Fluid Mechanics II 3 credit hour. Fluid flow through pipes-minor losses

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1 COURSE NUMBER: ME 323 Fluid Mechanics II 3 credit hour Fluid flow through pipes-minor losses Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1

2 Losses in Noncircular Conduits Good approximations can be made for the head loss in conduits with noncircular cross sections by using the hydraulic radius R, defined by where A is the cross-sectional area and P is the wetted perimeter, that perimeter where the fluid is in contact with the solid boundary. For a circular pipe flowing full the hydraulic radius is R=r 0 /2. Hence we simply replace the radius r 0 with 2R and use the Moody diagram with The head loss becomes To use this hydraulic radius technique the cross section should be fairly open, such as a rectangle with aspect ratio less than 4:1; an equilateral triangle, or an oval. For other shapes, such as an annulus, the error would be significant. 2

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4 Minor Losses in Pipe Flow Pipe systems do include valves, elbows, contractions, inlets, outlets, bends, and other fittings that cause additional losses, referred to as minor losses. Figure: Bends, elbows and different types of connectors 4

5 Figure: Different types of valves Each of these devices causes a change in the magnitude and/or the direction of the velocity vectors and result in losses, i.e. energy losses. 5

6 In general, if the flow is gradually accelerated by adevice, the losses are very small; relatively large losses are associated with sudden enlargements because of the separated regions (a separated flow occurs when the primary flow separates from the wall). A pipe fitting that has a relatively large loss coefficient with no change in cross-sectional sectional area is the pipe bend, or the elbow. 6

7 This results primarily from the secondary flow caused by the fluid flowing from the high-pressure region to the low-pressure region, as shown in Fig.7.14; this secondary flow is eventually dissipated after the fluid leaves the long sweep bend or elbow. In addition, a separated region occurs at the sharp corner in a standard d elbow. Energy is needed d to maintain such a secondary flow and the flow in the separated region. Thiswasted energy is measured intermsof a loss coefficient. i 7

8 A minor loss is expressed in terms of a loss coefficient K, defined by Values of K have been determined experimentally for the various fittings and geometry changes of interest in piping systems. One exception is the sudden expansion from area A 1 to area A 2, for which the loss can be calculated; this was done in Example 4.17, where we found that Thus, for the sudden expansion If A 2 is extremely large (e.g., a pipe exiting into a reservoir), K =1.0. 8

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13 Loss coefficients It is often the practice to express a loss coefficient as an equivalent length L e of pipe. This is done as follows: Which gives, Hence the square edged entrance of a 20 cm diameter pipe with a friction factor of f = 0.02 could be replaced by an equivalent pipe length of L e =5 m. In piping systems involving intermediate lengths (i.e., 100 diameters) of pipe pp between minor losses, the minor losses may be of thesameorderof magnitude as the frictional losses; for relatively short lengths the minor losses may be substantially greater than the fi frictionali llosses; and for long lengths (e.g., 1000 diameters) of pipe, the minor losses are usually neglected. 13

14 Hydraulic and Energy Grade Lines The hydraulic grade line (HGL), the dashed d line in Fig. 7.17, in apiping i system is formed by the locus of points located a distance p/γ above the center of the pipe, or p/γ +zaboveapreselecteddatum;theliquidina piezometer tube would rise to the HGL. The energy grade line (EGL), the solid line in Fig. 7.17, is formed by the locus of points a distance V 2 /2g above the HGL, or the distance V 2 /2g + p/γ + z above the datum; the liquid in a pitot tube would rise to the EGL. 14

15 The concepts of energy grade line and hydraulic grade line may also be applied to open channel flows. The HGL coincides with the free surface and the EGL is a distance V 2 /2gabovethefreesurface. The following points are noted relating to the HGL and the EGL: As the velocity goes to zero, the HGL and the EGL approach each other. Thus, in a reservoir, they are identical and lie on the surface (see Fig. 7.17). The EGL and, consequently, the HGL slope downward in the direction of the flow due to the head loss in the pipe. The greater the loss per unit length, the greater the slope. As the average velocity in the pipe increases, the loss per unit length increases. A sudden change occurs in the HGL and the EGL whenever a loss occurs due to a sudden geometry change, as represented by the valve or the sudden enlargement of Fig A jump occurs in the HGL and the EGL whenever useful energy is added to the fluid, as occurs with a pump, and a drop occurs if useful energy is extracted from the flow, as occurs with a turbine. 15

16 At points where the HGL passes through the centerline of the pipe, the pressure is zero. If the pipe lies above the HGL, there is a vacuum in the pipe, a condition that is often avoided, if possible, in the design of piping i systems; an exception would be in the design of a siphon. 16

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20 Simple Pipe System with a Pump The head produced by a centrifugal pump and its efficiency η P (see Eq ) dependdon the discharge, as shown by the pump characteristic i curves, the solid curves in Fig Companies furnish such characteristic curves for each centrifugal pump manufactured. Such a curve provides one equation relating the flow rate Q and pump head H P. 20

21 The system demand curve is provided by the energy equation, which can typically be written as This is the system demand curve along ao with the characteristic aa ei i curve must be solved simultaneously to yield the desired flow rate. To determine the power requirement of the pump, p the efficiency η P must be used. Note that for the piping system, the required pump energy head H P, demanded by the energy equation, increases with Q and from the pump characteristic i curve we see that H P decreases with ih Q; hence the two curves will intersect at a point, called the operating point of the system. 21

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28 Example: The flow rate is measured to be m 3 /s in the pipe shown. Find the loss coefficient of the valve if H = 4 cm. Ans: Example: Estimate t the flow rate to be expected throughh the plastic siphon shown if the diameter is 8 cm. Ans: m 3 /s 28