Fluid Mechanics: Fundamentals and Applications

Yunus Cengel

Chapter 13 OPEN-CHANNEL FLOW - all with Video Answers

Educators

Chapter Questions

Problem 1

How does open-channel flow differ from internal flow?

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Problem 2

What is the driving force for flow in an open channel? How is the flow rate in an open channel established?

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Problem 3

How does the pressure change along the free surface in an open-channel flow?

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Consider steady fully developed flow in an open channel of rectangular cross section with a constant slope of $5^{\circ}$ for the bottom surface. Will the slope of the free surface also be $5^{\circ}$ ? Explain.

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Problem 5

How does uniform flow differ from nonuniform flow in open channels? In what kind of channels is uniform flow observed?

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Problem 6

What is normal depth? Explain how it is established in open channels.

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Problem 7

What causes the flow in an open channel to be varied (or nonuniform)? How does rapidly varied flow differ from gradually varied flow?

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Problem 8

In open channels, how is hydraulic radius defined? Knowing the hydraulic radius, how can the hydraulic diameter of the channel be determined?

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Problem 9

Given the average flow velocity and the flow depth, explain how you would determine if the flow in open channels is tranquil, critical, or rapid.

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Problem 10

What is the Froude number? How is it defined? What is its physical significance?

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Problem 11

What is critical depth in open-channel flow? For a given average flow velocity, how is it determined?

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Problem 12

The flow in an open channel is observed to have undergone a hydraulic jump. Is the flow upstream from the jump necessarily supercritical? Is the flow downstream from the jump necessarily subcritical?

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Problem 13

Consider the flow of water in a wide channel. Determine the speed of a small disturbance in the flow if the flow depth is (a) 10 cm and (b) 80 cm . What would your answer be if the fluid were oil?

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Problem 14

Water at $20^{\circ} \mathrm{C}$ is flowing uniformly in a wide rectangular channel at an average velocity of $2 \mathrm{~m} / \mathrm{s}$. If the water depth is 0.2 m , determine (a) whether the flow is laminar or turbulent and (b) whether the flow is subcritical or supercritical.

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Problem 15

Water at $20^{\circ} \mathrm{C}$ flows in a partially full 2 -m-diameter circular channel at an average velocity of $2 \mathrm{~m} / \mathrm{s}$. If the maximum water depth is 0.5 m , determine the hydraulic radius, the Reynolds number, and the flow regime.

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Problem 16

Water at $15^{\circ} \mathrm{C}$ is flowing uniformly in a $2-\mathrm{m}$-wide rectangular channel at an average velocity of $4 \mathrm{~m} / \mathrm{s}$. If the water depth is 8 cm , determine whether the flow is subcritical or supercritical. Answer: supercritical

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Problem 17

After heavy rain, water flows on a concrete surface at an average velocity of $1.3 \mathrm{~m} / \mathrm{s}$. If the water depth is 2 cm , determine whether the flow is subcritical or supercritical.

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Problem 18

Water at $70^{\circ} \mathrm{F}$ is flowing uniformly in a wide rectangular channel at an average velocity of $6 \mathrm{ft} / \mathrm{s}$. If the water depth is 0.5 ft , determine (a) whether the flow is laminar or turbulent and (b) whether the flow is subcritical or supercritical.

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Problem 19

Water at $10^{\circ} \mathrm{C}$ flows in a 3 -m-diameter circular channel half-full at an average velocity of $2.5 \mathrm{~m} / \mathrm{s}$. Determine the hydraulic radius, the Reynolds number, and the flow regime (laminar or turbulent).

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Problem 20

A single wave is initiated in a sea by a strong jolt during an earthquake. Taking the average water depth to be 2 km and the density of seawater to be $1.030 \mathrm{~kg} / \mathrm{m}^3$, determine the speed of propagation of this wave.
Specific Energy and the Energy Equation

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Problem 21

How is the specific energy of a fluid flowing in an open channel defined in terms of heads?

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Problem 22

Consider steady flow of water through two identical open rectangular channels at identical flow rates. If the flow in one channel is subcritical and in the other supercritical, can the specific energies of water in these two channels be identical? Explain.

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Problem 23

For a given flow rate through an open channel, the variation of specific energy with flow depth is studied. One person claims that the specific energy of the fluid will be minimum when the flow is critical, but another person claims that the specific energy will be minimum when the flow is subcritical. What is your opinion?

