Channel, cross-section

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

For gradual changes in channel cross section and hquid depth, and for slopes less than 10°, the momentum equation for a rectangular channel of width b and liquid depth h may be written as a differential equation in the flow direction x. 

Detemiine tlie mean (superficial) fluid velocity, u, as tlie volumetric flowrate divided by die flow channel cross-section. 

This example shows how the distribution of filler particles can vary in the channel cross-section during injection moulding. 

Transferring Eq. (4.15) to divergent form and integrating this equation through the micro-channel cross-section we obtain ... 

The void fraction data obtained in micro-channels and conventional size channels showed significant differences depending on the channel cross-section and inlet geometry. For the micro-channel with a diameter of 100 pm, the effects of the inlet geometry and gas-liquid mixing method on the void fraction were seen to be quite strong, while the conventional size channels have shown a much smaller effect of inlet geometry on the void fraction. 

For laminar flow in channels of rectangular cross-section, the velocity profile can be determined analytically. For this purpose, incompressible flow as described by Fq. (16) is assumed. The flow profile can be expressed in form of a series expansion (see [100] and references therein), which, however, is not always useful for practical applications where often only a fair approximation of the velocity field over the channel cross-section is needed. Purday [101] suggested an approximate solution of the form... 

A number of authors have considered channel cross-sections other than rectangular [102-104]. Figure 2.17 shows some examples of cross-sections for which friction factors and Nusselt numbers were computed. In general, an analytical solution of the Navier-Stokes and the enthalpy equations in such channel geometries would be involved owing to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 

Figure 2.17 Channel cross-sections considered by Shah [103] (left)...

Figure 2.20 Streamline patterns of secondary flow in quadratic channels for fC=151 (above) and K = 202 (below), taken from [110]. Only the upper half of the channel cross-section is shown.

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. 

In addihon to those more or less regular channel cross-sections, Richardson et al. [104] studied heat transfer in some more exohc channels as displayed on the right side of Figure 2.17. They numerically computed Nusselt numbers and expressed them as a dimensionless enhopy generation rate, defined as... 

Figure 2.42 Micro mixer geometry with staggered groove structures on the bottom wall, as considered in [137], The top of the figure shows a schematic view of the channel cross-section with the vortices induced by the grooves. At the bottom, confocal micrographs showing the distribution of two liquids over the cross-section are displayed.

Rectangular channel cross-section 4.5 X 2 mm Center hole diameter 300 (j.m... 

P 30] A 120 parallel micro-channel device was employed (see [R 2] for a description of the corresponding single-channel device) [7]. The total flow rate was 1-10 ml h The contact length was 14 mm the channel cross-section was 3000 gm. A residence time of 2-20 s resulted. 

Using the above definitions and integrating over the channel cross section, with some manipulations, Zuber s kinematic equation results (Hsu and Graham, 1976) ... 

In the simplest and most often used form, the screw has a free channel cross-section that diminishes at a steady rate from the feed to the delivery end. The ratio of the channel depths from feed to die region along the screw is usually referred to as the compression ratio, since it gives a crude indication of the relative conveying capacities at feed and discharge. 

Figure 1. Labyrinth air flow calorimeter. Cross section A - A shows the measuring channel with four samples and the two outer insulating labyrinth channels. Cross section B - B shows one of the two piles of thermocouples which are placed at the channel inlet and the outlet to record the air temperature difference. (Reproduced with permission from ref. 10. Copyright 1989 De Gruyter.)...

Here the Ac is the channel cross-section flow area, and the factor, fRe, is a numerical constant computed and tabulated for various channel geometries. The characteristic dimension has been replaced by a hydraulic diameter defined as four times the flow area divided by the channel perimeter. 

The hydraulic models require topographic information in the form of the stream s longitudinal profile and channel cross-sections at several locations. 

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