Newtonian fluids smooth pipes/turbulent 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. 

There are insnfficient data in the literatnre to provide a reliable estimate of the effect of roughness on friction loss for non-Newtonian flnids in tnrbnlent flow. However, the influence of roughness is normally neglected, since the laminar bonndary layer thickness for such fluids is typically much larger than for Newtonian fluids (i.e., the flow conditions most often fall in the hydraulically smooth range for common pipe materials). An expression by Darby et al. (1992) for / for the power law flnid, which applies to both laminar and turbulent flow, is... 

HARTNETT and KOSTIC 26 have recently examined the published correlations for turbulent flow of shear-thinning power-law fluids in pipes and in non-circular ducts, and have concluded that, for smooth pipes, Dodge and Metzner S(27) modification of equation 3.11 (to which it reduces for Newtonian fluids) is the most satisfactory. 

The opening of the test pipe has been designed and tested to ensure fully developed turbulent flow in the test section. Hence, the inlet length of the pipe to the first pressure gap amounts to 220 x d. The device has been carfully tested and calibrated with water, which complies with the Newtonian theory of fluids for smooth pipes. Measurements are taken after stabilization occurs and then a magnetic device in the form of a swinging arm turns the outlet tube to the collecting vessel and at this moment a stop-watch is... 

For turbulent flow of Newtonian fluids in smooth pipes, two common correlations are those of Blasius [413] for 3000 < Re < 100 000 ... 

A large body of literature is available on estimating friction loss for laminar and turbulent flow of Newtonian and non-Newtonian fluids in smooth pipes. For laminar flow past solid boundaries, surface roughness has no effect (at least for certain degrees of roughness) on the friction pressure drop of either Newtonian or non-Newtonian fluids. In turbulent flow, however, die nature... 

Dodge and Metzner (16) presented an extensive theoretical and experimental study on the turbulent flow of non-Newtonian fluids in smooth pipes. They extended von Karman s (17) work on turbulent flow friction factors to include the power law non-Newtonian fluids. The following implicit expression for the friction factor was derived in terms of the Metzner-Reed modified Reynolds number and the power law index ... 

Universal velocity distribution turbulent flow of newtonian fluid in smooth pipe. 

Figure 5.10 is a friction-factor chart in which / is plotted against for the flow of power-law fluids in smooth pipes. A series of lines depending on the magnitude of n is needed for turbulent flow. For these lines the following equation, andogous to Eq. (5.43), for newtonian fluids, has been suggested ... 

Besides clarifying the strange shape of Fig. 6.2, Reynolds made the most celebrated application of dimensional analysis (Chap. 13) in the history of fluid mechanics. He showed that for smooth, circular pipes, for all newtonian fluids, and for all pipe diameters, the transition from laminar to turbulent flow occurs when the dimensionless group DVpIfjt, has a value of about 2000. Here D is the pipe diameter, V is the average fluid velocity in the pipe, p is the fluid density, and fi is the fluid viscosity. This dimensionless group is now called the Reynolds number For flows other than pipe flow, some other appropriate length is substituted for the pipe diameter in the Reynolds number, as discussed later. 

In a comprehensive study. Dodge and Metzner [1959] carried out a semi-empirical analysis of the fiilly developed turbulent flow of power-law fluids in smooth pipes. They used the same dimensional considerations for snch flnids, as Millikan [1939] for incompressible Newtonian fluids, and obtained an expression which can be re-arranged in terms of the apparent power law index, (eqnation 3.26) as follows ... 

The shear stresses within the fluid are responsible for the frictional force at the walls and the velocity distribution over the cross-section of the pipe. A given assumption for the shear stress at the walls therefore implies some particular velocity distribution. In line with the traditional concepts that have proved of value for Newtonian fluids, the turbulent flow of power-law fluids in smooth pipes can be considered by dividing the flow into three zones, as shown schematically in Figure 3.12. 

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