Fluid Mechanics

John F. Douglas, Janusz M. Gasiorek, John A. Swaffield

Chapter 15 Uniform Flow in Open Channels - all with Video Answers

Educators

Chapter Questions

01:52

Problem 1

A rectangular channel is 2.5 $\mathrm{m}$ wide and has a uniform bed slope of 1 in 500 . If the depth of flow is constant at $1.7 \mathrm{m}$ calculate $(a)$ the hydraulic mean depth,
(b) the velocity of flow, (c) the volume rate of flow. Assume that the value of the coefficient $C$ in the Chezy formula is 50 in SI units.
$$\left[(a) 0.72 \mathrm{m},(b) 1.9 \mathrm{m} \mathrm{s}^{-1},(c) 8.1 \mathrm{m}^{3} \mathrm{s}^{-1}\right]$$

Narayan Hari

Numerade Educator

03:02

Problem 2

An open channel has a vee-shaped cross-section with sides inclined at an angle of $60^{\circ}$ to the vertical. If the rate of flow is $80 \mathrm{dm}^{3} \mathrm{s}^{-1}$ when the depth at the centre is $0.25 \mathrm{m},$ what must be the slope of the channel assuming $C=45$ in SI units?
$$[1 \text { in } 401]$$

James Kiss

Numerade Educator

01:52

Problem 3

A channel $5 \mathrm{m}$ wide at the top and $2 \mathrm{m}$ deep has sides sloping 2 vertically in 1 horizontally. The slope of the channel is 1 in 1000 . Find the volume rate of flow when the depth of water is constant at $1 \mathrm{m}$. Take $C$ as 53 in SI units.
What would be the depth of water if the flow were to be doubled?
\[
\left[4.79 \mathrm{m}^{3} \mathrm{s}^{-1}, 1.6 \mathrm{m}\right]
\]

Narayan Hari

Numerade Educator

01:52

Problem 4

Water is conveyed in a channel of semicircular crosssection with a slope of 1 in 2500 . The Chezy coefficient $C$ has a value of 56 in SI units. If the radius of the channel is $0.55 \mathrm{m},$ what will be the volume in cubic decimetres per second flowing when the depth is equal to the radius?
If the channel had been rectangular in form with the same width of $1.1 \mathrm{m}$ and depth of flow of $0.55 \mathrm{m},$ what would be the discharge for the same slope and value of $C ?$
\[
\left[279 \mathrm{dm}^{3} \mathrm{s}^{-1}, 355 \mathrm{dm}^{3} \mathrm{s}^{-1}\right]
\]

Narayan Hari

Numerade Educator

09:12

Problem 5

A 900 mm diameter conduit 3600 m long is laid at a uniform slope of 1 in 1500 and connects two reservoirs. When the levels in the reservoirs are low the conduit runs partly full and it is found that a normal depth of $600 \mathrm{mm}$ gives a rate of flow of $0.322 \mathrm{m}^{3} \mathrm{s}^{-1}$
The Chezy coefficient $C$ is given by $K m^{n}$, where $K$ is a constant, $m$ is the hydraulic mean depth and $n=\frac{1}{6} .$ Neglecting losses of head at entry and exit obtain ( $a$ ) the value of $K$
(b) the discharge when the conduit is flowing full and the difference in level between the two reservoirs is $4.5 \mathrm{m}$
\[
\left[(a) 67.6,(b) 0.562 m^{3} s^{-1}\right]
\]

Ajay Singhal

Numerade Educator

03:37

Problem 6

An earth channel is trapezoidal in cross-section with a bottom width of $1.8 \mathrm{m}$ and side slopes of 1 vertical to 2 horizontal. Taking the friction coefficient $k$ in the Bazin formula as 1.3 and the slope of the bed as $0.57 \mathrm{m}$ per kilometre, find the discharge in cubic metres per second when the depth of water is $1.5 \mathrm{m}$
\[
\left[5.69 \mathrm{m}^{3} \mathrm{s}^{-1}\right]
\]

Khoobchandra Agrawal

Numerade Educator

01:27

Problem 7

The water supply for a turbine passes through a conduit which for convenience has its cross-section in the form of a square with one diagonal vertical. If the conduit is required to convey $8.5 \mathrm{dm}^{3} \mathrm{s}^{-1}$ under conditions of maximum discharge at atmospheric pressure when the slope of the bed is 1 in $4900,$ determine its size assuming that the velocity of flow is given by
\[
\begin{array}{c}
v=80 i^{1 / 2} m^{2 / 3} \\
{[0.222 \mathrm{m} \mathrm{side}]}
\end{array}
\]

