Question

Consider laminar flow in a circular tube. Will the friction factor be higher near the inlet of the tube or near the exit? Why? What would your response be if the flow were turbulent?

Short answer

Answer: In laminar flow, the friction factor is higher near the inlet and lower near the exit due to the developing velocity profile. In turbulent flow, there is no significant difference in the friction factor near the inlet and exit of the tube.

Step-by-step solution

Understand laminar flow and friction factor

Laminar flow is a type of flow in which fluid particles move in smooth, parallel layers without much mixing between layers. In laminar flow, the friction factor depends on the Reynolds number (Re), which is the ratio of inertial forces to viscous forces within the fluid. The friction factor (f) for a circular tube can be calculated using the Hagen-Poiseuille equation: f = \frac{16}{Re} where Re = \frac{ρuD}{μ}, with ρ being the fluid density, u the fluid velocity, D the diameter of the tube, and μ the dynamic viscosity of the fluid.

Discuss friction factor in laminar flow near the inlet and exit

As fluid enters the tube, the velocity profile is not fully developed, which means that the velocity near the wall is higher compared to a fully developed flow. Since the friction factor depends on the Reynolds number (which involves fluid velocity), it might vary near the inlet of the tube. As the flow becomes fully developed further downstream, the velocity profile becomes parabolic, and the friction force between the layers becomes more constant and uniform. Based on this, we expect the friction factor to be higher near the inlet of the tube and lower near the exit in laminar flow.

Understand turbulent flow and its effect on the friction factor

Turbulent flow is characterized by the random and chaotic motion of fluid particles, leading to significant mixing between layers. The friction factor in turbulent flow also depends on the Reynolds number, but the relationship is more complex and can be approximated by empirical formulas such as the Colebrook equation: \frac{1}{\sqrt{f}} = -2\log_{10}\left(\frac{\varepsilon}{3.7D} + \frac{2.51}{Re\sqrt{f}}\right) where ε is the roughness of the tube.

Discuss friction factor in turbulent flow near the inlet and exit

In turbulent flow, the velocity profile is more uniform throughout the tube, and the flow is dominated by random fluctuations. The friction factor depends on both the Reynolds number and the roughness of the tube walls. Therefore, the friction factor would not show significant variation near the inlet and exit of the tube for turbulent flow, but the overall friction factor would be higher in turbulent flow compared to laminar flow due to increased mixing and energy dissipation. In conclusion, for laminar flow, the friction factor is higher near the inlet and lower near the exit, whereas for turbulent flow, there is no significant difference in the friction factor near the inlet and exit.