Calculate the most economic area of a rectangular channel section with width ‘B’ and depth ‘y’.

Question

Question

This question was previously asked in

SSC JE Civil 9 Oct 2023 Shift 1 Official Paper-I

Answer 1

Explanation:

Concept:

Most economical hydraulic section:

The discharge from a channel section increases with increases in hydraulic radius or with decrease in the wetted perimeter.
From hydraulics viewpoint, therefore, the channel section having the least wetted perimeter for a given area has the maximum discharge; such a section is known as the best hydraulic section or most economical hydraulic section
So for most economical hydraulic section;
Wetted perimeter (P) = Minimum ⇒Q= maximum

Calculation:

Let us consider a rectangular channel section of width B, the depth of flow in the channel is y.

Full Test 2 (31-80) images Q.55

For P = minimum ⇒ \(\frac{dP}{dy}=0\)

Wetted perimeter P = B + 2y and Area A = B x y ⇒ \(B=\frac{A}{y}\)

\(P=\frac{A}{y}+2y\) and \(\frac{dP}{dy}=\frac{A}{y^2}+2\)

For maximum discharge, P=> minimum

\(\frac{dP}{dy}=0\) ⇒ \(\frac{dP}{dy}=\frac{A}{y^2}+2=0\) ⇒ \(A=2y^2\) ⇒\(B\times y=2y^2\) ⇒ \(B=2y\)

So, area A=2y²