Question

Given below are two statements:
Statement (I): A rectangular channel will be most economical when width is same as depth of flow.
Statement (II): In case of compressible flow, density of the fluid remains constant during flow.

In light of the above statements, choose the most appropriate answer from the options given below.

• A. Both Statement (I) and Statement (II) are correct.
• B. Both Statement (I) and Statement (II) are incorrect.
• C. Statement (I) is correct but Statement (II) is incorrect.
• D. Statement (I) is incorrect but Statement (II) is correct.

Hint:

For an open rectangular channel, the most economical section occurs when the cross-section represents a semi-square, meaning width \(b = 2 \times \text{depth } y\).
For compressible flow, think of air, whose density changes dynamically under pressure gradients.

Answer 1

The Correct Option is B

Step 1: Check Statement I using the most economical channel condition.
A rectangular open channel is most hydraulically economical when its wetted perimeter is minimized for a given flow area, and working through this optimization shows the condition is reached when the base width equals twice the depth of flow, \( b = 2y \), not when the width equals the depth. So Statement I, which claims width equal to depth, is incorrect.
Step 2: Check Statement II using the definition of compressible flow.
Compressible flow is, by definition, flow in which fluid density changes appreciably in response to pressure changes, such as high-speed gas flow. A flow where density stays constant throughout is instead called incompressible flow. So Statement II has the definition backwards, and is also incorrect.
Step 3: Combine the two findings.
Since neither statement holds up when checked against the standard hydraulic and fluid mechanics definitions, both must be marked incorrect together.
\[ \boxed{\text{Both Statement (I) and Statement (II) are incorrect.}} \]