Question

In the fully developed region of flow in a circular tube, will the velocity profile change in the flow direction? How about the temperature profile?

Short answer

Answer: No, in the fully developed region of flow in a circular tube, both the velocity profile and the temperature profile remain constant in the flow direction. This is because the flow has achieved a steady-state, wherein viscous forces and pressure forces are balanced for velocity, and convective heat transfer between the fluid and tube walls is in equilibrium for temperature.

Step-by-step solution

Understand Fully Developed Flow in a Circular Tube

A fully developed flow in a circular tube means the flow has reached a point where its properties, including velocity and temperature, no longer change in the axial direction along the tube. This occurs after a certain distance from the tube inlet, known as the entrance length, which depends on the Reynolds number.

Analyze the Velocity Profile

In a fully developed region of flow, by definition, the velocity profile does not change in the flow direction. This is because, after reaching the fully developed flow, the viscous forces and pressure forces are balanced, and the flow has achieved a steady-state velocity profile. Consequently, the velocity profile remains constant along the axial direction of the tube.

Analyze the Temperature Profile

Just like the velocity profile, the temperature profile in a fully developed flow also remains constant in the flow direction. It can be assumed that the viscous heating in the flow is negligible compared to the convective heat transfer between the fluid and the tube walls. Since at this steady-state, the heat transfer between fluid and tube walls is also in equilibrium, the temperature profile remains constant along the axial direction of the tube.

Conclusion

In the fully developed region of flow in a circular tube, the velocity profile doesn't change in the flow direction. The same applies to the temperature profile, which also remains constant along the axial direction of the tube.

Key concepts

Velocity Profile

Understanding the velocity profile is essential when studying fluid dynamics, especially within the context of flow through a pipe. When we refer to the fully developed flow in a circular tube, it indicates that the fluid has been flowing for a sufficient length, allowing its velocity profile to stabilize. What this means is that the layers of fluid move in a pattern that remains uniform along the tube's length, beyond a certain point known as the entrance region.

The profile itself is parabolic due to the no-slip condition at the tube walls, where the fluid has zero velocity, gradually increasing to the maximum at the center. The viscous forces acting on different layers of the fluid create this profile, and once fully developed, it's maintained because the forces are balanced. With this balanced state, there's no acceleration of fluid particles in the direction of flow, and the velocity distribution at any cross-section will be the same as at any other cross-section further along the tube.

Temperature Profile

Similar to the velocity profile, the temperature profile in a circular tube also stabilizes in fully developed flow. Initially, as the fluid enters the tube, it may have a uniform temperature. However, as it starts to interact with the tube walls, which might be at a different temperature, heat transfer begins. Over time, the fluid develops a temperature gradient from the wall to the center of the tube.

In the fully developed region, this temperature profile reaches an equilibrium and no longer changes as the fluid flows further down the tube. The distribution of temperature across any cross-section of the tube will mirror other sections, regardless of the distance from the inlet, assuming constant wall temperature or heat flux. It's important to note that this state is reached only after the thermal entrance length, which depends on both the flow rate and the thermal properties of the fluid.

Reynolds Number

The Reynolds number is a dimensionless quantity in fluid mechanics used to predict the flow regime within a pipe, be it laminar or turbulent. It's essentially a ratio of inertial forces to viscous forces and is given by the formula \(Re = \frac{\rho uD}{\mu}\), where \(\rho\) is the fluid density, \(u\) is the mean velocity, \(D\) is the tube diameter, and \(\mu\) is the dynamic viscosity of the fluid.

For flow in a pipe, a Reynolds number below 2100 typically indicates laminar flow, where fluid particles move in straight, parallel paths. Above 4000, the flow becomes turbulent, characterized by chaotic and eddying motions. The transitional flow occurs in the range between these values. The critical Reynolds number, the value at which flow transitions from laminar to turbulent, aids in determining the length required for a flow to become fully developed.

Convective Heat Transfer

Convective heat transfer in the context of flow in a circular tube is a mechanism through which heat is transported between the tube walls and the flowing fluid. This type of heat transfer is defined by the movement of fluid molecules and is crucial in understanding how the temperature stabilizes in a fully developed flow.

The heat transfer rate is governed by the temperature difference between the wall and the fluid, along with the properties of the fluid, like thermal conductivity, and the flow characteristics described by the Reynolds number and the Prandtl number. Convective heat transfer is quantified by the heat transfer coefficient, which encapsulates the aforementioned variables. The stability of the temperature profile is attributable to the balance between the thermal energy input or output at the walls and the thermal energy carried away by the fluid in motion.