a$)$ An above$-$ground flume, built from timber, is to convey water from a

mountain lake to a small hydroelectric plant. The flume is to deliver water

at $Q=2\frac{m^3}{s}$; the slope is $S_0=0\mathrm{.}002$ and $n=0\mathrm{.}013.$ Evaluate the required

flume size for $($a$)$ a rectangular section with $\frac{y}{b}=0\mathrm{.}5$ and $($b$)$ an equilateral

triangular section.

$Q=Av=A\frac{1}{n}{R}^{\frac{2}{3}}{S}_0^{\frac{1}{2}}$

b$)$ Water flows in a channel whose bottom slope is $0\mathrm{.}003$ and whose cross

section is a composite roughness section as shown in Figure $3\mathrm{.}1$ below.

The dimensions and the Manning coefficients for the surfaces of different

subsections are also given on the figure. Using Horton and Einstein

formula given below determine an equivalent Manning coefficient for the

section and then calculate flow rate on this basis.

$n_e=[\frac{{\displaystyle \underset{i=1}{\overset{M}{}}}{P}_i{n}_i^{\frac{3}{2}}}{P}{]}^{\frac{2}{3}}$

Figure $3\mathrm{.}1$

c$)$ A flow discharge of $35\frac{m^3}{s}$ is measured in a rectangular concrete channel

$4m$ wide with a flow depth of $3\mathrm{.}1m.$ Given the channel bed slope is $1$:$1200$ :

i$.$ Determine the Manning's roughness coefficient for the

channel.

ii$.$ It is decided to replace the above channel with a trapezoidal

channel with $45$ side slopes. The concrete finish will be the

same. Calculate the profile of the most economic section

for the above discharge rate given the following equation:

$b+2Ny=2y\sqrt[\phantom{2}]{1+N^2}$