Penny Riley

Numerade Educator

01:52

Problem 8

A trapezoidal channel is to be designed to carry $280 \mathrm{m}^{3}$ per minute of water. Determine the cross-sectional dimensions of the channel if the slope is 1 in $1600,$ side slopes $45^{\circ}$ and the cross-section is to be a minimum. Take $C=50 \text { in SI units. }$
\[
[D=1.53 \mathrm{m}, B=1.27 \mathrm{m}]
\]

Narayan Hari

Numerade Educator

04:15

Problem 9

A circular section open conduit conveys liquid under maximum velocity conditions. Show that the depth of liquid is 81 per cent of the diameter. Show, without complete calculation, that this will not be the maximum discharge condition.
Such a conduit having a diameter of $0.8 \mathrm{m}$ is to discharge $0.6 \mathrm{m}^{3} \mathrm{s}^{-1}$ at maximum velocity. Find the required channel slope if the Chezy constant is 90 in SI units.
$$[1 \text { in } 1050]$$

Chai Santi

Numerade Educator

02:39

Problem 10

It is required to excavate a canal out of rock. It is to be of rectangular cross-section and to bring $14.2 \mathrm{m}^{3}$ of water from a distance of $6.5 \mathrm{km}$ with a velocity of $2.25 \mathrm{m} \mathrm{s}^{-1}$ Determine the gradient and the most suitable section.
\[
[1 \text { in } 186, D=1.78 \mathrm{m}, B=3.56 \mathrm{m}]
\]

Kratika Bhadauria

Numerade Educator

06:59

Problem 11

An egg-shaped sewer has a section formed by circular arcs, the top being a semicircle of radius $R .$ The area and wetted perimeter of the section below the horizontal diameter of the semicircle are $3 R^{2}$ and $4.82 R,$ respectively. Prove that, if $C$ in the Chezy formula is constant, the maximum flow will occur when the water surface subtends
an angle of approximately $55^{\circ}$ at the centre of curvature of the semicircle.

Jacquelyn Trost

Numerade Educator

View

Problem 12

The upper portion of the cross-section of an open channel is a semicircle of radius $a ;$ the lower portion is a semiellipse of width $2 a,$ depth $2 a$ and perimeter $4.847 a$ whose minor axis coincides with the horizontal diameter of the semicircle. The channel is required to convey $14 \mathrm{m}^{3} \mathrm{s}^{-1}$ when running three-quarters full (i.e. with three-quarters of the vertical axis of symmetry immersed), the slope of the bed $i$ being $0.001 .$ Assuming that the mean velocity of flow is given by Manning's formula $v=80 i^{1 / 2} m^{2 / 3},$ determine the dimensions of the section and the depth under maximum flow conditions.
\[
[a=1.29 \mathrm{m}, 3.68 \mathrm{m}]
\]

Victor Salazar

Numerade Educator

01:50

Problem 13

The cross-section of a closed channel is a square with one diagonal vertical, $s$ is the side of the square and $y$ is the depth of the waterline below the apex. Show that for maximum discharge $y=0.127 s$ and that for maximum velocity $y=0.414 s$

Chai Santi

Numerade Educator

03:25

Problem 14

Find an expression for the theoretical depth for maximum velocity in a closed circular channel in terms of the diameter $d$
Compare the discharge at maximum velocity with that when the channel is running full, assuming that the Chezy coefficient is unaltered, and that the pressure remains atmospheric.
\[
[0.81 d, 0.964]
\]

Chai Santi

Numerade Educator

08:48

Problem 15

An open channel of economic trapezoidal crosssection with sides inclined at $60^{\circ}$ to the horizontal is required to give a discharge of $10 \mathrm{m}^{3} \mathrm{s}^{-1}$ when the slope of the bed is 1 in 1600 . Calculate the dimensions of the cross-section assuming $v=74 i^{1 / 2} m^{2 / 3}$
\[
[D=2.34 \mathrm{m}, B=0.72 \mathrm{m}]
\]

Khoobchandra Agrawal

Numerade Educator