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Problem 24

Consider steady supercritical flow of water through an open rectangular channel at a constant flow rate. Someone claims that the larger is the flow depth, the larger the specific energy of water. Do you agree? Explain.

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Problem 25

During steady and uniform flow through an open channel of rectangular cross section, a person claims that the specific energy of the fluid remains constant. A second person claims that the specific energy decreases along the flow because of the frictional effects and thus head loss. With which person do you agree? Explain.

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Problem 26

How is the friction slope defined? Under what conditions is it equal to the bottom slope of an open channel?

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Problem 27

Consider steady flow of a liquid through a wide rectangular channel. It is claimed that the energy line of flow is parallel to the channel bottom when the frictional losses are negligible. Do you agree?

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Problem 28

Consider steady one-dimensional flow through a wide rectangular channel. Someone claims that the total mechanical energy of the fluid at the free surface of a cross section is equal to that of the fluid at the channel bottom of the same cross section. Do you agree? Explain.

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Problem 29

How is the total mechanical energy of a fluid during steady one-dimensional flow through a wide rectangular
channel expressed in terms of heads? How is it related to the specific energy of the fluid?

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Problem 30

Express the one-dimensional energy equation for open-channel flow between an upstream section 1 and downstream section 2, and explain how the head loss can be determined.

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Problem 31

Water flows steadily in a $0.8-\mathrm{m}$-wide rectangular channel at a rate of $0.7 \mathrm{~m}^3 / \mathrm{s}$. If the flow depth is 0.25 m , determine the flow velocity and if the flow is subcritical or supercritical. Also determine the alternate flow depth if the character of flow were to change.

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Problem 32

Water at $15^{\circ} \mathrm{C}$ flows at a depth of 0.4 m with an average velocity of $6 \mathrm{~m} / \mathrm{s}$ in a rectangular channel. Determine the specific energy of water and whether the flow is subcritical or supercritical.

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Problem 33

Water at $15^{\circ} \mathrm{C}$ flows at a depth of 0.4 m with an average velocity of $6 \mathrm{~m} / \mathrm{s}$ in a rectangular channel. Determine (a) the critical depth, (b) the alternate depth, and (c) the minimum specific energy.

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Problem 34

Water at $10^{\circ} \mathrm{C}$ flows in a $6-\mathrm{m}$-wide rectangular channel at a depth of 0.55 m and a flow rate of $12 \mathrm{~m}^3 / \mathrm{s}$. Determine (a) the critical depth, (b) whether the flow is subcritical or supercritical, and (c) the alternate depth. Answers: (a) 0.742 m , (b) supercritical, (c) 1.03 m

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Problem 35

Water at $65^{\circ} \mathrm{F}$ flows at a depth of 0.8 ft with an average velocity of $14 \mathrm{ft} / \mathrm{s}$ in a wide rectangular channel. Determine (a) the Froude number, (b) the critical depth, and (c) whether the flow is subcritical or supercritical. What would your response be if the flow depth were 0.2 ft ?

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Problem 36

Repeat Prob. 13-35E for an average velocity of $10 \mathrm{ft} / \mathrm{s}$.

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Problem 37

Water flows through a 4-m-wide rectangular channel with an average velocity of $5 \mathrm{~m} / \mathrm{s}$. If the flow is critical, determine the flow rate of water.

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Problem 38

Water flows half-full through a $50-\mathrm{cm}$-diameter steel channel at an average velocity of $2.8 \mathrm{~m} / \mathrm{s}$. Determine the volume flow rate and whether the flow is subcritical or supercritical.

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Problem 39

Water flows half-full through a hexagon channel of bottom width 2 m at a rate of $45 \mathrm{~m}^3 / \mathrm{s}$. Determine (a) the average velocity and (b) whether the flow is subcritical and supercritical.

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Problem 40

Repeat Prob. 13-39 for a flow rate of $30 \mathrm{~m}^3 / \mathrm{s}$.
Uniform Flow and Best Hydraulic Cross Sections

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Problem 41

When is the flow in an open channel said to be uniform? Under what conditions will the flow in an open channel remain uniform?

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Problem 42

Consider uniform flow through a wide rectangular channel. If the bottom slope is increased, the flow depth will (a) increase, (b) decrease, or (c) remain constant.

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Problem 43

During uniform flow in an open channel, someone claims that the head loss can be determined by simply multiplying the bottom slope by the channel length. Can it be this simple? Explain.

