Dean numbers

Of much greater relevance in micro reactors are rectangular channels, which were the subject of a study by Cheng et al. [110], among others. They solved the Navier-Stokes equation for channel cross-sections with an aspect ratio between 0.5 and 5 and Dean numbers between 5 and 715 using a finite-difference method. The vortex patterns obtained as a result of their computations are depicted in Figure 2.20 for two different Dean numbers. 

As mentioned earlier, in curved channels a secondary flow pattern of two counter-rotating vortices is formed. Similarly to the situation depicted in Figrue 2.43, these vortices redistribute fluid volumes in a plane perpendicular to the main flow direction. Such a transversal mass transfer reduces the dispersion, a fact reflected in the dependence in Eq. (108) at large Dean numbers. For small Dean numbers, the secondary flow is negligible, and the dispersion in curved ducts equals the Taylor-Aris dispersion of straight ducts. 

Dean number Thermal conductivity Reaction rate constant Heat transfer coefficient Permeability... 

It is well known (15, 16, 17) that colled tube heat exchangers possess superior heat transfer characteristics because of secondary flow effects. At small curvatures, the Dean number, defin-N = N -V a... 

Dean number D. Re n i. . inertial force Flow in curved channels... 

Description of inertia forces which distort SAR flows - the Dean number... 

M 60] [P 54] An analysis of the role of secondary flows which distort the SAR flows owing to inertia forces was carried out by simulations [7]. This analysis was based on using the Dean number as a measure of how to achieve ideal SAR flows, following... 

A Dean number of -140 is a kind of threshold value [47,152], For lower values, two counter-rotating vortices are found, whereas for higher values, two additional counter rotating vortices appear which are close to the center of the outer channel wall. Means to achieve this are changes in the flow velocity, the hydraulic diameter and the radius of curvature. 

It can be seen that even if the residence time for the K = 141 flow is about a factor of 4 smaller than for K = 35 at the same position, mixing proceeded to a higher degree for the larger Dean number, as is clearly visible especially when comparing the images recorded at position 2. 

M 70a] [P 62] Computational flow simulation of the secondary flow, depicted by velocity vectors, was performed for Dean numbers of 10 and 100 [47]. The helical flow is weak for the smaller Dean number. The center of rotation is located close to the midpoint of the patch. For a Dean number of 100, a notable increase in the relative strength of the helical flow is observed the center of the vortex is shifted towards the outer channel wall. 

Species concentration and velocity fields along the flow passage at Dean numbers above and below the threshold value... 

Figure 1.146 Species concentration (encoded in gray) given for cross-sections at the inlet, outlet and two intermediate positions for two Dean numbers, K= 150 (left) and 300 (right). The initial condition is shown on the left, i.e. two lamellae. Additionally, the velocity fields of the secondary flow are shown [152],...

M 69] [P 61] In a curved channel, helical flows can be produced with four vortices, composed of two times two types, a small and large one (see Figure 1.147) [152], The total vorticity of the small vortices was integrated over the relevant part of the cross-section. It is found that these vortices start to develop at Dean numbers around 200. The strongest increase in the vorticity is observed at Dean numbers between 300 and 400. 

Figure 1.147 Total secondary vorticity as function of the Dean number [152],...

Figure 1.148 Interface stretching factor vs. element number for different Dean numbers ( ) K = 10 (A) K = 100 ( ) K = 200 [47],...

Figure 1.151 Relative transverse velocities 0,10 averaged over a certain zone for Dean numbers between K = 200 and 800 [152].

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