Turbulent flow smooth pipes, differences

The constants in this relation will be different for different critical Reynolds numbers. Also, the surfaces are assumed to be smooth, and the free stream to be turbulent free. For laminar flow, the friction coefficient depends on only the Reynolds number, and the surface roughness has no effect. For turbulent flow, however, surface roughness causes the friction coefficient to increase sevcralfold, to the point that in fully turbulent regime the friction coefficient is a function of surface roughness alone, and independent of the Reynolds number (Fig. 7-8). Tliis is also the case in pipe flow. 

A typical velocity distribution for a newtonian fluid moving in turbulent flow in a smooth pipe at a Reynolds number of 10,000 is shown in Fig. 5.3. The figure also shows the velocity distribution for laminar flow at the same maximum velocity at the center of the pipe. The curve for turbulent flow is clearly much flatter than that for laminar flow, and the difference between the average velocity and the maximum velocity is considerably less. At still higher Reynolds numbers the curve for turbulent flow would be even flatter than that in Fig. 5.3. 

The other curved lines in the turbulent-flow range represent the friction factors for various types of commercial pipe, each of which is characterized by a different value of k. The parameters for several common metals are given in the figure, Clean wrought-iron or steel pipe, for example, has a k value of 1.5 x 10 ft, regardless of the diameter of the pipe. Drawn copper and brass pipe may be considered hydraulically smooth. 

However, the curve of the sphere drag coefficient has some marked differences from the friction factor plot. It does not continue smoothly to higher and higher Reynolds numbers, as does the / curve instead, it takes a sharp drop at an of about 300,000. Also it does not show the upward jump that characterizes the laminar-turbulent transition in pipe flow. Both differences are due to the different shapes of the two systems. In a pipe all the fluid is in a confined area, and the change from laminar to turbulent flow affects all the fluid (except for a very thin film at the wall). Around a sphere the fluid extends in all directions to infinity (actually the fluid is not infinite, but if the distance to the nearest obstruction is 100 sphere diameters, we may consider it so), and no matter how fast the sphere is moving relative to the fluid, the entire fluid cannot be set in turbulent flow by the sphere. Thus, there cannot be the sudden laminar-turbulent transition for the entire flow, which causes the jump in Fig. 6.10. The flow very near the sphere, however, can make the sudden switch, and the switch is the cause of the sudden drop in Q at =300,(300. This sudden drop in drag coefficient is discussed in Sec. 11.6. Leaving until Chaps. 10 and 11 the reasons why the curves in Fig. 6.22 have the shapes they do, for now we simply accept the curves as correct representations of experimental facts and show how to use them to solve various problems. 

Thus plotting IIi as a function of 112, the Reynolds number, produces smooth curves with parametric lines described by 113/2, which is shown in Fig. 4.1. Fig. 4.1 shows Hi as a function of Reynolds number for a variety of spherical particles in a number of different diameter pipes. d(s) identifies sphere diameter. The curves are distinguished by the parametric 113/2, which is LchJD. Fig. 4.1 shows that IIi collapses to a common horizontal line in the turbulent flow regime, that is, at high Reynolds numbers. This horizontal line corresponds to the Burke—Plummer result. For low Reynolds numbers, the correlation for each size sphere is negatively sloped, which corresponds to the Blake—Kozeny result. However, unlike the Ergun equation, the different sized spheres each produce a different correlation in the laminar flow... 

For power law fluids, Tomlta (1959) extended his laminar flow model (discussed in Section 5.2.1.3) to turbulent flows in smooth pipes by applying Prandtl s mixing length concept, and developed a different implicit equation ... 

For turbulent flow, numerous correlations exist for both smooth and rough-walled pipes. A number of charts have been prepared such as those by Moody, and by Stanton and Pannell, in which friction factor is correlated against Reynolds number for differing pipe surface roughness. Itisimportantto note thatthisFanningfrictionfactorhasavalueof one-quarter of the Darcy friction factor. 

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