reactors, knowledge of the velocity profile in the channel cross-section is a necessary and sufficient prerequisite to describe the material transport. In microscopic dimensions down to a few micrometers, diffusion also has to be considered. In fact, without the influence of diffusion, extremely broad residence time distributions would be found because of the laminar flow conditions. Superposition of convection and diffusion is called dispersion. Taylor [91] was among the first to notice this strong dominating effect in laminar flow. It is possible to transfer his deduction to rectangular channels. A complete fluid dynamic description has been given of the flow, including effects such as the influence of the wall, the aspect ratio and a chemical wall reaction on the concentration field in the cross-section [37]. 

Hydrodynamic dispersion refers to the stretching of a solute band in the flow direction during its transport by a convecting fluid. Variation in the fluid velocity across the channel cross-section leads to such band broadening which is often quantifled in terms of the Taylor-Aris dispersion coefficient. 

Taylor " examined the case of injection of a small quantity of ink into a tube in which water flows at a constant rate. He measured the dispersion by measuring the absorption, by the ink, of light shining through the tube. He concluded that the symmetry came about through an interaction between convection (the mass movement, with a parabolic velocity profile) and diffusion. He also noted that the dispersion due to (convection + diffusion) was less than that due to convection alone (although it could be more than that due to diffusion alone). In other words, a random process (diffusion) decreases the total randomness (dispersion) of the system. 

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