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?

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.

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.

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.

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.

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.