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EGCE-435: Hydraulic Structures

Lecture 3

Phoolendra Mishra

September 9, 2014

Figures, tables and other contents in this material are freely borrowed from various sources solely forclassroom illustration purposes. Please do not redistribute or reproduce beyond class use.

Contents

3 Stability of bodies and pipe flow 23.1 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Stability of bodies in fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Flow in pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Minor losses in pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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3 Stability of bodies and pipe flow

3.1 Review

• Hydrostatic Forces on curved surface

• Projected areas in two planes

• Horizontal forces

• Vertical forces due to weight of water

• Metacenter and Stability of bodies

3.2 Stability of bodies in fluid

Example 3.1. A solid brass sphere of 30 cm diameter is used to hold a cylindrical buoy in place ( Figure 1)in seawater (s = 1.03). The buoy (S = 0.45) has a height of 2 m and is tied to the sphere at one end. Whatrise in tide, h, will be required to lift the sphere off the bottom?

Figure 1: Buoy in seawater

Example 3.2. The pontoon (15 ft(L) × 9 ft(W ) × 4 ft(H)) is built of uniform material γ = 45 lb/ft2.

(a) How much of it is submerged when floating in water ?

(b) If it is tilted by 12◦ as shown in figure2, what will be the moment of the righting couple?

2

Figure 2: Rectangular Pontoon

• GM = MB ±GB = IoV ol ±GB

• Righting Moment = WGM sin(θ)

Example 3.3. Subway tunnel is being constructed across the bottom of a harbor. The process involvestugboats that pull floating cylindrical sections (or tubes as they are often called) across the harbor and sinkthem in place, where they are welded to the adjacent section already on the harbor bottom. The cylindricaltubes are 50 ft long with a diameter of 36 ft. When in place for the tugboats, the tubes are submergedvertically to a depth of 42 ft, and of the tube is 8 ft above the water s = 1.02. To accomplish this, the tubesare flooded with 34 ft of water on the inside. Determine the metacentric height and estimate the rightingmoment when the tubes are tipped through a heel (list) angle of 4◦ by the tugboats. ( Hint: Assume thelocation of the center of gravity can be determined based on the water contained inside the tubes and thecontainer weight is not that significant.)

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3.3 Flow in pipes

• Continuity equation : Mass conservation

• Energy equations : E = pγ + z + αV

2

2g

• Momentum equation :∑Fi = Mout −Min

• E1 = E2 + hL

• In general hf = KQm (Page 71, Table 3.4)

• Darcy-Wiesbach: hf = fLV 2

2gD ; f = 64Re

(Laminar); Moody’s chart or Colebrook equation (Turbulent)

• For turbulent flow (Swami & Jain 1976) : f = 0.25[log

(e/D3.7 + 5.74

N0.9r

)]2

• Minor loss : hL = K V 2

2g

Figure 3: Moody’s Chart

Example 3.4. The elevated water tank shown in Figure 5 is being drained to an underground storage locationthrough a 10 in diameter pipe. The flow rate is 4488 gallons per minutes (gpm), and the total head loss is9.85 ft. Determine the water surface elevation in the tank.

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Figure 4: Roughness heights e for common pipe materials

Figure 5: Flow from tank

3.4 Minor losses in pipes

• Entrance loss : KeV 2

2g

• Sudden Contraction : KcV 22

2g

• Pipe Confusors (gradual change) : K ′cV 22

2g

• Sudden Expansion : (V1−V2)2

2g

• Pipe diffusor : K ′E(V1−V2)

2

2g

• Exit loss : special case of pipe expansion K ′E = 1 and V2 ≈ 0

• Pipe Bends : KbV 2

2g ; Pipe Valves :KvV 2

2g

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Figure 6: Coefficients for pipe entrances

Figure 7: Sudden contraction coefficients in pipes

Figure 8: Coefficients for pipe Confusors

Figure 9: Sudden expansion coefficients in pipes

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Figure 10: Coefficients for pipe bends

Figure 11: Coefficients for pipe valves

Figure 12: Coefficients for pipe valves (continued)

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