Pipe flow noncircular conduits

The cross section of the fluid in the partially full pipe will not be circular (see Fig. 7-6), so the methods used for flow in a noncircular conduit are applicable, i.e., the hydraulic diameter applies. Thus, Eq. (7-61) becomes... 

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. 

We are no more able to calculate the pressure drop in steady, turbulent flow in a noncircular conduit than we are in a circular one. However, it seems reasonable to expect that we could use the friction-loss results for circular pipes to estimate the results for other shapes. Let us assume that the shear stress at the wall of any conduit is the same for a given average fluid flow velocity independent of the shape of the conduit. Then, from a force balance on a horizontal section like that leading to Eq. 6.3, we conclude that in steady flow... 

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 ... 

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. 

The situation is often encountered in which a fluid flows through a conduit having a noncircular cross section, such as an annulus. The heat-transfer coefficients for turbulent flow can be determined by using the same equations that apply to pipes and tubes if the pipe diameter D appearing in these equations is replaced by an equivalent diameter De. Best results are obtained if... 

It will be our purpose in this chapter to develop those techniques that are needed to determine F/,. We will consider both laminar and turbulent flows, conduits with circular and noncircular cross sections, complex piping situations, expansions, and contractions. The overall result will be the ability to handle effectively many of the situations that can and do confront the engineering practitioner. 

---