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Problem 44

During uniform flow in open channels, the flow velocity and the flow rate can be determined from the Manning equations expressed as $V_0=(a / n) R_h^{2 / 3} S_0^{1 / 2}$ and $\dot{V}$ $=(a / n) A_c R_h^{2 / 3} S_0^{1 / 2}$. What is the value and dimension of the constant $a$ in these equations in SI units? Also, explain how the Manning coefficient $n$ can be determined when the friction factor $f$ is known.

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Problem 45

Show that for uniform critical flow, the general critical slope relation $S_c=\frac{g n^2 y_c}{a^2 R_h^{4 / 3}}$ reduces to $S_c=\frac{g n^2}{a^2 y_c^{1 / 3}}$ for film flow with $b \gg y_c$.

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Problem 46

Which is a better hydraulic cross section for an open channel: one with a small or a large hydraulic radius?

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Problem 47

Which is the best hydraulic cross section for an open channel: (a) circular, (b) rectangular, (c) trapezoidal, or (d) triangular?

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Problem 48

The best hydraulic cross section for a rectangular open channel is one whose fluid height is (a) half, (b) twice, (c) equal to, or ( $d$ ) one-third the channel width.

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Problem 49

The best hydraulic cross section for a trapezoidal channel of base width $b$ is one for which the length of the side edge of the flow section is (a) $b,(b) b / 2$, (c) $2 b$, or (d) $\sqrt{3} b$

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Problem 50

Consider uniform flow through an open channel lined with bricks with a Manning coefficient of $n=0.015$. If the Manning coefficient doubles ( $n=0.030$ ) as a result of some algae growth on surfaces while the flow cross section remains constant, the flow rate will (a) double, (b) decrease by a factor of $\sqrt{2}$, (c) remain unchanged, (d) decrease by half, or (e) decrease by a factor of $2^{1 / 3}$.

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Problem 51

Water is flowing uniformly in a finished-concrete channel of trapezoidal cross section with a bottom width of 0.6 m , trapezoid angle of $50^{\circ}$, and a bottom angle of $0.4^{\circ}$. If the flow depth is measured to be 0.45 m , determine the flow rate of water through the channel.

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Problem 52

Water flows uniformly half-full in a 2 -m-diameter circular channel that is laid on a grade of $1.5 \mathrm{~m} / \mathrm{km}$. If the channel is made of finished concrete, determine the flow rate of the water.

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Problem 53

A 6-ft-diameter semicircular channel made of unfinished concrete is to transport water to a distance of 1 mi uniformly. If the flow rate is to reach $150 \mathrm{ft}^3 / \mathrm{s}$ when the channel is full, determine the minimum elevation difference across the channel.

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Problem 54

A trapezoidal channel with a bottom width of 5 m , free surface width of 10 m , and flow depth of 2.2 m discharges water at a rate of $120 \mathrm{~m}^3 / \mathrm{s}$. If the surfaces of the channel are lined with asphalt ( $n=0.016$ ), determine the elevation drop of the channel per km.

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Problem 55

Reconsider Prob. 13-54. If the maximum flow height the channel can accommodate is 2.4 m , determine the maximum flow rate through the channel.

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Problem 56

Consider water flow through two identical channels with square flow sections of $3 \mathrm{~m} \times 3 \mathrm{~m}$. Now the two channels are combined, forming a $6-\mathrm{m}$-wide channel. The flow rate is adjusted so that the flow depth remains constant at 3 m . Determine the percent increase in flow rate as a result of combining the channels.

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Problem 57

A trapezoidal channel made of unfinished concrete has a bottom slope of $1^{\circ}$, base width of 5 m , and a side surface slope of $1: 1$, as shown in Fig. P13-57. For a flow rate of $25 \mathrm{~m}^3 / \mathrm{s}$, determine the normal depth $h$.

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Problem 58

Repeat Prob. 13-57 for a weedy excavated earth channel with $n=0.030$.

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Problem 59

A cast iron V-shaped water channel shown in Fig. P13-59 has a bottom slope of $0.5^{\circ}$. For a flow depth of 1 m at the center, determine the discharge rate in uniform flow. Answer: $3.59 \mathrm{~m}^3 / \mathrm{s}$

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Problem 60

Water is to be transported in a cast iron rectangular channel with a bottom width of 6 ft at a rate of $70 \mathrm{ft}^3 / \mathrm{s}$. The terrain is such that the channel bottom drops 1.5 ft per 1000 ft length. Determine the minimum height of the channel under uniform-flow conditions.

