Distribution and Friction Factor

Velocity Distribution and the Friction Factor. For a concentric annular duct with inner radius r, and outer radius r , the velocity distribution and friction factor for fully developed flow in a concentric annular duct are as follows [1] ... 

Fully Developed Flow. For a parallel plate duct with hydraulic diameter Dh = 4b (b being the half-distance between the plates) and the origin at the duct axis, the velocity distribution and friction factor are given by the following expression ... 

Isosceles Triangular Ducts. For isosceles triangular ducts like those shown in Fig. 5.27c, the velocity distribution and friction factors for fully developed laminar flow are expressed by the following set of equations suggested by Migay [173] ... 

Fully Developed Flow. Velocity distribution, the friction factor, and heat transfer for fully developed laminar flow in concentric annular ducts are described sequentially. 

The cases considered above, when the reaction rate is restricted by the distribution of elastic stresses in the polymer, do not describe in full the kinetics of complex multistage tribochemical reactions in pol3Tuers. The limiting stage may happen to be the displacement of reacting particles, formation of a new friction surface and so on. The reaction velocity constant may vary during the friction process as it depends upon the pol3mier permolecular structure, molecular-mass distribution and other factors. 

For the laminar boundary layer flows of incompressible Newtonian fluids over a wide plate, Schhchting (Boundary Layer Theory, 6th edn.. Me Graw Hill, New York, 1965) showed that the following two equations for the velocity distributions give values of the shear stress and friction factor which are comparable with those obtained using equation (7.10) ... 

The specific form of the distributed wall friction factor, Eq. [16.34], for natural-circulation flows, has been the subject of extensive investigations. Todreas and Kazimi (1990) present a summary to that time, including rod bundle data by Gruszynski and Viskanta (1983). Swapnalee and Vijayan (2011) and Ambrosini et al. (2004) are additional examples. The special consideration required for supercritical thermodynamic states has been noted earUer in this chapter (eg, Pioro and Duffey, 2003 Yadav et al., 2012b). Natural-circulation flows, having bulk motions, are somewhat different from natural convection and low-flow forced convection. The necessity for a continuous representation of the friction factor for wall-distributed resistance is an additional critical aspect of stability of NCLs as discussed in Section 16.10. 

An industrial chemical reacdor is a complex device in which heat transfer, mass transfer, diffusion, and friction may occur along with chemical reaction, and it must be safe and controllable. In large vessels, questions of mixing of reactants, flow distribution, residence time distribution, and efficient utilization of the surface of porous catalysts also arise. A particular process can be dominated by one of these factors or by several of them for example, a reactor may on occasion be predominantly a heat exchanger or a mass-transfer device. A successful commercial unit is an economic balance of all these factors. 

This creates the same type of cross-flow and improper distribution as was discussed for bubble cap tray operation. The recommendation of Hughmark and O Connell [31] includes corrections to the friction factor of Klein [39]. 

Models of the polymer coil are based on the end-to-end distance, which is generally not directly available as a quantitative feature. Coils in dilute solution can be characterized in terms of the radius of gyration, Rg, which is a statistical measure of the distribution of mass about the center of gravity or in terms of the hydrodynamic radius, Rh, that is usually determined through the use of Stokes law and a measurement of a drag coefficient or friction factor, /drag/ for the coil,... 

Derive the relation between the friction factor and Reynolds number in turbulent flow for smooth pipe [Eq. (6-34)], starting with the von Karman equation for the velocity distribution in the turbulent boundary layer [Eq. (6-26)]. 

For turbulent flow, with roughly uniform distribution, assuming a constant friction factor, the combined effect of friction and inertial (momentum) pressure recovery is given by... 

In laminar flow the velocity distribution, and hence the frictional energy loss, is governed entirely by the rheological constitutive relation of the fluid. In some cases it is possible to derive theoretical expressions for the friction factor. Where this is possible, a three-step procedure must be followed. 

Other errors, which could influence the results obtained, are, for example, wall effects ( slipping ), the dissipation of heat, and the increase in temperature due to shear. In a tube, the viscosity of a flowing medium is less near the tube walls compared to the center. This is due to the occurrence of shear stress and wall friction and has to be minimized by the correct choice of the tube diameter. In most cases, an increase in tube diameter reduces the influence of wall slip on the flow rate measured, but for Newtonian materials of low viscosity, a large tube diameter could be the cause of turbulent flow. ° When investigating suspensions with tube viscometers, constrictions can lead to inhomogeneous particle distributions and blockage. Due to the influence of temperature on viscosity (see Section Influence Factors on the Viscosity ), heat dissipated must be removed instantaneously, and temperature increase due to shear must be prevented under all circumstances. This is mainly a constructional problem of rheometers. Technically, the problem is easier to control in tube rheometers than in rotating instruments, in particular, the concentric cylinder viscometers. ... 

Figure 232. Pressure distribution in the roller nip during rolling of aluminum. Comparison between experimental and calculated results (1 kp/mm = 9.81 x lO N/m ). (a) Smooth roller surface, friction factor ft=0.14 (b) rough roller surface, friction factor ft = 0.4. /Zq = 2 mm, 1 mm, / =Z)/2 = 90mm...

Figure 3-4 illustrates the distribution of fluid energy with work done by the pump and heat added to the system. Table 3-6 gives friction factors for clean commercial steel pipes with flow in zones of complete turbulence. 

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