Question

Consider laminar forced convection in a circular tube. Will the heat flux be higher near the inlet of the tube or near the exit? Why?

Short answer

Answer: The heat flux is higher near the inlet of the circular tube than near the exit in a laminar forced convection scenario. This is because the heat transfer coefficient decreases along the flow due to the increasing thickness of the boundary layer, which reduces the fluid's ability to pick up heat from the tube's wall.

Step-by-step solution

Define Laminar Forced Convection

Laminar forced convection is a flow regime where an external force drives the fluid flow through a conduit or channel, and the fluid motion is smooth and uniform. In this case, the fluid flows through a circular tube where heat is transferred between the fluid and the tube's walls.

Understand Heat Flux

Heat flux is the rate of heat energy transferred per unit area. In the context of forced convection, it is determined by the heat transfer coefficient (h), the flow velocity, and the temperature difference between the fluid and the tube wall. Mathematically, heat flux (q) can be expressed as: \[q = h * (T_{wall} - T_{fluid})\]

Study Heat Transfer Behaviour Along the Tube

As the fluid flows along the tube, the heat transfer coefficient generally changes with the flow. In laminar flow, the heat transfer coefficient decreases along the flow because the boundary layer, where the fluid is slower and more thermally conductive, thickens as the fluid moves from the inlet to the exit of the tube. Consequently, the ability of the fluid to pick up heat from the tube's wall is reduced.

Determine Heat Flux Near Inlet and Exit

Since the heat flux (q) depends on the heat transfer coefficient (h), a higher h near the inlet will result in a higher heat flux near the inlet, while a lower h near the exit will result in a lower heat flux. Provided that the temperature difference between the fluid and the wall remains relatively constant, these differences become evident in the heat flux values along the tube.

Conclusion

The heat flux is higher near the inlet of the circular tube than near the exit in a laminar forced convection scenario. The reason is that the heat transfer coefficient decreases along the flow due to the thickness increase of the boundary layer, which reduces the ability of the fluid to pick up heat from the tube's wall.

Key concepts

Heat Flux

In the realm of physics and engineering, heat flux plays a key role in understanding how energy is transferred in the form of heat. Heat flux is essentially the rate at which heat energy passes through a given surface area. Imagine the warmth from a radiator spreading across a room—that's heat energy on the move, and by measuring the area of the radiator, you could calculate the heat flux.

Mathematically, it is represented as \( q = h * (T_{wall} - T_{fluid}) \), where \(h\) is the heat transfer coefficient, \(T_{wall}\) is the temperature of the tube wall, and \(T_{fluid}\) is the temperature of the fluid. In a scenario like laminar forced convection within a circular tube, as the fluid starts at the inlet, it has a higher capacity for heat absorption due to a thinner boundary layer, which leads to a higher heat flux. As the fluid travels towards the exit, its ability to absorb heat decreases, resulting in a lower heat flux.

Heat Transfer Coefficient

Diving deeper into the concept of heat flux, we encouter the heat transfer coefficient, symbolized as \(h\). This coefficient is a critical factor in the equation and represents the efficiency with which heat can be transferred between two mediums—in this case, from the tube wall to the flowing fluid. A higher heat transfer coefficient signifies that the transfer of heat is more efficient.

The determination of \(h\) is complex; it involves not just the material properties and the temperature gradient, but also the flow characteristics of the fluid. For example, in a laminar flow situation, the heat transfer coefficient is higher at the inlet where the flow is undeveloped and the boundary layer is thin. This coefficient tends to decrease along the length of the tube as the boundary layer grows thicker, which means less heat is transferred from the wall to the fluid per unit area as you move towards the tube's exit.

Boundary Layer

The boundary layer is a crucial concept in fluid dynamics and heat transfer. It refers to the thin layer of fluid that is in direct contact with the boundary surface of an object, like a tube wall. The behavior of a fluid within the boundary layer is significantly different from the fluid in the main flow away from the wall, primarily due to friction.

In forced convection scenarios, like the flow of fluid in a tube, the boundary layer starts to develop at the inlet and grows in thickness as the fluid progresses along the tube’s length. The increased thickness affects the fluid’s thermal conductivity—meaning its ability to transport heat. In essence, the thicker the boundary layer, the more resistant it is to heat transfer, as seen in the gradual decrease in the heat transfer coefficient from the tube inlet to the exit.

Laminar Flow Behavior

Laminar flow refers to a fluid motion in a smooth, orderly manner, where the fluid flows in parallel layers, with minimal mixing or disruption between them. It contrasts with turbulent flow, which is characterized by chaotic eddies and swirls.

Laminar flow behavior is essential in assessing heat transfer in scenarios like that inside a circular tube in forced convection. At the inlet of the tube, the flow begins laminar with a flat velocity profile. As the flow continues, it develops a parabolic velocity profile because of the no-slip condition at the wall, leading to a velocity gradient that stimulates the growth of the boundary layer. This behavior is predictable and steady, and it allows for precise calculations of heat transfer phenomena using established principles of fluid dynamics and thermodynamics.