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Problem 61

Water flows in a channel whose bottom slope is 0.002 and whose cross section is as shown in Fig. P13-61. The dimensions and the Manning coefficients for the surfaces of different subsections are also given on the figure. Determine the flow rate through the channel and the effective Manning coefficient for the channel.

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Problem 62

Consider a 1-m-internal-diameter water channel made of finished concrete ( $n=0.012$ ). The channel slope is 0.002 .

For a flow depth of 0.25 m at the center, determine the flow rate of water through the channel.

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Problem 63

Reconsider Prob. 13-62. By varying the flow depth-to-radius ratio $y / R$ from 0.1 to 1.9 while holding the flow area constant and evaluating the flow rate, show that the best cross section for flow through a circular channel occurs when the channel is half-full. Tabulate and plot your results.

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Problem 64

A clean-earth trapezoidal channel with a bottom width of 1.5 m and a side surface slope of $1: 1$ is to drain water uniformly at a rate of $8 \mathrm{~m}^3 / \mathrm{s}$ to a distance of 1 km . If the flow depth is not to exceed 1 m , determine the required elevation drop.

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Problem 65

A water draining system with a constant slope of 0.0015 is to be built of three circular channels made of finished concrete. Two of the channels have a diameter of 1.2 m and drain into the third channel. If all channels are to run half-full and the losses at the junction are negligible, determine the diameter of the third channel.

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Problem 66

Water is to be transported in an open channel whose surfaces are asphalt lined at a rate of $4 \mathrm{~m}^3 / \mathrm{s}$ in uniform flow. The bottom slope is 0.0015 . Determine the dimensions of the best cross section if the shape of the channel is (a) circular of diameter $D$, (b) rectangular of bottom width $b$, and (c) trapezoidal of bottom width $b$.

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Problem 67

A rectangular channel with a bottom slope of 0.0005 is to be built to transport water at a rate of $800 \mathrm{ft}^3 / \mathrm{s}$. Determine the best dimensions of the channel if it is to be made of (a) unfinished concrete and (b) finished concrete.

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Problem 68

Consider uniform flow in an asphalt-lined rectangular channel with a flow area of $2 \mathrm{~m}^2$ and a bottom slope of 0.0003 . By varying the depth-to-width ratio $y / b$ from 0.1 to 2.0 , calculate and plot the flow rate, and confirm that the best flow cross section occurs when the flow depth-to-width ratio is 0.5 .

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Problem 69

How does nonuniform or varied flow differ from uniform flow?

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Problem 70

How does gradually varied flow (GVF) differ from rapidly varied flow (RVF)?

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Problem 71

Someone claims that frictional losses associated with wall shear on surfaces can be neglected in the analysis of rapidly varied flow, but should be considered in the analysis of gradually varied flow. Do you agree with this claim? Justify your answer.

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Problem 72

Consider steady flow of water in a horizontal channel of rectangular cross section. If the flow is subcritical, the flow depth will (a) increase, (b) remain constant, or (c) decrease in the flow direction.

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Problem 73

Consider steady flow of water in a downwardsloped channel of rectangular cross section. If the flow is subcritical and the flow depth is greater than the normal depth $\left(y>y_n\right.$ ), the flow depth will (a) increase, (b) remain constant, or (c) decrease in the flow direction.

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Problem 74

Consider steady flow of water in a horizontal channel of rectangular cross section. If the flow is supercritical, the flow depth will (a) increase, (b) remain constant, or (c) decrease in the flow direction.

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Problem 75

Consider steady flow of water in a downwardsloped channel of rectangular cross section. If the flow is subcritical and the flow depth is less than the normal depth $\left(y<y_n\right)$, the flow depth will (a) increase, (b) remain constant, or (c) decrease in the flow direction.

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Problem 76

Consider steady flow of water in an upward-sloped channel of rectangular cross section. If the flow is supercritical, the flow depth will (a) increase, (b) remain constant, or (c) decrease in the flow direction.

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Problem 77

Is it possible for subcritical flow to undergo a hydraulic jump? Explain.

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Problem 78

Why is the hydraulic jump sometimes used to dissipate mechanical energy? How is the energy dissipation ratio for a hydraulic jump defined?

