Question

An ideal gas passes isothermally through a long horizontal uniform cross-section pipe under steady flow. Consider that the pressure gradient in the pipe is sufficient for a finite change in the density of the gas. If the flow of the gas is purely pressure driven and subsonic throughout, then the average flow velocity

• A. increases along the flow
• B. decreases along the flow
• C. does not change throughout the pipe
• D. increases in the hydrodynamic entrance region and then decreases thereafter

Hint:

Use mass conservation and see how density must fall along the pipe because of friction.

Answer 1

The Correct Option is A

Step 1: Write the mass conservation condition.
For steady flow through a pipe of uniform cross-section area \(A\), the mass flow rate \(\dot{m} = \rho V A\) must stay the same at every section, since mass cannot build up or vanish inside.

Step 2: See how density changes along the pipe.
Friction in a long pipe steadily drops the pressure as the gas moves along it. Since the flow is isothermal, the ideal gas law \(P = \rho R T\) at constant \(T\) means \(\rho\) is directly proportional to \(P\). So as \(P\) falls along the pipe, \(\rho\) falls too.

Step 3: Combine mass conservation with the falling density.
From \(\rho V A = \text{constant}\) and \(A\) fixed, \(V\) must go up whenever \(\rho\) goes down, to keep the product \(\rho V\) unchanged.

Final Answer:
Since density keeps dropping along the pipe from the pressure drop, the average flow velocity keeps rising along the flow. \[ \boxed{V \text{ increases along the flow}} \]