Question

How does the friction factor \(f\) vary along the flow direction in the fully developed region in (a) laminar flow and (b) turbulent flow?

Short answer

Answer: In fully developed laminar flow, the friction factor (f) is constant along the flow direction. In fully developed turbulent flow, the friction factor (f) varies with pipe roughness and diameter along the flow direction.

Step-by-step solution

(a) Laminar Flow

In laminar flow, the fluid particles move in parallel layers, with little to no mixing or fluctuation. Laminar flow is characterized by a low Reynolds number (typically below 2000). In this type of flow, the fluid layers smoothly slide across each other, resulting in a constant friction factor along the flow direction in the fully developed region. In a fully developed laminar flow through a pipe, the friction factor (f) can be calculated using the Hagen-Poiseuille equation: \[f = \frac{16}{Re}\] Here, Re is the Reynolds number. Since the Reynolds number remains constant in the fully developed region of laminar flow, the friction factor (f) is also constant.

(b) Turbulent Flow

In turbulent flow, the fluid particles have random and chaotic motion, resulting in significant mixing and non-uniform velocities. Turbulent flow is characterized by a high Reynolds number (typically above 4000). In the fully developed region of a turbulent flow through a pipe, the friction factor (f) depends on the pipe's roughness and the Reynolds number, which can be calculated from the Moody Chart or using the following empirically derived Colebrook equation: \[\frac{1}{\sqrt{f}} = -2\log_{10} \left(\frac{k_s}{3.7D} + \frac{2.51}{Re\sqrt{f}}\right)\] Here, \(k_s\) is the pipe wall roughness, and D is the pipe diameter. Note that the Reynolds number can still be assumed constant in the fully developed region of turbulent flow, but due to the different processes occurring in a turbulent flow, the friction factor (f) is no longer constant. As per the given equation, it varies with the changing values of pipe roughness and diameter along the flow direction. In summary: - In fully developed laminar flow: the friction factor (f) is constant along the flow direction. - In fully developed turbulent flow: the friction factor (f) varies with pipe roughness and diameter along the flow direction.