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Problem 79

Water flows uniformly in a rectangular channel with finished-concrete surfaces. The channel width is 3 m , the flow depth is 1.2 m , and the bottom slope is 0.002 . Determine if the channel should be classified as mild, critical, or steep for this flow.

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Problem 80

Consider uniform water flow in a wide brick channel of slope $0.4^{\circ}$. Determine the range of flow depth for which the channel is classified as being steep.

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Problem 81

Consider the flow of water through a 12 -ft-wide unfinished-concrete rectangular channel with a bottom slope of $0.5^{\circ}$. If the flow rate is $300 \mathrm{ft}^3 / \mathrm{s}$, determine if the slope of this channel is mild, critical, or steep. Also, for a flow depth of 3 ft , classify the surface profile while the flow develops.

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Problem 82

Water is flowing in a $90^{\circ} \mathrm{V}$-shaped cast iron channel with a bottom slope of 0.002 at a rate of $3 \mathrm{~m}^3 / \mathrm{s}$. Determine if the slope of this channel should be classified as mild, critical, or steep for this flow.

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Problem 83

Water discharging into an 8-m-wide rectangular horizontal channel from a sluice gate is observed to have undergone a hydraulic jump. The flow depth and velocity before the jump are 1.2 m and $9 \mathrm{~m} / \mathrm{s}$, respectively. Determine (a) the flow depth and the Froude number after the jump, (b) the head loss and the dissipation ratio, and (c) the mechanical energy dissipated by the hydraulic jump.

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Problem 84

Water flowing in a wide horizontal channel at a flow depth of 35 cm and an average velocity of $12 \mathrm{~m} / \mathrm{s}$ undergoes a hydraulic jump. Determine the head loss associated with hydraulic jump.

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Problem 85

During a hydraulic jump in a wide channel, the flow depth increases from 0.6 to 3 m . Determine the velocities and Froude numbers before and after the jump, and the energy dissipation ratio.

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Problem 86

Consider the flow of water in a $10-\mathrm{m}$-wide channel at a rate of $70 \mathrm{~m}^3 / \mathrm{s}$ and a flow depth of 0.50 m . The water now undergoes a hydraulic jump, and the flow depth after the jump is measured to be 4 m . Determine the mechanical power wasted during this jump.

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Problem 87

The flow depth and velocity of water after undergoing a hydraulic jump are measured to be 2 m and $3 \mathrm{~m} / \mathrm{s}$, respectively. Determine the flow depth and velocity before the jump, and the fraction of mechanical energy dissipated.

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Problem 88

Water flowing in a wide channel at a depth of 2 ft and a velocity of $40 \mathrm{ft} / \mathrm{s}$ undergoes a hydraulic jump. Determine the flow depth, velocity, and Froude number after the jump, and the head loss associated with the jump.

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Problem 89

Draw a flow depth-specific energy diagram for flow through underwater gates, and indicate the flow through the gate for cases of (a) frictionless gate, (b) sluice gate with free outflow, and (c) sluice gate with drowned outflow (including the hydraulic jump back to subcritical flow).

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Problem 90

For sluice gates, how is the discharge coefficient $C_d$ defined? What are typical values of $C_d$ for sluice gates with free outflow? What is the value of $C_d$ for the idealized frictionless flow through the gate?

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Problem 91

What is the basic principle of operation of a broadcrested weir used to measure flow rate through an open channel?

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Problem 92

Consider steady frictionless flow over a bump of height $\Delta z$ in a horizontal channel of constant width $b$. Will the flow depth $y$ increase, decrease, or remain constant as the fluid flows over the bump? Assume the flow to be subcritical.

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Problem 93

Consider the flow of a liquid over a bump during subcritical flow in an open channel. The specific energy and the flow depth decrease over the bump as the bump height is increased. What will the character of flow be when the specific energy reaches its minimum value? Will the flow become supercritical if the bump height is increased even further?

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Problem 94

What is a sharp-crested weir? On what basis are the sharp-crested weirs classified?

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Problem 95

Water is released from a $14-\mathrm{m}$-deep reservoir into a $5-\mathrm{m}$-wide open channel through a sluice gate with a 1-m-high opening at the channel bottom. If the flow depth downstream from the gate is measured to be 3 m , determine the rate of discharge through the gate.

