Fluid Mechanics
Fluid Mechanics
8th Edition
ISBN: 9780073398273
Author: Frank M. White
Publisher: McGraw-Hill Education
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Chapter 10, Problem 10.6CP
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To determine

(a)

To compute:

The flow rate follows from the sluice gate correlation Q.

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Expert Solution
Check Mark

Answer to Problem 10.6CP

The flow rate follows from the sluice gate correlation Q=379ft3/s.

Explanation of Solution

Figure:

Fluid Mechanics, Chapter 10, Problem 10.6CP

Given:

b=8ft, H=3ft, y1=12ft, g=32.2ft/s2.

Concept Used:

Annuity problem requires the use of the formula as follows:

Q=CdHb2gy1

Cd=0.611+0.61(H1y1)

Here,

Acceleration gravity =g, height =H, width =b.

Calculation:

As per the given problem:

b=8ft, H=3ft

Annuity problem requires the use of this formula:

Cd=0.611+0.61(H1y1)

Substitute these values in the formula

Cd=0.611+0.61(3ft12ft)=0.568

Cd=0.568

Annuity problem requires the use of this formula:

Cd=0.568, b=8ft, H=3ft, y1=12ft, g=32.2ft/s2

Q=CdHb2gy1

Substitute these values in the formula:

Q=(0.568)(3ft)(8ft)2(32.2ft/s2)(12ft)=379ft3/s

Q=379ft3/s

Conclusion:

The flow rate follows from the sluice gate correlation Q=379ft3/s..

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To determine

(b)

To compute:

The normal depth.

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Expert Solution
Check Mark

Answer to Problem 10.6CP

It has no normal depth.

Explanation of Solution

Given Information:

This channel is horizontal.

If you know the flow rate, you can find the normal depth, but this channel is horizontal, It has no normal depth.

Conclusion:

It has no normal depth.

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To determine

(c)

To compute:

(y2).

y2=1.83ft

Given:

Q=379ft3/s, b=8ft, g=32.2ft/s2, y1=12ft

Concept Used:

Annuity problem requires the use of the formula as follows:

E1=y1+V122g

E2=y2+V222g

Annuity problem requires the use of the formula as follows:

Qb=V1y1

Qb=V2y2

Here,

Velocity =V, Depth =y, Flow rate =Q

Annuity problem requires the use of the formula as follows

Fr2=V2gy2

Calculation:

As per the given problem:

Q=379ft3/s, b=8ft, g=32.2ft/s2, y1=12ft

Annuity problem requires the use of this formula:

Qb=V1y1

Substitute these values in the formula:

Qb=379ft38ft=47.39ft2/s

47.39ft2/s=V1y1

V1=47.39ft2/s(12ft)=3.95ft/s

V1=3.95ft/s

Annuity problem requires the use of this formula:

V1=3.95ft/s, y1=12ft, g=32.2ft/s2

E1=y1+V122g

Substitute these values in the formula:

E1=12ft+(3.95ft/s)22(32.2ft/s2)=12.24ft

E1=12.24ft

According to Bernoulli equation:

E1=E2

E2=12.24ft

Annuity problem requires the use of this formula:

Qb=47.39ft2/s, E2=12.24ft, g=32.2ft/s2.

Qb=V2y2.

Substitute these values in the formula:

V2=47.39ft2/sy2

Annuity problem requires the use of this formula:

E2=y2+V222g

Substitute these values in the formula:

12.24ft=y2+(47.39ft2/sy2)22(32.2ft/s2)

y2=1.83ft

Conclusion:

y2=1.83ft.

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Expert Solution
Check Mark

Answer to Problem 10.6CP

y2=1.83ft

Explanation of Solution

Given:

Q=379ft3/s, b=8ft, g=32.2ft/s2, y1=12ft

Concept Used:

Annuity problem requires the use of the formula as follows:

E1=y1+V122g

E2=y2+V222g

Annuity problem requires the use of the formula as follows:

Qb=V1y1

Qb=V2y2

Here,

Velocity =V, Depth =y, Flow rate =Q

Annuity problem requires the use of the formula as follows

Fr2=V2gy2

Calculation:

As per the given problem:

Q=379ft3/s, b=8ft, g=32.2ft/s2, y1=12ft

Annuity problem requires the use of this formula:

Qb=V1y1

Substitute these values in the formula:

Qb=379ft38ft=47.39ft2/s

47.39ft2/s=V1y1

V1=47.39ft2/s(12ft)=3.95ft/s

V1=3.95ft/s

Annuity problem requires the use of this formula:

V1=3.95ft/s, y1=12ft, g=32.2ft/s2

E1=y1+V122g

Substitute these values in the formula:

E1=12ft+(3.95ft/s)22(32.2ft/s2)=12.24ft

E1=12.24ft

According to Bernoulli equation:

E1=E2

E2=12.24ft

Annuity problem requires the use of this formula:

Qb=47.39ft2/s, E2=12.24ft, g=32.2ft/s2.

