Pipe flow wall stress

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. 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

In addition to the measurement of the viscosity, this technique also allows the yield stress to be estimated. For a typical yield stress type material, there is a critical shear stress below which the material does not deform and above which it flows. In pipe flow, the shear stress is linear with the radius, being zero at the center and a maximum at the wall. Hence, the material would be expected to yield at some intermediate position, where the stress exceeds the yield stress. The difficulty with this method is in the determination of the point at which yielding occurs and, indeed, whether the material is appropriately modeled as having a yield stress or is... 

The die-swell (extrudate swell) effect describes the significant expansion of the diameter of the fluid column after exiting from a small pipe (Figure 4.3.8(b)). Some polymer fluids can have a swelling of up to two or three times the exit diameter. A simple proposition for the mechanism of the die-swell phenomenon is that while the fluid is inside the exit pipe, it is subject to a velocity shear, similar to the pipe flow with a maximum shear stress at the wall [18]. This velocity shear stretches... 

The wall shear stress can be calculated on the basis of the fully developed pipe flow correlation (the hydraulic diameter concept). 

Fanning (Darcy) friction factor f(f or fD) e, D 2 V2L fo = 4f TW yv2 e, = friction loss (energy/mass) rw = wall stress (Energy dissipated)/ (KE of flow x 4L/D) or (Wall stress)/ (momentum flux) Flow in pipes, channels, fittings, etc. 

Reynolds number flows /vRe N -°Vp /vRe — pV2 pV/D AQp izDp PV2 Tw/8 Pipe flow rw =wall stress (inertial momentum flux)/ (viscous momentum flux) Pipe/internal flows (Equivalent forms for external flows)... 

Because the friction loss and wall stress are related by Eq. (5-47), the loss coefficient for pipe flow is related to the pipe Fanning friction factor as follows ... 

Because the shear stress is always zero at the centerline in pipe flow and increases linearly with distance from the center toward the wall [Eq. (6-4)], there will be a finite distance from the center over which the stress is always less than the yield stress. In this region, the material has solid-like properties and does not yield but moves as a rigid plug. The radius of this plug (r0) is, from Eq. (6-4),... 

Consider, first, the simple Reynolds analogy for pipe flows. The pipe wall temperature will be assumed to be uniform. If y is the distance from the wall to the point in the flow being considered as shown in Fig. 7.2, the total shearing stress and heat transfer rate are again written as ... 

Now, for fully developed pipe flow it is usual to work with the friction factor, /, rather than the wall shearing stress. The friction factor is defined in terms of the pressure gradient by ... 

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

Silverman [21] derived velocity correlations between a rotating cylinder (mO, pipe flow (m2), annulus flow (1/3), and an impinging jet (wall jet region only, 1/4), as listed in Table 2. These equations assume that the appropriate transformations are to be made on the basis of equal mass-transfer rates for the different geometries. Silverman [21] also explored the case where the equality of surface shear stress is the appropriate criterion, on the basis that the equality of the shear stress will ensure the same corrosion processes for the various geometries. We stress that the equations listed in Table 2 must be used with great caution because they are based on the... 

Fig. 4 shows Toms / - data for both a Newtonian solvent and a DR polymer solution. Drag reduction onset occurs at 10,000 in the 0.202 cm tube, while the critical conditions for DR are reached in the laminar region for the 0.0645 cm tube and there is no apparent onset. Rather, friction factors depart gradually from the laminar equation as A Re increases (see also the section Concentration). Virk proposed that onset of DR in turbulent pipe flows, at which the friction factor departs abruptly from Eq. (4), occurs at a distinct wall-shear stress, which does not depend on the pipe size and is only weakly dependent on concentration for a given polymer-solvent combination, but depends strongly on the radii of gyration, Rq, of the polymer molecules ... 

For DR surfactant solutions with thread-like micelles, Qi proposed a possible DR mechanism in pipe flow as shown in Fig. 16. At rest, thread-like micelles are distributed randomly in the solution. As Arb increases, the thread-like micelles near the wall are extended and start to align along the flow direction because of high wall-shear stress and the solution starts to show DR. Turbulent fluctuations decrease in the radial direction because of the micelle alignment and turbulent energy dissipation is reduced. 

Pipe Flow. The rheological properties of cement slurries have also been characterized using pipe viscometers (13,15-18). The experimental results show different flow patterns. In some circumstances, relatively large pipe diameters and high shear rates, the flow curves, and shear-stress at the wall versus Newtonian shear rate at the wall are independent of pipe diameter (16, 17). The reverse situation is observed when pipe diameters are relatively small, of the order of 5 mm (13, 15). The diameter dependency of the flow curves can be explained, as discussed... 

From Fig. 16.8 we see that in ordinary pipe flow for regions away from the wall the eddy viscosity is typically about 100 times the molecular viscosity (i.e., the Reynolds stresses are about 100 times the stresses due to molecular viscosity), that the eddy viscosity is a strong function of position and Reynolds number, and that it is difficult to calculate values of the eddy viscosity near the center of the pipe. From Eq. 16.15 we see that the sum of the eddy and molecular viscosities is equal to Tl dVJdy) at the center of the pipe both quantities are zero. To obtain the correct limit in this ratio as both numerator and denomnator approach zero requires more precise experimental measurements of and y than are currently available. We may infer from Fig. 16.8 that in this type of pipe flow the heat transfer and mixing will be of the order of 100 times the heat transfer and mixing due to molecular thermal conductivity and molecular diffusion. 

Friction loss or drag is synonymous with the dissipationof energy. For flow in a pipe, the rate of energy dissipation per unit mass of fluid can be expressed in terms of the mean- flow velocity and either the wall stress, pressure drop, or Fanning friction factor, as follows ... 

The boundary constraints used in pipe flow regimes are inlet velocity profile, zero velocity on solid non-slip walls, and stress free (or for long pipes developed flow) exit conditions. In shell and tube systems with solid and porous walls, used in thickening of suspensions by cross-flow filtration, a different set of boundary conditions must be given. These are the inlet velocity profile, zero velocity on outer shell s solid walls, stress-free conditions at the exit, and the following Darcy flow conditions on porous wall ... 

The power-law equation does not hold, as y goes to zero at the wall. Another useful relation is the Blasius correlation for shear stress for pipe flow, which is consistent at the wall for the wall shear stress Tq. For boundary-layer flow over a flat plate, it becomes... 

Blasius conducted tests on turbulent flows in pipes and developed an equation for the shear wall stress in terms of the maximum velocity outside the boundary layer. 

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