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Unformatted text preview: Example 101 Water flows under a sluice gate, as shown. The downstream velocity is assumed to be uniform across the crosssection and the upstream velocity is assumed to be negligibly small. Find the velocity V 2 by using the top streamline from 1 to 2. Show that this is equal to velocity V 4 by using the bottom streamline from 3 to 4. Calculate the value of V 2 in ft/s and the volumetric flow rate per foot of sluice gate width in ft3/s. Solution: The Bernoulli equation along the top streamline gives
P 1 P2 V 22 V 12 + + gz1 = + + gz2 ρ 2 ρ 2 Patm V 22 V 12 + + gH1 = + + gH 2 ρ 2 ρ 2 Patm Since V 1 ≈ 0
V 22 gH1 = + gH 2 2 V 2 = 2 g ( H1 − H 2 )
The Bernoulli equation along the bottom streamline gives
P3 V 32 P4 V 42 + + gz3 = + + gz4 ρ 2 ρ 2 Patm + ρ gH1 ρ V 32 Patm + ρ gH 2 V 42 + + gz3 = + + gz4 2 2 ρ Since V 3 ≈ 0 and z3 = z4
V 42 gH1 = + gH 2 2 V 4 = 2 g ( H1 − H 2 ) = V
⎛ ft ⎞ ⎛ 2 ft ⎟ V 2 = 2 ⎜ 32.2 2 ⎟ ⎜ 1.75ft − s ⎠⎝ 12 ⎠ ⎝ 2.5 ⎞ V 2 = 9.96 ft s ft ⎞ ⎛ 2.5 ⎞ ⎛ V = V 2 A = ⎜ 9.96 ⎟ ⎜ ft ⎟ (1ft ) s ⎠ ⎝ 12 ⎠ ⎝
ft 3 V = 2.08 s ...
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 Spring '08
 BORCATASCIUC
 Fluid Dynamics, Gate, Mass flow rate, sluice gate, gh1

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