Question

Someone claims that the average velocity in a circular pipe in fully developed laminar flow can be determined by simply measuring the velocity at \(R / 2\) (midway between the wall surface and the centerline). Do you agree? Explain.

Short answer

Answer: No, the claim is not valid. The velocity at R/2 is not equal to the average velocity in a circular pipe with fully developed laminar flow.

Step-by-step solution

Understand laminar flow in a circular pipe

Laminar flow occurs when fluid flows in parallel layers with no disruption between them. In a circular pipe, the velocity of the flow varies across the cross-sectional area, with the maximum flow at the centerline and minimum (zero) at the wall of the pipe.

Identify the velocity profile for fully developed laminar flow

The velocity profile for a fully developed laminar flow in a circular pipe can be described by the Hagen-Poiseuille equation: \(\frac{u}{U} = 1-\frac{r^2}{R^2}\) where \(u\) represents the velocity at any radial distance \(r\) from the centerline, \(U\) is the maximum velocity at the centerline, and \(R\) is the radius of the pipe.

Calculate the velocity at \(R / 2\)

To analyze the claim, we need to find the value of \(u\) at \(R / 2\). By substituting \(r = R / 2\) in the Hagen-Poiseuille equation, we get: \(\frac{u_{R/2}}{U}=1-\frac{(R/2)^2}{R^2}=1-\frac{1}{4}=\frac{3}{4}\) This means that the velocity at \(R / 2\) is \(3/4\) times the maximum velocity at the centerline, or \(u_{R/2} = \frac{3}{4}U\).

Calculate the average velocity

The average velocity (\(U_{avg}\)) in fully developed laminar flow can be calculated using the following relation: \(U_{avg} = \frac{1}{2}U\)

Compare the velocity at \(R / 2\) with the average velocity

From Steps 3 and 4, we have: \(u_{R/2} = \frac{3}{4}U\) and \(U_{avg} = \frac{1}{2}U\) Comparing the two, it is clear that: \(u_{R/2} \neq U_{avg}\) Thus, the claim that the average velocity in a circular pipe in fully developed laminar flow can be determined by simply measuring the velocity at \(R / 2\) is not valid.