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

This is a compressible pipe flow question in disguise. Even with an ideal gas and an isothermal process, letting density change means velocity cannot stay fixed just because the pipe cross section is uniform.

1. Increases along the flow: correct. Friction drops pressure along the pipe, and at constant temperature the ideal gas law ties density directly to pressure, so density falls too. Mass flux $\rho V A$ must stay constant with $A$ fixed, so a falling $\rho$ forces $V$ to climb.
2. Decreases along the flow: wrong, this would need density to rise along the pipe, but pressure and hence density can only fall as friction acts on the flow.
3. Does not change throughout the pipe: wrong, that only holds for genuinely incompressible flow with constant density, but the question states the density undergoes a finite change.
4. Rises then falls after the entrance region: wrong, that describes a developing boundary layer profile, not the bulk average velocity trend driven by density change in this long, pressure driven pipe.

Mass conservation with falling density along the pipe means the average velocity must increase along the flow, option A.