Liquid mercury at \(250^{\circ} \mathrm{C}\) is flowing with a velocity of \(0.3 \mathrm{~m} / \mathrm{s}\) in parallel over a \(0.1-\mathrm{m}\)-long flat plate where there is an unheated starting length of \(5 \mathrm{~cm}\). The heated section of the flat plate is maintained at a constant temperature of \(50^{\circ} \mathrm{C}\). Determine \((a)\) the local convection heat transfer coefficient at the trailing edge, \((b)\) the average convection heat transfer coefficient for the heated section, and \((c)\) the rate of heat transfer per unit width for the heated section.

The Reynolds number is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It compares the relative significance of viscous forces and inertial forces and is vital in determining whether a fluid flow is laminar or turbulent. In the exercise, the Reynolds number calculation involves properties of liquid mercury at a high temperature and a defined velocity flowing over a flat plate.

The formula for the Reynolds number over a flat plate is expressed as: \[ Re_{x}=\frac{\rho u x}{\mu} \] where \(\rho\) is the fluid density, \(u\) the flow velocity, \(x\) the distance along the plate, and \(\mu\) the dynamic viscosity. The solution shows that the calculated Reynolds number is well below the threshold for turbulent flow, indicating laminar flow conditions.

The calculation of the Reynolds number is critical in the subsequent steps of determining the heat transfer coefficients, as the flow type impacts the correlations used for these calculations.

After calculating the Reynolds number, the exercise involves analyzing the flow regime. Laminar flow, characterized by smooth and orderly fluid motion, is confirmed when the Reynolds number falls below a certain threshold, typically around 500,000 for flow over a flat plate. Laminar flow is significant for several reasons:

• It affects the local and average convection heat transfer coefficients since different empirical correlations are used for laminar and turbulent flows.
• It's associated with lower frictional losses compared to turbulent flow.
• Heat transfer in laminar flow relies strongly on the temperature gradient and thermal conductivity of the fluid.

In the solution, the presence of laminar flow over the heated section of the flat plate leads to the utilization of specific formulas for heat transfer coefficient calculations tailored to laminar conditions.

The Prandtl number (Pr) is a dimensionless number that signifies the ratio of momentum diffusivity to thermal diffusivity, essentially comparing the rate at which momentum and heat are transported within a fluid. In the context of the exercise, the Prandtl number is a crucial parameter for calculating the local convection heat transfer coefficient in a laminar flow situation.

The Prandtl number is calculated with fluid properties, including viscosity and thermal conductivity, and for liquid mercury at \(250^\circ C\), it is given as a very low value. This implies that thermal diffusivity is significantly higher than momentum diffusivity, a characteristic that influences the convective heat transfer in the fluid. The Prandtl number is used in the formula: \[ h_{x}=\frac{0.332k}{x}Re_{x}^{1/2}Pr^{1/3} \] illustrating its impact on heat transfer alongside the Reynolds number and thermal conductivity.

Thermal conductivity (\(k\)) measures a material's ability to conduct heat. In the exercises' context, the thermal conductivity of liquid mercury at a specific temperature significantly influences the heat transfer process. It appears in the formulas used for calculating the local and average convection heat transfer coefficients.

High thermal conductivity indicates that the material can transfer heat efficiently. For substances like liquid mercury with high thermal conductivity, heat is rapidly transferred through the fluid, which increases the convection heat transfer coefficient. The property is thus integral to determining the effectiveness with which heat can be exchanged between the mercury and the flat plate in the given scenario.

The heat transfer rate calculation is a critical step in thermal engineering applications, as it determines the amount of heat energy transferred per unit time and area. In our exercise, we calculate the rate of heat transfer per unit width of the heated section of the flat plate, with the local and average convection heat transfer coefficients already determined.

The heat transfer rate per unit width (\(q''_{x}\)) is given by the formula: \[ q''_{x} = h_{avg}(T_{s} - T_{\infty}) \] where \(h_{avg}\) is the average convection heat transfer coefficient, \(T_{s}\) is the surface temperature, and \(T_{\infty}\) is the free-stream fluid temperature. The resulting value provides insight into the thermal performance of the heated surface. The calculation underscores the importance of understanding the mechanisms of heat exchange and being able to estimate the heat transfer rates for practical thermal management.