A vertical \(0.9\)-m-high and \(1.8\)-m-wide double-pane window consists of two sheets of glass separated by a \(2.2-\mathrm{cm}\) air gap at atmospheric pressure. If the glass surface temperatures across the air gap are measured to be \(20^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C}\), the rate of heat transfer through the window is (a) \(19.8 \mathrm{~W}\) (b) \(26.1 \mathrm{~W}\) (c) \(30.5 \mathrm{~W}\) (d) \(34.7 \mathrm{~W}\) (e) \(55.0 \mathrm{~W}\) (For air, use \(k=0.02551 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7296, v=\) \(1.562 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\). Also, the applicable correlation is \(\mathrm{Nu}\) \(\left.=0.42 \mathrm{Ra}^{1 / 4} \mathrm{Pr}^{0.012}(H / L)^{-0.3}\right)\) (For air, use \(k=0.02588 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=0.7282, v=1.608 \times\) \(10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) )

The convective heat transfer coefficient (h) is crucial for understanding how heat moves between a surface and a fluid in contact with it. It quantifies the rate at which heat is transferred per unit area per unit temperature difference. Essentially, this value tells us how effective the fluid is at carrying heat away from the surface.

In the context of a double-pane window with an air gap, the convective heat transfer coefficient helps to quantify the heat transfer due to the natural convection occurring in the trapped air. Since natural convection can play a significant role in heat loss or gain through windows, knowing the convective heat transfer coefficient assists in designing more energy-efficient buildings.

To calculate the convective heat transfer coefficient, the Nusselt number (Nu), which is a dimensionless quantity, is used. It relates the convective heat transfer to the conductive heat transfer within the fluid. The higher the Nusselt number, the higher the convective heat transfer coefficient, leading to more effective transfer of heat.

The Grashof Number (Gr) is another dimensionless quantity that helps determine if a flow will be dominated by buoyancy forces or by viscosity forces. It comes into play in situations where natural convection is present, as is the case in the air gap of our double-pane window scenario.

Grashof number is especially important because it can indicate the presence and intensity of natural convection. When a fluid such as air is heated, it expands and becomes less dense, causing it to rise while cooler and denser air sinks down. This natural movement of air can transport heat.

When calculating Gr, we use the temperature difference between the glass surface and the surrounding air, the height of the window, and the kinematic viscosity of air. As Grashof number increases, it typically suggests stronger convection currents, which can lead to higher rates of heat transfer through natural convection.

The Nusselt Number (Nu) is pivotal for finding the convective heat transfer coefficient. It is a dimensionless ratio that suggests how effective the convective heat transfer is compared to conductive heat transfer across a boundary layer. In other words, it helps in assessing the enhancement of heat transfer through a fluid by means of convection in contrast to pure conduction.

For our window with an air gap, using the Nusselt number lets us bridge the gap between the theoretical heat transfer (if it were happening only through conduction) and the actual heat transfer, which includes convection. To determine Nu, we can use empirical correlations that take into account the Rayleigh and Prandtl numbers, and the geometry of the system. High Nusselt numbers signify efficient convection, which in practical terms, could mean more heat is lost through a window on a cold day.

The Rayleigh Number (Ra) combines the effects of buoyancy and viscosity on convective heat transfer through a fluid. By multiplying the Grashof number (which represents the buoyancy forces) and the Prandtl number (which represents the fluid's properties), Ra gives a comprehensive picture of the convection process.

In situations where the driving force for natural convection is the temperature difference, such as the air between two panes of glass in a window, the Rayleigh number helps predict the flow regime and the efficiency of convection. When Ra is high, it typically suggests vigorous convection, potentially resulting in a significant amount of heat transfer.

Lastly, the overall heat transfer coefficient (U) is an inclusive measure of how well a series of materials can transfer heat. It encompasses all forms of heat transfer: conduction, convection, and radiation. In the specific case of a double-pane window, the U-factor measures the window's ability to resist heat flow. The lower the U-value, the better the window is at being an insulator.

The U-factor considers the conductive properties of the glass panes and the convective properties of the air gap between them. When calculating U, it becomes clear that every layer and every kind of heat transfer interaction affects the overall thermal performance of the window. Knowing the U-factor is instrumental in making informed choices about window materials and design for thermal efficiency in buildings.