A trapezoidal channel has side slope 1 in 1. It is desired to convey discharge of 13.75 $m^3/sec$ of water with bed gradient 1 in 1000. If unlined Chezy’s constant is taken as 44 and if lined, Chezy’s constant is 60. The cost per m3 of excavation is 4 times cost of per $m^2$ of lining.

The channel has to be designed efficiently, find whether lined canal or unlined canal will be cheaper, Also find the dimensions of most efficient section.

Solution :

Given :

Side slope, $\quad n=\frac{1}{1}=1$

Discharge, $\quad Q=13.75 \mathrm{~m}^{3} / \mathrm{s}$

Slope of bed, $\quad i=\frac{1}{1000}$

For unlined, $\quad C=44 $

For lined, $\quad C=60$

Let the cost per $\mathrm{m}^{2}$ of lining $=x$

Then cost per $\mathrm{m}^{3}$ of excavation $=4 x$

As the channel is most efficient, …