Water enters a circular pipe of length L = 5.0 m and diameter D = 0.20 m with Reynolds number ReD = 500. The velocity profile at the inlet of the pipe is uniform while it is parabolic at the exit. The Reynolds number at the exit of the pipe is___________.

The correct answer is 500.

Step-by-Step Explanation:

1. Definition of Reynolds Number:
The Reynolds number ($\mathrm{Re}$) for flow through a circular pipe of diameter $D$ is defined as:
$\mathrm{Re}=\frac{\rho V_{\mathrm{avg}}D}{\mu }$
where:
• $\rho$ is the fluid density,
• $V_{\mathrm{avg}}$ is the average velocity of the fluid flow,
• $D$ is the inner diameter of the pipe,
• $\mu$ is the dynamic viscosity of the fluid.

2. Conservation of Mass (Continuity Equation):
For steady, incompressible flow of water through a pipe of constant cross-sectional area (constant diameter $D$), the mass flow rate ($\stackrel{·}{m}$) must be conserved between the inlet and the exit:
${\stackrel{·}{m}}_{\mathrm{inlet}}={\stackrel{·}{m}}_{\mathrm{exit}}$
Since the fluid is water (incompressible) and the cross-sectional area $A$ is constant:
$V_{\mathrm{avg},\mathrm{inlet}}=V_{\mathrm{avg},\mathrm{exit}}$
Thus, the average velocity does not change along the length of the pipe.

3. Independence of Velocity Profile:
At the inlet, the velocity profile is uniform, and at the exit, it develops into a parabolic profile. While the local velocity distribution (shape of the profile) changes, the average velocity $V_{\mathrm{avg}}$ remains constant because the total volumetric flow rate must be conserved.

Since the density ($\rho$), dynamic viscosity ($\mu$), pipe diameter ($D$), and average velocity ($V_{\mathrm{avg}}$) are all constant from inlet to exit, the Reynolds number does not change:
${\mathrm{Re}}_{D,\mathrm{exit}}={\mathrm{Re}}_{D,\mathrm{inlet}}=500$