Qb=V2y2.

Substitute these values in the formula:

V2=47.39ft2/sy2

Annuity problem requires the use of this formula:

E2=y2+V222g

Substitute these values in the formula:

12.24ft=y2+(47.39ft2/sy2)22(32.2ft/s2)

y2=1.83ft

Conclusion:

y2=1.83ft.

<
To determine

(d)

To compute:

(y3).

y3=7.86ft

Given:

y2=1.83ft, V2=47.39ft2/sy2

Calculation:

Annuity problem requires the use of this formula:

V2=47.39ft2/sy2

Substitute these values in the formula:

V2=47.39ft2/s1.83ft=25.9ft/s

V2=25.9ft/s

Annuity problem requires the use of this formula:

V2=25.9ft/s, g=32.2ft/s2, y2=1.83ft

Fr2=V2gy2

Substitute these values in the formula:

Fr2=25.9(32.2)1.83=3.37

Fr2=3.37

The flow in section 2 is highly supercritical, Now use hydraulic jump theory

Annuity problem requires the use hydraulic jump theory as follow:

Fr2=3.37, y2=1.83ft

2y3y2=1+1+8Fr22

Substitute these values in the formula:

2y3(1.83)=1+1+8(3.37)2

y3=7.86ft

Conclusion:

y3=7.86ft.

<
Expert Solution
Check Mark

Answer to Problem 10.6CP

y3=7.86ft

Explanation of Solution

Given:

y2=1.83ft, V2=47.39ft2/sy2

Calculation:

Annuity problem requires the use of this formula:

V2=47.39ft2/sy2

Substitute these values in the formula:

V2=47.39ft2/s1.83ft=25.9ft/s

V2=25.9ft/s

Annuity problem requires the use of this formula:

V2=25.9ft/s, g=32.2ft/s2, y2=1.83ft

Fr2=V2gy2

Substitute these values in the formula:

Fr2=25.9(32.2)1.83=3.37

Fr2=3.37

The flow in section 2 is highly supercritical, Now use hydraulic jump theory

Annuity problem requires the use hydraulic jump theory as follow:

Fr2=3.37, y2=1.83ft

2y3y2=1+1+8Fr22

Substitute these values in the formula:

2y3(1.83)=1+1+8(3.37)2

y3=7.86ft

Conclusion:

y3=7.86ft.

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To determine

(e)

To compute:

(y4)..

(y4)=11.53ft

Given:

Calculation:

yweir=6ft, Q=379ft3/s, g=32.2ft/s2 s Annuity problem requires the use of this formula:

Cd,shapecrested=0.564+0.0846H4Yweir

Substitute these values in the formula:

Cd,shapecrested=0.564+0.0846H46ft

Annuity problem requires the use of this formula:

Q=379ft3/s, Cd,shapecrested=0.564+0.0846H46ft, g=32.2ft/s2

Q=CdbgH43/2

Substitute these values in the formula:

Q=(0.564+0.0846H46ft)(8ft)(32.2ft/s2)H43/2

379ft3/s=(0.564+0.0846H46ft)(8ft)(32.2ft/s2)H43/2

H4=5.53ft

Annuity problem requires the use of this formula:

H4=5.53ft

Cd,shapecrested=0.564+0.0846H46ft

Substitute these values in the formula

Cd,shapecrested=0.564+0.08465.536ft=0.642

Cd=0.642

Annuity problem requires the use of this formula

(y4)=H4+Yweir

Substitute these values in the formula:

(y4)=5.53+6ft=11.53ft

(y4)=11.53ft

Conclusion:

(y4)=11.53ft.

<
Expert Solution
Check Mark

Answer to Problem 10.6CP

(y4)=11.53ft

Explanation of Solution

Given:

Calculation:

yweir=6ft, Q=379ft3/s, g=32.2ft/s2 s Annuity problem requires the use of this formula:

Cd,shapecrested=0.564+0.0846H4Yweir

Substitute these values in the formula:

Cd,shapecrested=0.564+0.0846H46ft

Annuity problem requires the use of this formula:

Q=379ft3/s, Cd,shapecrested=0.564+0.0846H46ft, g=32.2ft/s2

Q=CdbgH43/2

Substitute these values in the formula:

Q=(0.564+0.0846H46ft)(8ft)(32.2ft/s2)H43/2

379ft3/s=(0.564+0.0846H46ft)(8ft)(32.2ft/s2)H43/2

H4=5.53ft

Annuity problem requires the use of this formula:

H4=5.53ft

Cd,shapecrested=0.564+0.0846H46ft

Substitute these values in the formula

Cd,shapecrested=0.564+0.08465.536ft=0.642

Cd=0.642

Annuity problem requires the use of this formula

(y4)=H4+Yweir

Substitute these values in the formula:

(y4)=5.53+6ft=11.53ft

(y4)=11.53ft

Conclusion:

(y4)=11.53ft.

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