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Problem 96

Water flowing in a wide channel encounters a 22 -cm-high bump at the bottom of the channel. If the flow depth is 1.2 m and the velocity is $2.5 \mathrm{~m} / \mathrm{s}$ before the bump, determine if the flow is chocked over the bump, and discuss.

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Problem 97

Consider the uniform flow of water in a wide channel with a velocity of $8 \mathrm{~m} / \mathrm{s}$ and flow depth of 0.8 m . Now water flows over a $30-\mathrm{cm}$-high bump. Determine the change (increase or decrease) in the water surface level over the bump. Also determine if the flow over the bump is sub- or supercritical.

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Problem 98

The flow rate of water in a 4 -m-wide horizontal channel is being measured using a $0.75-\mathrm{m}$-high sharp-crested rectangular weir that spans across the channel. If the water depth upstream is 2.2 m , determine the flow rate of water.

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Problem 99

Repeat Prob. 13-98 for the case of a weir height of 1 m .

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Problem 100

Water flows over a 2-m-high sharp-crested rectangular weir. The flow depth upstream of the weir is 3 m , and water is discharged from the weir into an unfinished-concrete channel of equal width where uniform-flow conditions are established. If no hydraulic jump is to occur in the downstream flow, determine the maximum slope of the downstream channel.

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Problem 101

A full-width sharp-crested weir is to be used to measure the flow rate of water in a 10 - ft -wide rectangular channel. The maximum flow rate through the channel is $150 \mathrm{ft}^3 / \mathrm{s}$, and the flow depth upstream from the weir is not to exceed 5 ft . Determine the appropriate height of the weir.

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Problem 102

Consider uniform water flow in a wide rectangular channel with a depth of 2 m made of unfinished concrete laid on a slope of 0.0022 . Determine the flow rate of water per m width of channel. Now water flows over a $15-\mathrm{cm}$-high bump. If the water surface over the bump remains flat (no rise or drop), determine the change in discharge rate of water per meter width of the channel. Hint: Investigate if a flat surface over the bump is physically possible.

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Problem 103

Consider uniform water flow in a wide channel made of unfinished concrete laid on a slope of 0.0022 . Now water flows over a $15-\mathrm{cm}$-high bump. If the flow over the bump is exactly critical ( $\mathrm{Fr}=1$ ), determine the flow rate and the flow depth over the bump per m width.

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Problem 104

The flow rate of water through a 5-m-wide (into the paper) channel is controlled by a sluice gate. If the flow depths are measured to be 1.1 and 0.45 m upstream and downstream from the gates, respectively, determine the flow rate and the Froude number downstream from the gate.

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Problem 105

Water flows through a sluice gate with a 1.1 -fthigh opening and is discharged with free outflow. If the upstream flow depth is 5 ft , determine the flow rate per unit width and the Froude number downstream the gate.

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Problem 106

Repeat Prob. 13-105E for the case of a drowned gate with a downstream flow depth of 3.3 ft .

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Problem 107

Water is to be discharged from a 6-m-deep lake into a channel through a sluice gate with a $5-\mathrm{m}$ wide and 0.6 m -high opening at the bottom. If the flow depth downstream from the gate is measured to be 3 m , determine the rate of discharge through the gate.

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Problem 108

Consider water flow through a wide channel at a flow depth of 8 ft . Now water flows through a sluice gate with a 1 -ft-high opening, and the freely discharged outflow subsequently undergoes a hydraulic jump. Disregarding any losses associated with the sluice gate itself, determine the flow depth and velocities before and after the jump, and the fraction of mechanical energy dissipated during the jump.

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Problem 109

The flow rate of water flowing in a 3-m-wide channel is to be measured with a sharp-crested triangular weir 0.5 m above the channel bottom with a notch angle of $60^{\circ}$. If the flow depth upstream from the weir is 1.5 m , determine the flow rate of water through the channel. Take the weir discharge coefficient to be 0.60 .

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Problem 110

Repeat Prob. 13-109 for an upstream flow depth of 1.2 m .

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Problem 111

A sharp-crested triangular weir with a notch angle of $100^{\circ}$ is used to measure the discharge rate of water from a large lake into a spillway. If a weir with half the notch angle $\left(\theta=50^{\circ}\right)$ is used instead, determine the percent reduction in the flow rate. Assume the water depth in the lake and the weir discharge coefficient remain unchanged.

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Problem 112

A 1-m-high broad-crested weir is used to measure the flow rate of water in a $5-\mathrm{m}$-wide rectangular channel. The flow depth well upstream from the weir is 1.6 m . Determine the flow rate through the channel and the minimum flow depth above the weir.

