Economical channel: unlined versus concrete lined

For a given area and side slope, the most efficient (least wetted perimeter) trapezoid is used for each case. Then the cost per metre of channel is compared in terms of the unit cost of lining.

Given data

$Q=15$ m³/s, $z=1$ (1H:1V), $S_0=1/1000$, $C=45$ (unlined) or $70$ (lined). Let cost of lining per m² $=1$ unit; excavation cost per m³ $=3$ units.

Most efficient trapezoidal section ($z=1$)

$$b=2y(\sqrt{1+z^2}-z)=2y(\sqrt2-1)=0.8284\,y,\qquad R=\frac y2$$ $$A=(b+zy)y=1.8284\,y^2,\qquad P=b+2y\sqrt{1+z^2}=3.6569\,y$$

Chezy's equation $Q=CA\sqrt{RS_0}$

$$15=C\,(1.8284\,y^2)\sqrt{\frac y2\times0.001}$$

Cost per metre length (assuming the excavated section equals the flow section)

• Unlined: excavation only $=3\times A=3\times9.798=29.39$ units.
• Lined: excavation $=3\times6.881=20.64$ units, plus lining $=1\times P=7.09$ units; total $=27.74$ units.

Answer: lined channel costs 27.74 units/m against 29.39 units/m for the unlined channel, so the concrete-lined channel (y = 1.94 m, b = 1.61 m) is more economical (cheaper by about 5.6%).