A horizontal \(1.5\)-m-wide, \(4.5\)-m-long double-pane window consists of two sheets of glass separated by a \(3.5-\mathrm{cm}\) gap filled with water. If the glass surface temperatures at the bottom and the top are measured to be \(60^{\circ} \mathrm{C}\) and \(40^{\circ} \mathrm{C}\), respectively, the rate of heat transfer through the window is (a) \(27.6 \mathrm{~kW}\) (b) \(39.4 \mathrm{~kW}\) (c) \(59.6 \mathrm{~kW}\) (d) \(66.4 \mathrm{~kW} \quad(e) 75.5 \mathrm{~kW}\) (For water, use \(k=0.644 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \operatorname{Pr}=3.55, v=\) \(0.554 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \beta=0.451 \times 10^{-3} \mathrm{~K}^{-1}\). Also, the applicable correlation is \(\mathrm{Nu}=0.069 \mathrm{Ra}^{1 / 3} \operatorname{Pr}^{0.074}\) ).

The concept of thermal conductivity is pivotal when studying heat transfer, as it represents a material's ability to conduct heat. In the context of a double-pane window, we're typically dealing with two materials: glass and a filler substance, such as air or another gas, which in our case study is water.

Thermal conductivity, denoted by the symbol \( k \), is quantified as the amount of heat, in Watts (W), that passes through a material with a thickness of one meter (m) for each square meter of surface area and for each degree Kelvin (K) temperature difference across the material. For example, water has a thermal conductivity of \( 0.644 \frac{W}{m \times K} \). This property tells us that if we have a temperature difference across a layer of water, the material allows heat to transfer at a rate governed by its thermal conductivity.

When analyzing the rate of heat transfer through a window, we need to consider the thermal conductivity of the filler material in combination with the window's dimensions and temperature difference to estimate how much heat is being lost or gained.

While thermal conductivity tells us about the material's intrinsic ability to conduct heat, the Nusselt number offers insight into the convective heat transfer occurring in a fluid. The Nusselt number, \( Nu \), is a dimensionless parameter that correlates the convective to the conductive heat transfer at a boundary in a fluid. In our double-pane window example, the water between the panes serves as our fluid.

To calculate the Nusselt number, we employ empirical correlations, which are developed through experimental data and theoretical analysis. The correlation provided for our double-pane window problem is \( Nu = 0.069 Ra^{1/3}Pr^{0.074} \), which ties the Nusselt number to both the Rayleigh and Prandtl numbers. This calculated Nusselt number is then utilized to figure out the convective heat transfer coefficient, essential for determining the total heat transfer through the window.

The Rayleigh number \( (Ra) \) is another dimensionless quantity crucial in the study of convection. It essentially assesses the potential for convection given a specific set of conditions. Rayleigh number is influenced by a fluid's buoyancy-driven movement resulting from temperature differences, which in turn is affected by factors like the fluid's coefficient of thermal expansion \( (\beta) \), the gravitational acceleration \( (g) \), the temperature difference across the fluid \( (\Delta T) \), the characteristic length \( (L) \), and the kinematic viscosity of the fluid \( (v) \).

Using the formula \( Ra = \frac{g \beta L^3 \Delta T}{v^2} \), we include these variables to gain a Rayleigh number that helps estimate how the fluid will behave in response to the temperature gradient imposed by the window's hot and cold surfaces. A high Rayleigh number suggests a turbulent flow and therefore more effective convection, while a low number indicates a more stable, less convective flow.

Last but not least, we come to the Prandtl number \( (Pr) \), named after the German physicist Ludwig Prandtl. This dimensionless figure expresses the ratio of momentum diffusivity (kinematic viscosity, \( v \)) to thermal diffusivity. Essentially, the Prandtl number tells us about the fluid's thermal and velocity boundary layers.

The Prandtl number is calculated as \( Pr = \frac{\mu C_p}{k} \), where \( \mu \) is the fluid's dynamic viscosity, \( C_p \) is the specific heat capacity at constant pressure, and \( k \) is the previously discussed thermal conductivity. The Prandtl number plays a role in determining how a fluid transfers heat and is put to practical use when calculating the Nusselt number. For water, \( Pr \) typically falls around 3.55, indicating that the momentum and thermal diffusivity are of a similar order of magnitude, leading to comparable thicknesses of the velocity and thermal boundary layers.