Generalized Channel Cross-Sections

Besides channels with different cross-sections, channels with specific periodic shapes were considered by several authors, mostly in the framework of 2-D simulations. Three different channel shapes having been studied are shown in 

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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. 

The Reynolds number, based on a hydraulic diameter, is Reo = puD/fi. In general the hydraulic diameter is given as D = 4AC/P, where Ac is the channel cross-sectional area. For a laminar flow, an analytic solution for flow in cylindrical tubes provides... 

Channels are of course also formed by all porins. A general porin contains 16 /1-strands and has a shear number of 20 and a nearly circular cross section (Table II). Three parallel barrels associate to form trimers. The type of residues outlining the channel determines the specificity of a porin which, however, is usually not very strict. The two 18-stranded porins are very specific. Their channel cross-sections are actually smaller than those of the general porins in agreement with their higher selectivity. The 22-stranded barrels of the iron transporter proteins have circular cross sections and would form a very wide channel if they were not filled with the globular N-terminal 150-residue domain. 

Another advantage of the fins is that a liquid that wets the solid surface (as is generally the case with oil and a porous catalyst) tends to occupy the grooves between the fins, as depicted in Fig. 6. Thus, an organized division of the available channel cross section for liquid flow and gas flow is obtained, with a relatively large part available for liquid flow in combination with a relatively large liquid/solid contact area. In the case of countercurrent flow, the fins may stabilize the relatively thick liquid layer against formation... 

Due to the shapes of channel cross sections, pressure losses can reach values of several bars for usual lengths. This leads to small flow velocities (some mm/s or cm/s) and low Reynolds numbers. The flow is then generally laminar or transitional. For very low Reynolds numbers (Re 1) the flow is said to be creeping and, neglecting the inertia term, the momentum equation becomes... 

It is possible to simulate complete velocity and temperature fields through the screw channel cross-section and not simply be limited to mean fluxes. Here one generalizes Eqs. (119a), (119b), and (120) to... 

A general (diagrammatic) cross-sectional view of the mould is presented in Fig. 2, from which it may be seen that the mould consists of three essential parts, viz., (1) an aluminium alloy split female mould, (2) an intermediate silicone rubber male core, and (3) a tapered central aluminium alloy plug. Important features to note are the spacer inserted between the split halves of the outer mould, the available space below the central plug, and the channels moulded into the silicone rubber (top and bottom) to form a reservoir for the resin. Prior to use with epoxide resins, the mould surfaces are treated with a silicone release agent (Tego 290 -Ambersil Ltd.) and cured for 3 hours at 230 C. 

In this section, the general expression for the determination of the velocities of spontaneous capillary flows in composite, confined microchannels of arbitrary shapes is presented. This expression generalizes the conventional Lucas-Washburn-Rideal model, which is valid for cylindrical channels. It will be shown that the use of an equivalent hydraulic diameter in the Lucas-Washburn-Rideal model introduces a bias when the shape of the channel cross section differs notably from a circle. 

It is generally preferable to meter each of the individual components of a two-phase mixture separately prior to mixing, since it is difficult to meter such mixtures accurately. Problems arise because of fluctuations in composition with time and variations in composition over the cross section of the channel. Information on metering of such mixtures can be obtained from the following sources. 

The constant pattern concept has also been extended to circumstances with nonplug flows, with various degrees of rigor, including flow profiles in tubes [Sartory, Jnd. Eng. Chem. Fundam., 17, 97 (1978) Tereck et al., Jnd. Eng. Chem. Res., 26, 1222 (1987)], wall effects [Vortmeyer and Michael, Chem. Eng. ScL, 40, 2135 (1985)], channeling [LeVan and Vermeulen in Myers and Belfort (eds.). Fundamentals of Adsorption, Engineering Foundation, New York (1984), pp. 305-314, AJChE Symp. Ser No. 233, 80, 34 (1984)], networks [Aviles and LeVan, Chem. Eng. Sci., 46, 1935 (1991)], and general structures of constant cross section [RudisiU and LeVan, Jnd. Eng. Chem. Res., 29, 1054 (1991)]. 

In order to make the derivation completely general, the term dc is used to represent the cross-sectional dimensions of any channel. In the case of a round tube, dc equals d.)... 

The data on pressure drop in irregular channels are presented by Shah and London (1978) and White (1994). Analytical solutions for the drag in micro-channels with a wide variety of shapes of the duct cross-section were obtained by Ma and Peterson (1997). Numerical values of the Poiseuille number for irregular microchannels are tabulated by Sharp et al. (2001). It is possible to formulate the general features of Poiseuille flow as follows ... 

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... 

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