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Problem 113

Repeat Prob. 13-112 for an upstream flow depth of 2.2 m .

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Problem 114

Consider water flow over a 0.80 -m-high sufficiently long broad-crested weir. If the minimum flow depth above the weir is measured to be 0.50 m , determine the flow rate per meter width of channel and the flow depth upstream of the weir.

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Problem 115

A trapezoidal channel with a bottom width of 4 m and a side slope of $45^{\circ}$ discharges water at a rate of $18 \mathrm{~m}^3 / \mathrm{s}$. If the flow depth is 0.6 m , determine if the flow is subcritical or supercritical.

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Problem 116

A rectangular channel with a bottom width of 2 m discharges water at a rate of $8 \mathrm{~m}^3 / \mathrm{s}$. Determine the flow depth below which the flow is supercritical.

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Problem 117

Water flows in a canal at an average velocity of $4 \mathrm{~m} / \mathrm{s}$. Determine if the flow is subcritical or supercritical for flow depths of (a) 0.2 m , (b) 2 m , and (c) 1.63 m .

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Problem 118

Water flows through a $1.5-\mathrm{m}$-wide rectangular channel with a Manning coefficient of $n=0.012$. If the water is 0.9 m deep and the bottom slope of the channel is $0.6^{\circ}$, determine the rate of discharge of the channel in uniform flow.

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Problem 119

A 5-m-wide rectangular channel lined with finished concrete is to be designed to transport water to a distance of 1 km at a rate of $12 \mathrm{~m}^3 / \mathrm{s}$. Using EES (or other) software, investigate the effect of bottom slope on flow depth (and thus on the required channel height). Let the bottom angle vary from 0.5 to $10^{\circ}$ in increments of $0.5^{\circ}$. Tabulate and plot the flow depth against the bottom angle, and discuss the results.

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Problem 120

Repeat Prob. 13-119 for a trapezoidal channel that has a base width of 5 m and a side surface angle of $45^{\circ}$.

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Problem 121

A trapezoidal channel with brick lining has a bottom slope of 0.001 and a base width of 4 m , and the side surfaces are angled $30^{\circ}$ from the horizontal, as shown in Fig. P13-121. If the normal depth is measured to be 2 m , estimate the flow rate of water through the channel

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Problem 122

A 2-m-internal-diameter circular steel storm drain ( $n=0.012$ ) is to discharge water uniformly at a rate of $12 \mathrm{~m}^3 / \mathrm{s}$ to a distance of 1 km . If the maximum depth is to be 1.5 m , determine the required elevation drop.

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Problem 123

Consider water flow through a V-shaped channel. Determine the angle $\theta$ the channel makes from the horizontal for which the flow is most efficient.

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Problem 124

A rectangular channel with unfinished concrete surfaces is to be built to discharge water uniformly at a rate of $200 \mathrm{ft}^3 / \mathrm{s}$. For the case of best cross section, determine the bottom width of the channel if the available vertical drop is (a) 8 and (b) 10 ft per mile.

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Problem 125

Repeat Prob. 13-124E for the case of a trapezoidal channel of best cross section.

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Problem 126

Water flows in a channel whose bottom slope is $0.5^{\circ}$ and whose cross section is as shown in Fig. P13-126. The dimensions and the Manning coefficients for the surfaces of different subsections are also given on the figure. Determine the flow rate through the channel and the effective Manning coefficient for the channel.

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Problem 127

Consider two identical channels, one rectangular of bottom width $b$ and one circular of diameter $D$, with identical flow rates, bottom slopes, and surface linings. If the flow height in the rectangular channel is also $b$ and the circular channel is flowing half-full, determine the relation between $b$ and $D$.

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Problem 128

Consider the flow of water through a parabolic notch shown in Fig. P13-128. Develop a relation for the flow rate, and calculate its numerical value for the ideal case in which the flow velocity is given by Toricelli's equation $V=\sqrt{2 g(H-y)}$.

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Problem 129

( $\in$ S In practice, the V-notch is commonly used to measure flow rate in open channels. Using the idealized Toricelli's equation $V=\sqrt{2 g(H-y)}$ for velocity, develop a relation for the flow rate through the V-notch in terms of the angle $\theta$. Also, show the variation of the flow rate with $\theta$ by evaluating the flow rate for $\theta=25,40,60$, and $75^{\circ}$, and plotting the results.

