upstream of the pump under the assumption that this segment is short hence has

# Upstream of the pump under the assumption that this

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upstream of the pump, under the assumption that this segment is short (hence, has a small L/D ). But this could easily be done with an additional application of the Bernoulli equation, another useful exercise for the reader. Head Loss Modification for Non-Circular Cross Sections The Darcy–Weisbach formula, Eq. (4.67), holds only for pipes having circular cross section. But in engineering practice we often must deal with flow in square and rectangular ducts, sometimes even ducts with triangular cross sections; in fact, situations may arise in which geometry of the duct cross section may be very complicated. In this section we introduce a very straightforward approach to approximately handle these more difficult geometries. We start by recalling the Darcy–Weisbach formula: h f = f L D U 2 avg 2 g . The modification we introduce here involves simply replacing the pipe diameter D with an “equiv- alent diameter,” termed the hydraulic diameter and denoted D h , that at least partially accounts for effects arising from the non-circular cross section. The hydraulic diameter is defined as D h 4 A P , (4.71) where A is the cross-sectional area, and P is the “wetted perimeter” (around the cross section) of the duct. For example, for a duct having rectangular cross section with height h and width w , the hydraulic diameter is D h = 4 wh 2( w + h ) . Thus, to estimate the head loss for a duct of essentially arbitrary cross-sectional geometry, we use Eq. (4.71) to calculate the hydraulic diameter. Then we use this diameter in the Darcy– Weisbach formula, and in addition for calculating Re and surface roughness factor needed to find the friction factor f . That is, we use Re h = ρUD h μ and ε D h . This approach works reasonably well for cross sections that do not deviate too much from circular; one can easily show that for a circular cross section D h = D . But as the geometry departs significantly from circular, use of D h as described leads to large errors. In particular, additional flow physics can arise even in not very complicated geometries if, e.g. , sharp corners are present. We will see some effects of this sort in the next subsection, but there they will be quantified only as they occur with respect to the streamwise direction. In the present context they may also influence flow behavior in the cross stream direction, and this simply is not being taken into account by analyses such as discussed here.
148 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS Minor Losses We begin this section by again reminding the reader that all of the preceding analyses arose either directly, or indirectly through necessary empiricism, from the Hagen–Poiseuille solution to the N.–S. equations. In particular, application of the preceding formulas requires an assumption of fully-developed flow. In the present section we will treat situations in which the flow cannot possibly be fully developed, and introduce formulas for additional head loss arising in such cases.

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