Flow curved channel

For friction loss in laminar flow through semicircular ducts, see Masliyah and Nandakumar AlChE J., 25, 478-487 [1979]) for curved channels of square cross section, see Cheng, Lin, and On ]. Fluids Eng., 98, 41-48 [1976]). 

The exit region of a die used to extrude a plastic section is 10 mm long and has the cross-sectional dimensions shown below. If the channel is being extruded at the rate of 3 m/min calculate the power absorbed in the die exit and the melt temperature rise in the die. Flow curves for the polymer melt are given in Fig. 5.3. The product pCp for the melt is 3.3 x 10. ... 

A polyethylene moulding material at I70°C passes along the channel shown at a rate of 4 X 10 m /s. Using the flow curves given and assuming n = 0.33 calculate the pressure drop along the chatmel. 

In the articles cited above, the studies were restricted to steady-state flows, and steady-state solutions could be determined for the range of Reynolds numbers considered. Experimental work on flow and heat transfer in sinusoidally curved channels was conducted by Rush et al. [121]. Their results indicate heat-transfer enhancement and do not show evidence of a Nusselt number reduction in any range... 

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. 

Figure 3.42 Evolution of a pulse at the entrance of a micro channel for different diffusion coefficients. Calculated concentration profile (left) and cumulative residence time distribution curve (channel 300 pm x 300 pm x 20 mm flow velocity 1 m s f = 10 s) [27],...

Check the system (or loop) instability by using the Ledinegg criterion with an average lumped channel pressure drop. If it does not satisfy the Ledinegg stability criterion, one or more of the three remedies can be taken orifice the inlet, increase the steepness of the pump head-versus-flow curve or increase the resistance of the downcomer of a natural-circulation loop. 

Fig. 10.9 Photograph of the sensor chip and the flow through cuvette. The chip contains 11 four channel YI sensors. The light is coupled into the chip by means of an optical fiber at the top right hand side. The light is transported to the sensing area by means of a curved channel waveguide, visible as a white curved line. Ruler indicates the dimensions in centimeters. Reprinted from Ref. 28 with permission. 2008 American Chemical Society...

With the development of modern computation techniques, more and more numerical simulations occur in the literature to predict the velocity profiles, pressure distribution, and the temperature distribution inside the extruder. Rotem and Shinnar [31] obtained numerical solutions for one-dimensional isothermal power law fluid flows. Griffith [25], Zamodits and Pearson [32], and Fenner [26] derived numerical solutions for two-dimensional fully developed, nonisothermal, and non-Newtonian flow in an infinitely wide rectangular screw channel. Karwe and Jaluria [33] completed a numerical solution for non-Newtonian fluids in a curved channel. The characteristic curves of the screw and residence time distributions were obtained. 

Numerical solution of equations (13)—(19) for polypropylene extrusion was made in 29,34> using approximation of the flow function (flow curve) by a piecewise power function. To find the root of b(f, M) of Eq. (13), the authors used a formal search algorithm compiled as a standard program for computer M-20 (USSR). Figure 2 gives dependency of b/f upon M (M is the specific moment of a core s rotation, i.e., the moment related to the length of the channel). It can be seen in Fig. 2 that (b/f) is a strictly decreasing function. 

Fig. 12. Tentative flow curves of low-density polyethylene with MFI = 2.0 g/10 min extruded at 170 °C through channels with a two-angle ellipse Wber cross section with a length of 50 (a), 75 (b), and 100 (c) mm with reciprocating-rotary vibration of the element in the zone upstream of the inlet to the channel (according to the diagram given in Fig. 9) ...

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

Figure 1.81 Two different horizontally multi-laminating mixers with different co-flow schemes curved-channel design (P-type, left), straight-channel design (middle). In addition, a part of a V-type mixer with minimized dead volume is shown [Pfeifer et at, Chem. Ing. Tech. 76, 5 (2004) 607],...

The generation of secondary helical flows in suitably curved channels, known as Dean vortices, is not an entirely new fluidic phenomenon discovered at the micro scale actually it was found already also for wound tubings of conventional diameter (see a summary in [152]). In-depth studies concerning Dean vortices in curved channels were made in the framework of various applications such as filtration, heat exchange, friction and mixing. 

M 69] [P 61 ] For a curved channel design, species concentration and velocity fields of the secondary flow were given (see Figure 1.146) [152], These fields each were... 

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. 

Periodic switching of helical flow in curved channels - chaotic advection... 

Hence alternating flows can also be achieved by repetitive induction of entrance flow effects [152], This relies to a constant re-direction of flows from one curved channel into another. 

Let us study the flow curve in the form Ig t[ = f(lg r), where t is shear stress on the channel wall, p is viscosity. The displacement vector along which the FCs are matched can be presented as consisting of 4 components ... 

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