Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 13, Problem 142P
To determine

The flow rate of water through the gate per meter width.

The flow depth y1 and y2.

The energy dissipation ratio of the jump.

Expert Solution & Answer
Check Mark

Answer to Problem 142P

The flow rate of water through the gate per meter width is 12m3/s.

The flow depth y1 and y2 are 9.6m and 1.969m.

The energy dissipation ratio of the jump is 0.012.

Explanation of Solution

Given information:

Water is flowing through a sluice gate, the velocity of water before reaching the gate is 1.25m/s, the velocity of water after the jump is 4m/s, the flow depth at (3) is 3m.

The flow is considered per meter width so b=1m.

The figure below shows the flow.

  Fluid Mechanics: Fundamentals and Applications, Chapter 13, Problem 142P

  Figure-(1)

Write the expression for the cross-sectional area at (3).

  Ac,3=by3  ...... (I)

Here, the width is b, the flow depth at (3) is y3.

Write the expression for the cross-sectional area at (3).

  V˙=V3Ac,3  ...... (II)

Here, the velocity at section (3) is V3 and area at section (3) is Ac,3.

Write the expression for the conservation of mass.

  V1y1=V3y3y1=V3V1y3  ...... (III)

Here, the velocity at section (3) is V3, the velocity at section

Write the expression for the Froude number at section (3).

  Fr,3=V3gy3  ...... (IV)

Here, the acceleration due to gravity is g.

Write the expression for the flow depth.

  y2=0.5y3(1+1+8Fr32)  ...... (V)

Write the expression for the conservation of mass.

  V2y2=V3y3V2=V3y3y2 (VI)

The flow depth at section (2) is y2.

Write the expression for the Froude number at section (2).

  Fr,2=V2gy2  ...... (VII)

Write the expression for the head loss during jumping.

  hL=y2y3+V22V322g  ...... (VIII)

Here, the velocity at section (2) is V2 and the velocity at section (3) is V3.

Write the expression for the specific energy before the jump.

  Es2=y2+V222g  ...... (IX)

Write the expression for the dissipation ratio.

  Ds=hLEs2....... (X)

Calculation:

Substitute 1m for b and 3m for y3 in Equation (I).

  Ac3=1m×3m=3m2

Substitute 4m/s for V3, 3m2 for Ac3 in Equation (II).

  V˙=4m/s×3m2=12m3/s

Substitute 4m/s for V3, 1.25m/s for V1 in Equation (III).

  y1=4m/s1.25m/s(3m)=3.2(3m)=9.6m

Substitute 4m/s for V3

  9.81m/s2 for g and 3m for y3 in Equation (IV).

  Fr3=4m/s 9.81m/ s 2 ×3m=4m/s5.425m/s=0.7373

Substitute 3m for y3

  0.7373 for Fr,3 in Equation (V).

  y2=0.5(3m)(1+ 1+8 ( 0.7373 ) 2 )=1.5m(1.3127)=1.969m

Substitute 3m for y3, 1.969m for y2 and 4m/s for V3 in Equation (VI).

  V2=3m1.969m(4m/s)=1.5236(4m/s)=6.094m/s

Substitute 6.094m/s for V2, 9.81m/s2 for g and 1.969m for y2 in Equation (VII).

  Fr,2=6.094m/s ( 9.81 m/s 2 )( 1.969m )=6.094m/s4.39498m/s=1.387

Here Froude number is greater than 1 thus the flow before jump is super critical.

Substitute 1.969m for y2, 3m for y3

  6.094m/s for V2, 4m/s for V3, 9.81m/s2 for g in Equation (VIII).

  hL=1.969m3m+ ( 6.9094m/s )2 ( 4m/s )22( 9.81m/ s 2 )=1.969m3m+1.6177m=0.0463m

Substitute 1.9639m for y2, 6.094m/s for V2 and 9.81m/s2 for g in Equation (IX).

  Es2=1.9639m+ ( 6.094m/s )22×( 9.81m/ s 2 )=1.9639m+1.892m=3.862m

Substitute 0.0463m for hL and 3.862m for Es,2 in Equation (X).

  Ds=0.0463m3.862m=0.012m

Conclusion:

The flow rate of water through the gate per meter width is 12m3/s.

The flow depth y1 and y2 are 9.6m and 1.969m.

The energy dissipation ratio of the jump is 0.012.

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Chapter 13 Solutions

Fluid Mechanics: Fundamentals and Applications

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