Discussion
Consider the steady, incompressible, and fully developed laminar flow of a fluid through a circular pipe. Here, \(\Delta P\) is the pressure drop in the direction of the flow and \(V\) is the average axial velocity of the fluid at any cross-section. The relation between \(\Delta P\) and \(V\) is:
\[ \Delta P = K V^{n} \]
Here, \(K\) and \(n\) are constants.
Which one of the following options is the correct value of \(n\)?
Answer: A
The exponent is \(n = 1\).
Fully developed laminar flow in a circular pipe obeys the Hagen-Poiseuille equation:
\[ \Delta P = \frac{32\mu L V}{D^{2}} \]
The pressure drop is directly proportional to the average velocity, so \(K = 32\mu L/D^{2}\) and \(n = 1\). Equivalently, the laminar friction factor \(f = 64/Re\) falls as \(1/V\), cancelling one power of \(V\) in the Darcy-Weisbach \(V^{2}\).
\(n = 2\) is the fully rough turbulent limit, \(n = 1.75\) is smooth-pipe turbulent flow (Blasius friction factor), and 0.5 has no physical basis here.
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