Circular cross-section pipes, frictional

We now have to thank Stanton and PanneU, and also Moody for their studies of flow using numerous fluids in pipes of various diameters and surface roughness and for the evolution of a very useful chart (see Fig. 48.6). This chart enables us to calculate the frictional pressure loss in a variety of circular cross-section pipes. The chart plots Re)molds numbers (Re), in terms of two more dimensionless groups a friction factor < ), which represents the resistance to flow per unit area of pipe surface with respect to fluid density and velocity and a roughness factor e/ID, which represents the length or height of surface prelections relative to pipe diameter. 

Another simple relationship was developed by Blasius for flow in smooth pipes at Reynolds numbers between 2.5 x 10 and 1 x 10. In this region the friction factor (]) for smooth circular cross-section pipes approximates to... 

Skin friction loss. Skin friction loss is the loss from the shear forces on the impeller wall caused by turbulent friction. This loss is determined by considering the flow as an equivalent circular cross section with a hydraulic diameter. The loss is then computed based on well-known pipe flow pressure loss equations. 

So far, only the frictional pressure drop in straight lengths of pipe of circular cross-section has been discussed. The pressure drop in pipelines containing valves and fittings can be calculated from equation 2.13 but with fittings represented by the length of plain pipe that causes the same pressure drop. 

FRICTION FACTOR IN FLOW THROUGH CHANNELS OF NONQRCULAR CROSS SECTION. The friction in long straight channels of constant noncircular cross section can be estimated by using the equations for circular pipes if the diameter in the Reynolds number and in the definition of the friction factor is taken as an equivalent diameter, defined as four times the hydraulic radius. The hydraulic radius is denoted by r and in turn is defined as the ratio of the cross-sectional area of the channel to the wetted perimeter of the channel ... 

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. 

All the relationships presented in Chapter 6 apply directly to circular pipe. However, many of these results can also, with appropriate modification, be applied to conduits with noncircular cross sections. It should be recalled that the derivation of the momentum equation for uniform flow in a tube [e.g., Eq. (5-44)] involved no assumption about the shape of the tube cross section. The result is that the friction loss is a function of a geometric parameter called the hydraulic diameter ... 

Noncircular Channels Calculation of frictional pressure drop in noncircular channels depends on whether the flow is laminar or turbulent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter DH should be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraulic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraulic diameter for a circular pipe is DH = D, for an annulus of inner diameter d and outer diameter D,DH = D-d, for a rectangular duct of sides a, b, DH=ab/[2(a+b)]. The hydraulic radius Rh is defined as one-fourth of the hydraulic diameter. 

For turbulent flow in a conduit of noncircular cross section, an equivalent diameter can be substituted for the circular-section diameter, and the equations for circular pipes can then be applied without introducing a large error. This equivalent diameter is defined as four times the hydraulic radius RH, where the hydraulic radius is the ratio of the cross-sectional flow area to the wetted perimeter. When the flow is viscous, substitution of 4RH for D does not give accurate results, and exact expressions relating frictional pressure drop and velocity can be obtained only for certain conduit shapes. 

This result can be used to apply the previous equations for circular pipes to conduits of any other shape, by replacing D in the appropriate equation with for the noncircular conduit. This gives excellent results for turbulent flows, for which the boundary layer is generally thin relative to the flow area dimensions, since the wall resistance (i.e., friction loss) is confined to a region very near the wall and, consequently, is not very sensitive to the shape of the cross section. 

As shown in Fig. 5.11, the orifice meter consists of a flat orifice plate with a circular hole drilled in it. There is a pressure tap upstream from the orifice plate and another just downstream. If the flow direction is horizontal and we apply Bernoulli s equation, ignoring friction from point 1 to point 2, we find Eq. 5.30, exactly the same equation we found for a venturi meter. However, in this case we cannot assume frictionless flow and uniform flow across any cross section of the pipe as easily as we can in the case of the venturi meter. 

In Fig. 3.1-5 the friction factor X for pipes with a siuface roughness k (k is the mean height of protuberances) is plotted against the Reynolds number. The equations presented here in combination with the diagram are general tools to calculate the pressure drop Ap in circular tubes with constant cross-sectional area. 

Whereof and Reef are constriction based friction factor and Reynolds number respectively, and Po is the Poiseuille number specific to the cross-sectional geometry of the channel, 16 for circular pipes, 14.23 for a square channel and 24 for flow between two parallel plates. 

This estimates the mean friction exerted by the flow on the wall. The value coincides with the local value at all points on the wall when the cross-section of the pipe is circular, or in the case of a plane flow. On the other hand, friction is inhomogeneous when the profile of the pipe has angles, in which case relation [4.8] estimates the mean friction over the perimeter of the pipe. 

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