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 146P
To determine

The velocity over the bump.

The flow depth over the bump.

The Froude number over the bump.

Expert Solution & Answer
Check Mark

Answer to Problem 146P

The velocity over the bump is 1.42794m/s.

The flow depth over the bump is 1.5757m.

The Froude number over the bump is 0.3632.

Explanation of Solution

Given information:

The height of bump is 20cm, initial velocity of flow is 1.25m/s, the flow depth is 1.8m.

Write the expression for Froude number.

  Fr=Vgy...... (I)

Here, velocity is V, acceleration due to gravity is g, flow depth is y.

Write the expression for critical flow of depth.

  yc=[ y 1 2 V 1 2g]1/3...... (II)

Here, velocity before bump is V1, depth before bump is y1, acceleration due to gravity is g.

Write the expression for specific energy before bump.

  Es1=y1+V122g...... (III)

Write the expression for specific energy after the bump.

  Es2=Es1Δzb...... (IV)

Here, height of bump is Δzb.

Write the expression for critical specific energy.

  Ec=32yc...... (V)

Write the relation for flow depth after the bump.

  y23(Es1Δzb)y22+V122gy12=0...... (VI)

Write the expression for velocity after the bump.

  V2=y1y2V1...... (VII)

Calculation:

Substitute Fr1 for Fr, 1.25m/s for V1, 9.81m/s2 for g and 1.8m for y1 in Equation (I)

  Fr1=1.25m/s 9.81m/ s 2 ×1.8m=1.25m/s 17.658 m 2 / s 2 =0.2975

The Froude number is less than 1, so flow is sub-critical and surface level over bump drops.

Substitute 1.25m/s for V1, 9.81m/s2 for g and 1.8m for y1 in Equation (II).

  yc=[ ( 1.8m ) 2 ( 1.25m/s ) 2 9.81m/ s 2 ]1/3=[ 5.0625 m 4 / s 2 9.81m/ s 2 ]1/3=0.8023m

Substitute 1.25m/s for V1, 9.81m/s2 for g and 1.8m for y1 in Equation (III).

  Es1=1.8m+ ( 1.25m/s )22×9.81m/ s 2=1.8m+0.07963m=1.8796m

Substitute 1.8796m for Es1 and 20cm for Δzb in Equation (IV).

  Es2=1.8796m20cm=1.8796m20cm×1m100cm=1.8796m0.2m=1.6796m

Substitute 0.8023m for yc in Equation (V).

  Ec=32×0.8023m=1.20345m

Substitute 1.8796m for Es1, 20cm for Δzb, 1.25m/s for V1, 9.81m/s2 for g and 1.8m for y1 in Equation (VI).

  y23(1.8796m0.20m)y22+ ( 1.25m/s )22×9.81m/ s 2(1.8m)2=0y23(1.6796)y22+0.258=0

Solve for y2.

  y2=1.5757m

  y2=0.4599m

  y2=0.356m

Here, depth of flow should be greater than critical depth.

  y2=1.5757m

Substitute 1.25m/s for V1, 1.5757m for y2 and 1.8m for y1 in Equation (VII).

  V2=1.8m1.5757m×1.25m/s=2.83626×1.25m/s=1.42794m/s

Substitute Fr2 for Fr, 1.42794m/s for V2, 9.81m/s2 for g and 1.5757m for y2 in Equation (I)

  Fr2=1.42794m/s 9.81m/ s 2 ×1.5757m=1.42794m/s 15.4576 m 2 / s 2 =0.3632

Conclusion:

The velocity over the bump is 1.42794m/s.

The flow depth over the bump is 1.5757m.

The Froude number over the bump is 0.3632.

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Chapter 13 Solutions

Fluid Mechanics: Fundamentals and Applications

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