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Problem 130

Water flows uniformly half-full in a 1.2-m-diameter circular channel laid with a slope of 0.004 . If the flow rate of water is measured to be $1.25 \mathrm{~m}^3 / \mathrm{s}$, determine the Manning coefficient of the channel and the Froude number.

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Problem 131

Water flowing in a wide horizontal channel approaches a $20-\mathrm{cm}$-high bump with a velocity of $1.25 \mathrm{~m} / \mathrm{s}$ and a flow depth of 1.8 m . Determine the velocity, flow depth, and Froude number over the bump.

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Problem 132

Reconsider Prob. 13-131. Determine the bump height for which the flow over the bump is critical $(\mathrm{Fr}=1)$.

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Problem 133

Consider water flow through a wide rectangular channel undergoing a hydraulic jump. Show that the ratio of the Froude numbers before and after the jump can be expressed in terms of flow depths $y_1$ and $y_2$ before and after the jump, respectively, as $\mathrm{Fr}_1 / \mathrm{Fr}_2=\sqrt{\left(y_2 / y_1\right)^3}$.

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Problem 134

A sluice gate with free outflow is used to control the discharge rate of water through a channel. Determine the flow rate per unit width when the gate is raised to yield a gap of 30 cm and the upstream flow depth is measured to be 1.8 m . Also determine the flow depth and the velocity downstream.

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Problem 135

Water flowing in a wide channel at a flow depth of 45 cm and an average velocity of $8 \mathrm{~m} / \mathrm{s}$ undergoes a hydraulic jump. Determine the fraction of the mechanical energy of the fluid dissipated during this jump.

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Problem 136

Water flowing through a sluice gate undergoes a hydraulic jump, as shown in Fig. P13-136. The velocity of the water is $1.25 \mathrm{~m} / \mathrm{s}$ before reaching the gate and $4 \mathrm{~m} / \mathrm{s}$ after the jump. Determine the flow rate of water through the gate per meter of width, the flow depths $y_1$ and $y_2$, and the energy dissipation ratio of the jump.

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Problem 137

Repeat Prob. 13-136 for a velocity of $2 \mathrm{~m} / \mathrm{s}$ after the hydraulic jump.

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Problem 138

Water is discharged from a $5-\mathrm{m}$-deep lake into a finished concrete channel with a bottom slope of 0.004 through a sluice gate with a $0.5-\mathrm{m}$-high opening at the bottom. Shortly after supercritical uniform-flow conditions are established, the water undergoes a hydraulic jump. Determine the flow depth, velocity, and Froude number after the jump. Disregard the bottom slope when analyzing the hydraulic jump.

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Problem 139

Water is discharged from a dam into a wide spillway to avoid overflow and to reduce the risk of flooding. A large fraction of the destructive power of water is dissipated by a hydraulic jump during which the water depth rises from 0.50 to 4 m . Determine the velocities of water before and after the jump, and the mechanical power dissipated per meter width of the spillway.

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Problem 140

The flow rate of water in a $6-\mathrm{m}$-wide rectangular channel is to be measured using a $1.1-\mathrm{m}$-high sharp-crested rectangular weir that spans across the channel. If the head above the weir crest is 0.60 m upstream from the weir, determine the flow rate of water.

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Problem 141

Consider two identical 12 - ft -wide rectangular channels each equipped with a 2 -ft-high full-width weir, except that the weir is sharp-crested in one channel and broad-crested in the other. For a flow depth of 5 ft in both channels, determine the flow rate through each channel. Answers: $244 \mathrm{ft}^3 / \mathrm{s}, 79.2 \mathrm{ft}^3 / \mathrm{s}$

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Problem 142

Using catalogs or websites, obtain information from three different weir manufacturers. Compare the different weir designs, and discuss the advantages and disadvantages of each design. Indicate the applications for which each design is best suited.

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Problem 143

Consider water flow in the range of 10 to $15 \mathrm{~m}^3 / \mathrm{s}$ through a horizontal section of a 5 -m-wide rectangular channel. A rectangular or triangular thin-plate weir is to be installed to measure the flow rate. If the water depth is to remain under 2 m at all times, specify the type and dimensions of an appropriate weir. What would your response be if the flow range were 0 to $15 \mathrm{~m}^3 / \mathrm{s}$ ?

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