Trapezoidal channel

Wu and Cheng (2003) measured the friction factor of laminar flow of de-ionized water in smooth silicon micro-channels of trapezoidal cross-section with hydraulic diameters in the range of 25.9 to 291.0 pm. The experimental data were found to be in agreement within 11% with an existing theoretical solution for an incompressible, fully developed, laminar flow in trapezoidal channels under the no-slip boundary condition. It is confirmed that Navier-Stokes equations are still valid for the laminar flow of de-ionized water in smooth micro-channels having hydraulic diameter as small as 25.9 pm. For smooth channels with larger hydraulic diameters of 103.4-103.4-291.0pm, transition from laminar to turbulent flow occurred at Re = 1,500-2,000. 

A variety of studies can be found in the literature for the solution of the convection heat transfer problem in micro-channels. Some of the analytical methods are very powerful, computationally very fast, and provide highly accurate results. Usually, their application is shown only for those channels and thermal boundary conditions for which solutions already exist, such as circular tube and parallel plates for constant heat flux or constant temperature thermal boundary conditions. The majority of experimental investigations are carried out under other thermal boundary conditions (e.g., experiments in rectangular and trapezoidal channels were conducted with heating only the bottom and/or the top of the channel). These experiments should be compared to solutions obtained for a given channel geometry at the same thermal boundary conditions. Results obtained in devices that are built up from a number of parallel micro-channels should account for heat flux and temperature distribution not only due to heat conduction in the streamwise direction but also conduction across the experimental set-up, and new computational models should be elaborated to compare the measurements with theory. 

This electrohydrodynamic (EHD) mixer (Figure 1.6) device provides a simple flowthrough chamber which has an upper and lower electrode for generating a electromagnetic field. The chamber channel is given by a sandwich of two plates, one being microstructured [94], The bottom plate contains a trapezoid channel. Two electrode layers are deposited on parts of the channel bottom and channel top and on the top part of this plate so that they reach the outside for external electrical contact. 

Consider a trapezoidal channel having side walls sloped outward from the channel bottom s flat floor. This slope angle ( ) is measured from a vertical line that begins at the channel s floor, for both side walls. Then ... 

It has already been pointed out that the cross section most commonly encountered in open-channel hydraulics is not rectangular but trapezoidal. As repeated trial-and-error solutions of Eq. (10.119) become very tedious, practicing hydraulic engineers avail themselves of numerous tables and curves which have been prepared for finding the critical depth in trapezoidal channels of any bottom width and side slopes. 

Fig. 6.47 Two intermeshing screws with the channel segments of one of the screws filled with silicone rubber, reproducing the shape of the channel segments. Two segments are pulled out, one retaining the original curved shape and the other flattened out into a trapezoidal channel.

As illustrated in Figure 8.13 the CE capillary channel is etched into the bottom plate and a second plate is bonded on top. The channel dimensions are on the order of 10-12 mm deep and 30-70 mm wide. Most designs are rectangular in shape, although trapezoidal channels have also been constructed that are 70 jam wide at the bottom of the channel and 90 nm... 

C. Chiranjivi, and P. S. Rao, Laminar and Turbulent Convection Heat Transfer in a Symmetric Trapezoidal Channel, Indian Journal of Technology, (9) 416-420,1971. 

Anonymous (1979). Hydrology president elect Jaime Amarocho. Eos 60(35) 640. P Babb, A, Amorocho, J. (1965). Plow conveyance efficiency of transitions and check structures in a trapezoidal channel. Dept. Irrigation, University of California Davis CA. de Vries, J.J., Amorocho, J., Hartman, W.J. (1980). Sediment modeling for the Sacramento... 

Edson, C.G. (1952). Two nomographs developed for trapezoidal channels. Civil Engineering 22(2) 148-149. 

The above equations can be solved analytically to yield / for simple geometries such as a tube, an ellipse, and a rectangular channel with no sidewalls. For more complicated geometries such as a trapezoidal channel or one arising from isotropic etching fabrication methods, one has to resort to numerical solutions. In these cases, commercially available solvers such as FEMLAB and PDE Toolbox in MATLAB may be used to rapidly determine the dispersivity function/via numerical techniques. 

Hydrodynamic Dispersion, Fig. 4 Effect of the different geometric parameters on solnte dispersion in a trapezoidal channel for a fixed depth in its central section... 

It is important to note that while the dispersivity function / in a triangular profile assumes a value of about 4 for 0 = 0, this quantity decays off as (Z//) for trapezoidal channels... 

The aspect ratio P of a double-trapezoidal channel is defined as the ratio between the height (h) and the maximum width (a) P can assume all the values between 0, the parallel plate configuration, and 1.414, the rhombic configuration. 

For rectangular and trapezoidal channels, the dimensionless area of the cross section is a function of the aspect ratio (a) and of the apex angle (0) it can be expressed as... 

P.E. Geyer, D.F. Fletcher, B.S. Haynes, Laminar flow and heat transfer in a periodic trapezoidal channel with semicircular cross-section, Int. J. Heat Mass Transfer. 2007, 50. 3571-3480. 

M. Renksizbulut, H. Niamand, Laminar flow and heat transfer in the entrance region of trapezoidal channels with constant wall temperature. Journal of Heat Transfer -Transactions of the ASME, 2006, 128, 63-74. 

Recent research in packed beds [38], ceramic monoliths [20] and microreactors [30] also revealed an excellent performance of a Ce-Zr support mixture, which was explained by an increase in surface area by the addition of zirconium oxide [38]. A zirconium oxide-supported rhodium catalyst also revealed similar ignition performance to a mixed Ce-Zr oxide supported rhodium catalyst on the surface of Fecralloy monoliths with trapezoidal channels [9]. A novel route to a support, which might be useful for CPO and OSR, is the synthesis of silicon carbide foam, recently used for steam reforming [44] (Figure 25.2). This support would also be less acidic and is suitable for building a compact foam catalyst... 

One approach to reactors with additional supports is the use of metallic monoliths with trapezoidal channels of 1.2 mm hydraulic diameter for methane CPO for catalytically rich combustion in a gas turbine ]9]. Fuel is turned partly into hydrogen before combustion. The catalyst was Rh/Zr02 coated on a Fecralloy monolith. 

Leading laboratories for micro-heat pumps include Battelle in the US (see Appendix 5) and SINTEF in Norway (Munkejord et al., 1999). The work at SINTEF is on devices with main dimensions of the order of a few centimetres with trapezoidal channels of 0.5 mm width concentrated on the heat exchangers and their performance. Illustrated in Figure 11.10, the miCTo-condenser so constructed showed that a heat flux of 135kW/m, based on the refrigerant-side area, was possible. The overall U value was lOkW/m K. 

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