E1 Questions on Uniform Flow and Critical Flow.  Calculation of normal and critical depth.

E1.1
A rectangular channel is $3.0m$ wide, has a 0.01 slope, flow rate of $5.3m^{3}/s$, and $n=0.011$. Find its normal depth $y_{n}$ and critical depth $y_{c}$.
(Answer: $y_{n}=0.41m$, $y_{c}=0.683m$) 
E1.2
Water flows in a long rectangular channel at a depth of $1.22m$ and discharge of $Q=5.66m^{3}/s$. Determine the minimum channel width if the channel is to be subcritical.
(Answer: $1.34m$) 
E1.3
A rectangular channel has a bottom width of $B=8m$ and Manning’s $n=0.025$

(a)
Determine the slope to give a normal depth of $y_{n}=2m$ when the discharge is $12m^{3}/s$

(b)
Determine the critical slope and the critical depth when the discharge is $12m^{3}/s$

(c)
Determine the critical slope to give a the critical depth of $y_{c}=1.5m$ and compute the corresponding discharge.
(Answer:(a) $S_{o}=0.00024$, (b) $So_{c}=0.0087$, $y_{c}=0.61m$, (c) $So_{c}=46.03m^{3}/s$, $Q=0.00818$)

(a)

E1.4
For a trapezoidal channel with a base width $b=3.0m$, Manning’s $n=0.025$ and side slope $s=2$ (i.e. 1 vertical: 2 horizontal), calculate the critical depth, critical velocity, and critical slope if its discharge $Q=10m^{3}/s$.
(Answer:$y_{c}=0.855m$, $v_{c}=2.483m/$s, $So_{c}=0.00777$) 
E1.5
A rectangular channel $9m$ wide carries $7.6m^{3}/s$ of water when flowing $1.0m$ deep. Work out the flow’s specific energy. Is the flow subcritical or supercritical?
(Answer: $1.036$m and flow is subcritical) 
E1.6
Two engineers observed two rivers and recorded the following flow parameters: River 1: flow discharge $Q=130m^{3}/s$, flow velocity $V=1.6m/s$, water surface width $B=80m$; River 2: flow discharge $Q=1530m^{3}/s$, flow velocity $V=5.6m/s$, water surface width $B=90m$. Decide the flow regime of two rivers, i.e. subcritical or supercritical.
(Answer: River 1 is subcritical and River 2 is supercritical) 
E1.7
A concrete, trapezoidal channel has a bottom slope of $S_{o}=0.0009$ and a Manning roughness factor of $n=0.013$. The bottom width of the channel is $b=2.5m$, and the side slopes are 1 in 2. Determine the velocity and discharge when the flow is normal at a depth of $1.8m$.
(Answer: $v=2.37m/s$, $Q=26.01m^{3}/s$) 
E1.8
A trapezoidal channel has a bottom slope of $S_{o}=1$ in $40$ and a Manning roughness factor of $n=0.016$. The bottom width of the channel is $b=6.0m$, and the side slopes are 1 in 3. Determine the normal depth in this channel for $Q=42.3m^{3}/s$.
(Answer: $y_{n}=0.75m$). 
E1.9
The flow discharge in uniform flow in a rectangular channel $4.6m$ wide is $11.3m^{3}/s$ when the slope is 1:100. Is the flow subcritical or supercritical? Calculate the slope, $S_{c}$, that would give critical depth. The Manning roughness coefficient is $n=0.012$.
(Answer: supercritical, $Fr=2.1$, $S_{c}=0.002268$).Compound Channels

E1.10
The crosssection of a stream can be approximated by the compound channel shown in figure 2. The bottom slope is $S_{o}=0.0009$. The Manning roughness factor $n=0.025$ for the main channel and $n=0.035$ for the overbank areas. Determine the normal depth for a discharge of $197m^{3}/s$. Also, calculate the energy coefficient $\alpha$ and the momentum coefficient $\beta$ for the channel with this flow condition.
(Answer: $y_{n}=5.507m$, $\alpha=1.23$, $\beta=1.09$.)

E1.11
The total width of the channel considered in Question E1.10 is to be decreased by reducing the overbank portions symmetrically; however, this reduction must not cause an increase of more than $0.15m$ in the flow depth for the discharge of $197m^{3}/s$. Assuming normal depth still is present in the channel, determine the minimum allowable channel total width, $B$.
(Answer: $B=16.157m$.) 
E1.12
The crosssection of a river with flood plains flowing in uniform flow may be idealized as shown in Fig. 3. Determine the discharge carried by the river when its dimensions and roughness parameters are:
Bed slope: $S_{o}=2\times 10^{4}$
Manning’s ns: $n_{1}=n_{2}=n_{3}=0.02$
Side slopes: $s_{1}=s_{2}=s_{3}=1$
Bed widths: $B_{1}=3m$, $B_{2}=5m$, $B_{3}=4m$
Main channel depth: $y_{main}=3.0m$and
Normal depth $y_{n}=4.5m$
(Answer: $Q=69.42m^{3}/s$.) 
E1.13
For the channel of question E1.12 calculate the flow, if all dimensions, including the normal depth were the same, but the slope of the channel is 0.002.
(Answer $Q=219.53m^{3}/s$).Efficient Channels

E1.14
A trapezoidal channel has side slopes of 1:3/4 and the slope of the bed is 1 in 2000. Determine the optimum dimensions of the channel if it is to carry water at $0.5m^{3}/s$. Use the Chezy formula, assuming that $C=80m^{1/2}/s$.
(Answer: $y_{n}=0.552m$, $b=0.552m$). 
E1.15
An open channel with $n=0.011$ is to be designed to carry $1.0m^{3}/s$ of water at a slope of 0.0065. Find the most efficient crosssection for a rectangular section.
(Answer: $b=2y=0.869m$). 
E1.16
A rectangular channel has width $B=3m$ and normal depth $y=1m$. What is the diameter of a semicircular channel that will have the same discharge as in the rectangular channel, when flowing just full in uniform flow? Assume that $n$ and $S_{o}$ are the same in the two cases. Compare the two wetted perimeters.
(Answer: $D=2.057m$, $P_{rectangular}=5.0m$, $P_{circular}=6.463m$) 
E1.17
What are the dimensions of the most efficient rectangular channel section to carry $5m^{3}/s$ at a slope of 1 in 900. The surface of the channel is of concrete.
(Answer: $y=1.21m$, $b=2y=2.42m$ using $n=0.012$) 
E1.18
What is the most efficient depth for a brick channel of a trapezoidal section with sides sloping at $45^{\circ}$ to the horizontal to carry $3m^{3}/s$. The bed slope is 0.0009.
(Answer: $y=1.104m$, using $n=0.015$)