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: Understanding the Concept:
We analyze the geometric conditions for the most economical open channel cross-sections and the basic properties defining compressible fluid flows.

Step 3: Detailed Explanation:
1.

Statement (I): A rectangular channel is most economical when its hydraulic radius \(R\) is half the depth of flow (\(R = y/2\)).
This condition is satisfied when the width of the channel (\(b\)) is twice the depth of flow (\(b = 2y\)).
Since the statement claims the channel is most economical when the width is equal to the depth of flow (\(b = y\)), Statement (I) is incorrect.
2.

Statement (II): In compressible fluid flow, the density of the fluid (\(\rho\)) changes significantly throughout the flow field as pressure changes.
For constant density flows, the flow is classified as incompressible.
Therefore, Statement (II) is incorrect.

Step 4: Final Answer:
The correct option is 2, which corresponds to "Both Statement (I) and Statement (II) are incorrect".