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 a laminar flow, the friction factor is constant along the length of the tube and does not depend on the position, either near the inlet or the exit of the tube. In a turbulent flow, the friction factor will be higher near the exit of the tube than near the inlet due to fully developed turbulence and eddies.

Step-by-step solution

Understand Laminar and Turbulent Flows

In laminar flow, the fluid particles move in parallel layers without any mixing or significant turbulence. The flow is characterized by a low Reynolds number (Re < 2000). The laminar flow is smooth and well ordered. On the other hand, turbulent flow occurs when the fluid experiences chaotic, unpredictable behavior and significant mixing. The flow is characterized by a high Reynolds number (Re > 4000). Turbulent flow is disorderly and full of eddies and swirls.

Define Friction Factor

The friction factor (f) is a dimensionless quantity representing the ratio of shear stress at the tube wall to the dynamic pressure. It indicates the resistance to flow caused by wall friction. In pipe flow, the friction factor depends on the flow regime (laminar or turbulent), and it is also a function of the Reynolds number and the roughness of the pipe wall.

Discuss Friction Factor in Laminar Flow

In laminar flow, the fluid layers slide over one another, and the resistance to flow is mainly due to viscous forces. The friction factor for laminar flow in a circular pipe is given by the Hagen-Poiseuille equation: f = 16 / Re where Re is the Reynolds number. Since the Reynolds number is constant along the length of the tube, we can conclude that the friction factor is also constant in laminar flow and does not depend on the position, either near the inlet or the exit of the tube.

Discuss Friction Factor in Turbulent Flow

In turbulent flow, the resistance to flow is primarily due to turbulence and eddies. Near the inlet of the tube, the flow may not have fully developed its turbulence, which can make the local friction factor slightly lower than in the fully developed turbulent flow region. Contrarily, near the exit, the turbulent flow will be fully developed, and the friction factor will be higher. So, the friction factor will be higher near the exit of the tube in a turbulent flow than near the inlet.

Conclusion

In laminar flow, the friction factor is constant along the length of the tube and does not depend on the position, either near the inlet or the exit of the tube. However, in turbulent flow, the friction factor will be higher near the exit of the tube than near the inlet, as the flow becomes fully developed and experiences greater turbulence and eddies.

Key concepts

Laminar Flow

Laminar flow occurs when a fluid flows in parallel layers with no disruption between them. This type of flow is typically smooth and consistent, where the fluid particles move along well-defined paths. The layers glide over one another, creating minimal mixing. This occurs when the flow has a Reynolds number (Re) less than 2000. In this regime, the viscous forces are dominant, and the flow is orderly.

A familiar example of laminar flow is the movement of honey out of a jar. You can observe how it flows in straight, consistent streams. This occurs because of the high viscosity of the fluid, which slows the movement and maintains the organized flow. Due to its predictable patterns, laminar flow is less resistant to changes and is generally more energy-efficient compared to turbulent flow.

Turbulent Flow

Turbulent flow is characterized by chaotic changes in pressure and flow velocity. Unlike laminar flow, turbulent flow involves random eddies, fluctuations, and swirls, leading to significant mixing within the fluid. This type of flow typically occurs at high Reynolds numbers (Re > 4000). In turbulent flow, the inertial forces overpower the viscous forces, causing disorder.

Think about the water flowing down a steep hill. It becomes unpredictable and swirls around, carrying along any small debris it encounters. This increased mixing and irregularity result in higher energy loss and resistance in the fluid. Consequently, turbulent flow can cause more friction in pipes and conduits, especially in sections where the turbulence is fully developed.

Friction Factor

The friction factor is a crucial dimensionless figure that describes the resistance to flow within a pipe due to wall friction. It essentially measures how easily a fluid can flow through a pipe, and it varies depending on whether the flow is laminar or turbulent.

For laminar flow, the friction factor is calculated using the Hagen-Poiseuille equation, which is expressed as: \[ f = \frac{16}{Re} \] Here, as the Reynolds number remains constant along the pipe in laminar flow, the friction factor does not change along the pipe's length.

In turbulent flow, the friction factor is influenced by both the Reynolds number and the roughness of the pipe's interior surface. Initially, it may be lower when turbulence is not fully established near the inlet, but it typically increases as the flow becomes fully turbulent towards the pipe's exit.

Reynolds Number

The Reynolds number is a fundamental concept in fluid mechanics used to predict the flow regime in different situations. It's a dimensional analysis value that compares the inertial forces to the viscous forces within a fluid. It is calculated as: \[ Re = \frac{\rho u D}{\mu} \] where \(\rho\) is the density of the fluid, \(u\) is the flow velocity, \(D\) is the characteristic length, and \(\mu\) is the dynamic viscosity.

With values less than 2000, flows are typically laminar. As the value increases beyond 4000, the flow is classified as turbulent. Values in between these ranges can indicate transitional flow where some regions of flow may begin to exhibit turbulent characteristics while others remain laminar.

Understanding the Reynolds number is vital for predicting flow behaviors. It helps engineers and scientists determine the appropriate design and analysis strategy for piping systems, as well as anticipating changes in resistance and efficiency as flow conditions shift.