Circular section channel flow

It will be shown in Chapter 5 that the pressure drop, AP, for isothermal flow in a circular section channel is given by... 

A power law plastic is injected into a circular section channel using a constant pressure, P. Derive an expression for the flow length assuming that... 

A polymer melt is injected into a circular section channel under constant pressure. What is the ratio of the maximum non-isothermal flow length to the isothermal flow length in the same time for (a) a Newtonian melt and (b) a power law melt with index, n = 0.3. 

We consider a co-extrusion die consisting of an outer circular distribution channel of rectangular cross-section, connected to an extrusion slot, which is a slowly tapering narrow passage between two flat, non-parallel plates. The polymer melt is fed through an inlet into the distribution channel and flows into... 

Vfjp is the friction velocity and =/pVV2 is the wall stress. The friction velocity is of the order of the root mean square velocity fluctuation perpendicular to the wall in the turbulent core. The dimensionless distance from the wall is y+ = yu p/. . The universal velocity profile is vahd in the wall region for any cross-sectional channel shape. For incompressible flow in constant diameter circular pipes, = AP/4L where AP is the pressure drop in length L. In circular pipes, Eq. (6-44) gives a surprisingly good fit to experimental results over the entire cross section of the pipe, even though it is based on assumptions which are vahd only near the pipe wall. 

The objectives of the study by Kawahara et al. (2002) were to experimentally investigate the probability of appearance of different flow patterns in a circular micro-channel. The test section was a circular transparent channel made of fused silica with an internal diameter of 100 pm and length of 64.5 mm, providing an L/d ratio of 645. 

Frequently the civil engineer encounters flow in circular sections, as in sewers, which are not under pressure and must therefore be treated as open channels. The maximum rate of discharge in such a section occurs at slightly less than full depth. Let the following variables represent this analysis ... 

As Fig. 12.1 indicates, the manifold cross section may be bead shaped and not circular. Thus, pressure flow in an elliptical cross-section channel may be more appropriate for the solution of the manifold flow. Such a problem, for Newtonian incompressible fluids, has been solved analytically. (J. G. Knudsen and D. L. Katz, Fluid Dynamics and Heat Transfer, McGraw-Hill, New York, 1958). See also, Table 12.4 and Fig. 12.51. 

Hartwick (17) aligned uniformly sized fibers into a densely packed hexagonal array. The interstices between the fibers represented the flow channels. There was no transport between the channels. The performance of the device was low relative to its permeability. This is not unexpected A key property of a packed bed is the radial mass transfer, which evens out flow nonuniformities. Tto is not possible in a device consisting of parallel independent flow paths. In an array of circular parallel channels, the breakthrough time for an unretained sample is inversely proportional to the square of the diameter of the channel. To obtain a plate count of 10,000 plates, it would be necessary that the relative standard deviation of the channel diameter is under 0.5% (see also the footnote in Section 2.1.4). This is clearly a tall order. For retained peaks, similar demands would need to be placed on the uniformity of the stationary phase from channel to channel. 

Fluids are often pumped hydrodynamically to exert the flow. Various pumps are used, including syringe pumps, peristaltic pumps, piezoelectric pumps, and gas-pressure-driven hydrodynamic pumps. In the case of hydrodynamic pumps, an inert gas is pressurized in the headspace of the vial containing the sample or carrier fluid. The force exerted by the gas on the liquid phase sustains flow of the liquid in the channel. When a liquid moves along the circular cross-section channel, the Poiseuille equation can be used to relate the... 

Cell Design Albery and coworkers [9-14] used tubular electrodes for ex situ electrochemical EPR experiments. The tubular electrode is equivalent to the channel electrode in all respects, except that the cross section is circular rather than rectangular [82, 137]. Like the later-developed channel flow cell, this setup (shown in Fig. 23) permits the interrogation of electrode reaction mechanisms of relatively long-lived radical species, [9-14] since the convective-diffusion equations are mathematically well defined, which at steady state are given by Eq. (37)... 

For a channel of circular cross-section, turbulent flow occurs when the Reynolds number is greater than 2,300. The coefficient of heat transfer of the cooling system continues to increase as turbulence increases, so the design limit of the Reynolds number for cooling channels should be at least 5,000 and preferably 10,000. If the volume flow rate of the coolant remains... 

For flow in an open channel, only turbulent flow is considered because streamline flow occurs in practice only when the liquid is flowing as a thin layer, as discussed in the previous section. The transition from streamline to turbulent flow occurs over the range of Reynolds numbers, updm/p = 4000 — 11,000, where dm is the hydraulic mean diameter discussed earlier under Flow in non-circular ducts. 

We consider the problem of liquid and gas flow in micro-channels under the conditions of small Knudsen and Mach numbers that correspond to the continuum model. Data from the literature on pressure drop in micro-channels of circular, rectangular, triangular and trapezoidal cross-sections are analyzed, whereas the hydraulic diameter ranges from 1.01 to 4,010 pm. The Reynolds number at the transition from laminar to turbulent flow is considered. Attention is paid to a comparison between predictions of the conventional theory and experimental data, obtained during the last decade, as well as to a discussion of possible sources of unexpected effects which were revealed by a number of previous investigations. 

The convective and nucleate boiling heat transfer coefficient was the subject of experiments by Grohmann (2005). The measurements were performed in microtubes of 250 and 500 pm in diameter. The nucleate boiling metastable flow regimes were observed. Heat transfer characteristics at the nucleate and convective boiling in micro-channels with different cross-sections were studied by Yen et al. (2006). Two types of micro-channels were tested a circular micro-tube with a 210 pm diameter, and a square micro-channel with a 214 pm hydraulic diameter. The heat transfer coefficient was higher for the square micro-channel because the corners acted as effective nucleation sites. 

The special flow conditions in circular (capillaries, tubes) or rectangular channels cause very different stresses depending on the position of the particles in the flow cross section. With laminar flow, for example the following applies to velocity gradient (see e.g. [37]) ... 

C = 8 applies to circular cross sections. For rectangular channels with large width-to-height ratios b/D >l, C = 6. Equation (14) is valid for pipe flow for Re = u D/v < 2300, the transition point for rectangular channels is at Re = 1500. 

The mass transfer coefficient is usually obtained from correlations for flow in non-porous ducts. One case is that of laminar flow in channels of circular cross-section where the parabolic velocity profile is assumed to be developed at the channel entrance. Here the solution of LfivfiQUE(7), discussed by Blatt et al.(H>, is most widely used. This takes the form ... 

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