Question

How is the thermal entry length defined for flow in a tube? In what region is the flow in a tube fully developed?

Short answer

Answer: The relationship between thermal entry length and fully developed flow in a tube is that thermal entry length determines the point at which the temperature profile fully develops in the flow. When the flow reaches the thermal entry length, the temperature profile will no longer change and remain constant along the tube's length, corresponding to the fully developed flow in the tube.

Step-by-step solution

Definition of Thermal Entry Length

The thermal entry length is defined as the distance from the tube entrance where the temperature profile reaches a steady state or fully developed the condition. Specifically, it is the distance required for the temperature profile to remain unchanged in the axial direction, which means the axial heat conduction becomes negligible.

Fully Developed Flow in a Tube

In a fluid flowing through a tube, fully developed flow refers to a state where both the velocity and the temperature profiles have reached their steady states. For a fully developed flow in a tube, the fluid motion is entirely parallel to the tube walls and exhibits the following characteristics: 1. No variation in velocity profile across the axial direction of the tube (axial velocity gradient is zero). 2. No acceleration or deceleration of fluid particles as they move along the tube (radial velocity gradient is zero). 3. Constant temperature profile throughout the entire length of the tube (no axial temperature gradient).

Connection between Thermal Entry Length and Fully Developed Flow

The thermal entry length is related to the fully developed flow because it determines the point at which the temperature profile fully develops in the flow. When the flow reaches the thermal entry length, the temperature profile will no longer change and remain constant along the tube's length. This corresponds to the fully developed flow in the tube.

Key concepts

Fully Developed Flow

In the context of fluid flow through a tube, "fully developed flow" refers to a state of balance in both velocity and temperature profiles. Here's what you need to understand:

• For velocity, it means that the fluid moves in a steady way, parallel to the tube walls.
• There is no speed change in the axial direction, or along the length of the tube.
• The radial direction, or across the tube, also shows constant flow behavior.
• Temperature-wise, a fully developed flow has no temperature changes along the axial direction.

This balanced state occurs after the fluid has traveled some distance into the tube, past any initial disturbances at the tube entrance. At this distance, known as the thermal entry length, the system has adjusted to the tube's conditions and flows uniformly.

Temperature Profile

The temperature profile in a tube describes how heat distributes across the fluid as it flows. Here's what you need to know:

• Initially, when the fluid enters the tube, the temperature profile of the fluid is not uniform.
• It gradually changes while moving down the tube, trying to reach an equilibrium.
• As it reaches the thermal entry length, the temperature differences across the flow become stable and predictable, forming a steady profile.
• A steady temperature profile is one where the heat does not change with more travel along the tube.

Reaching a steady temperature profile means the heat is equally spread out, leading to efficient thermal transfer. This consistency helps in better thermal management in engineering processes.

Axial Direction

The axial direction refers to the lengthwise movement of fluid as it flows through the tube:

• This direction is crucial in understanding how flow and heat behave over a distance.
• In fully developed flow, both the velocity and temperature remain unchanged along this axial path.
• No variations mean there's no heat being conducted along the length of the tube anymore, simplifying calculations and predictions.

Understanding the axial direction helps engineers design systems where the thermal characteristics are predictable and manageable. It ensures efficient and steady flow of heat energy along the tube's length.

Heat Conduction

Heat conduction in a tube is the process by which thermal energy is transferred through the material:

• Early in the flow, heat is conducted both in the radial direction (in and out of the tube walls) and axially (along the tube length).
• As the flow reaches the thermal entry length, axial conduction becomes negligible.
• At this point, the primary heat transfer occurs radially, ensuring stable temperature profiles.
• Effective conduction management within the tube ensures that the system operates efficiently, minimizing energy loss.

By understanding heat conduction, particularly the change from axial to radial focus as the flow develops, engineers can optimize tubes for heating and cooling applications.