Consider a horizontal \(0.7\)-m-wide and \(0.85\)-m-long plate in a room at \(30^{\circ} \mathrm{C}\). Top side of the plate is insulated while the bottom side is maintained at \(0^{\circ} \mathrm{C}\). The rate of heat transfer from the room air to the plate by natural convection is (a) \(36.8 \mathrm{~W}\) (b) \(43.7 \mathrm{~W}\) (c) \(128.5 \mathrm{~W}\) (d) \(92.7 \mathrm{~W}\) (e) \(69.7 \mathrm{~W}\) (For air, use \(k=0.02476 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7323, v=\) \(\left.1.470 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right)\)

The Grashof number (Gr) is a dimensionless value in fluid dynamics and heat transfer that is essential to the analysis of natural convection scenarios. Natural convection occurs when a fluid moves due to density differences that result from temperature gradients within the fluid. In the context of the plate example, the Grashof number helps us predict how much the difference in temperature between the cold plate and the warmer surrounding air will drive the air motion due to buoyancy forces.

To calculate Gr, an understanding of several factors is necessary: the acceleration due to gravity, the characteristic length of the object, the temperature difference between the object's surface and the ambient fluid, and the fluid's kinematic viscosity. To help students grasp this concept, it's pivotal to emphasize the importance of using a consistent set of units, such as the International System of Units (SI), converting all temperatures to Kelvin, and expressing all numerical values to an appropriate number of significant digits.

In the exercise, the Grashof number is computed based on the temperature of the insulated bottom side of the plate and the room's air temperature. The temperature difference provides the driving force for convection, while the kinematic viscosity suggests how easily the air moves. The resulting Grashof number indicates the relative strength of the buoyancy to viscous forces in the air surrounding the plate. A high Gr number signifies that buoyancy forces significantly influence the fluid flow, leading to more pronounced natural convection.

The Rayleigh number (Ra) is another dimensionless quantity that combines the effects of buoyancy-driven flow and thermal diffusion, represented by the Grashof and Prandtl numbers, respectively. The Prandtl number (Pr) is a dimensionless number that represents the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. Thus, Ra plays a key role in characterizing the convection heat transfer in fluids.

Ra is particularly important as it tells us whether convection will occur. For instance, above a certain critical value of Ra, convection is likely to be the dominant heat transfer mode. In the provided solution, the Rayleigh number is first calculated by multiplying the Grashof number with the Prandtl number. Understanding the relationship between the Gr, Pr, and Ra can provide insights into how effectively heat is transferred through convection.

When educating students, it’s beneficial to articulate that the Rayleigh number is a measure of the driving force (due to temperature differences) for convection relative to the resisting forces (due to fluid viscosity and heat diffusion). Explaining the physical meaning behind dimensionless numbers like Ra makes them less abstract and helps students to better understand the mathematical steps involved in their calculation.

The Nusselt number (Nu) is the final piece of the puzzle that bridges the theoretical dimensionless analysis with the practical calculation of heat transfer rates in convection processes. Nu is a dimensionless number that represents the ratio of convective to conductive heat transfer across a fluid boundary. In essence, it quantifies the enhancement of heat transfer through a fluid layer as a result of convection relative to pure conduction.

For the case of the insulated plate, the Nusselt number is calculated using a correlation that takes into account the plate’s orientation (horizontal in this case) and the previously calculated Rayleigh number. It serves as a scaling factor in the formula that relates the temperature difference, thermal conductivity, and geometry of the plate to the actual heat transfer rate (Q). Additionally, when guiding students, it is important to explain that different empirical or semi-empirical correlations for Nu exist for various geometries and flow conditions.

By obtaining the Nusselt number and combining it with the thermal conductivity and temperature difference, we are able to calculate the heat transfer rate Q from the plate to the air. This calculated rate of heat transfer provides a tangible result that can validate the temperature maintenance, thermal design, or processing efficiency in numerous applications from industrial heat exchangers to maintaining the thermal comfort in living spaces.