Question

Consider laminar natural convection from a vertical hot-plate. Will the heat flux be higher at the top or at the bottom of the plate? Why?

Short answer

Answer: The heat flux is higher at the bottom of the vertical hot-plate due to the boundary layer formation and the varying temperature difference between the plate and fluid which impacts the heat transfer rate. As we move from the bottom to the top of the plate, the boundary layer thickness grows, and the temperature difference decreases, resulting in a reduced heat transfer rate at the top of the plate.

Step-by-step solution

Understand Laminar Natural Convection

Laminar natural convection occurs when the fluid motion is driven by buoyancy forces. The fluid is heated near the vertical hot-plate, causing it to expand and become less dense. Due to this density gradient, the warmer fluid rises, and the cooler fluid takes its place. This process of heat transfer is the natural convection.

Gravitational Effects

In natural convection, it is essential to account for the gravitational force acting in the vertical direction. The fluid motion is driven by the buoyancy force, which acts opposite to gravity. Due to the continuous rise in the fluid, the heat transfer efficiency is expected to vary along the vertical hot plate.

Boundary Layer Formation

As the fluid comes in contact with the hot-plate, it forms a boundary layer. The boundary layer thickness, denoted by δ, refers to how far the fluid velocity has reached from 0% to 99% of the free-stream velocity. In the case of natural convection, the boundary layer thickness increases along the plate from the bottom to the top.

Heat Flux Calculation

To predict the heat flux at different positions along the plate, we must compute the non-dimensional parameter- the Nusselt number (Nu). The relation between Nusselt number, Reynolds number (Re), and Prandtl number (Pr) is given by: Nu = f(Re, Pr) where, Nu = \frac{hL}{k} h = heat transfer coefficient L = characteristic length k = thermal conductivity The heat flux (q) along the plate can then be calculated using the heat transfer coefficient, h, and the temperature difference, ΔT: q = hΔT.

Comparing Heat Flux at the Top and Bottom

Since the boundary layer thickness grows along the direction of the flow, the heat transfer rate decreases as we move from the bottom to the top of the plate. The fluid near the bottom of the plate will be denser than the fluid at the top, which means that the heat transfer rate is higher at the bottom than at the top. Conclusion: After examining the concepts of laminar natural convection, boundary layer formation, and heat flux calculation, we can conclude that the heat flux would be higher at the bottom of the vertical hot-plate, as the boundary layer and its thickness have a significant impact on heat transfer. The temperature difference between the plate and the fluid decreases as you move from the bottom to the top, resulting in a reduced heat transfer rate at the top of the plate.

Key concepts

Boundary Layer Formation

In the context of laminar natural convection, the boundary layer is a crucial concept. When a fluid comes into contact with a hot surface, such as a vertical hot-plate, a thin layer of fluid known as the boundary layer forms. This layer is where most of the viscous effects occur and plays a significant role in heat transfer. There are a few things to keep in mind about the boundary layer:

• Formation: The boundary layer is formed due to the interaction between the fluid and the heated surface. This interaction causes the fluid closest to the surface to slow down, resulting in a velocity gradient perpendicular to the surface.
• Thickness: As we move upward along the plate, the boundary layer becomes thicker because the fluid has more time to interact with the heated surface.
• Impact on Heat Transfer: A thicker boundary layer generally reduces the rate of heat transfer because the temperature gradient between the surface and the fluid decreases.

This concept is essential to understanding why the heat flux varies along the length of the hot-plate.

Heat Flux

Heat flux is a measure of how much heat energy is transferred per unit area over time. In laminar natural convection, heat flux is influenced by several factors, including the boundary layer properties and temperature differences.

Key points about heat flux in this context:

• Dependence on Temperature Difference: The heat flux is directly proportional to the temperature difference between the hot surface and the cooler fluid.
• Calculation: Heat flux ( q ) can be calculated using the formula, q = hΔT, where is the heat transfer coefficient, and ΔT is the temperature difference.
• Variability Along the Plate: Due to the varying thickness of the boundary layer, heat flux is not uniform along the plate. It’s generally higher at the bottom where the boundary layer is thinner, and reduces as one moves upwards.

This variability is important for predicting how effectively heat is transferred in different sections of the hot-plate.

Nusselt Number

The Nusselt number (Nu) is a dimensionless number that is important in the context of heat transfer. It represents the ratio of convective to conductive heat transfer across a boundary.

Understanding the Nusselt Number:

• Role: It helps determine the efficiency of heat transfer in convective systems. A higher Nu indicates that convection dominates over conduction, meaning better heat transfer.
• Formula: Nu is calculated as Nu = \( \frac{hL}{k} \), where is the heat transfer coefficient, L is the characteristic length, and is the thermal conductivity of the fluid.
• Relation to Other Numbers: It is related to the Reynolds number (Re) and Prandtl number (Pr) through empirical correlations, allowing for the prediction of heat transfer in various flow situations.

Understanding Nu is key when analyzing heat transfer rates in systems like the vertical hot-plate.

Gravitational Effects

Gravitational effects are a driving force behind natural convection. In natural convection, fluid motion results from buoyancy forces that are directly influenced by gravity.

Here's how gravity affects this process:

• Buoyancy Force: When a fluid is heated, it expands and its density decreases. This less-dense, warmer fluid rises due to the buoyancy force, which acts against gravity.
• Variable Heat Transfer: As the warm fluid rises, it creates a continuous flow pattern, allowing cooler fluid to take its place near the heat source like a vertical hot-plate. This flow is critical for maintaining heat transfer in the system.
• Influence on Heat Flux: Due to the gravitational pull, the efficiency of heat transfer changes from the bottom to the top of the plate. At the top, the heated fluid has already risen and spent the majority of its energy, resulting in a lower heat transfer rate compared to the bottom.

Gravitational effects are essential for understanding the dynamics of fluid flow and heat transfer in natural convection scenarios.