Question

A vertical 4-ft-high and 6-ft-wide double-pane window consists of two sheets of glass separated by a 1 -in air gap at atmospheric pressure. If the glass surface temperatures across the air gap are measured to be $65^{\circ} \mathrm{F}\( and \)40^{\circ} \mathrm{F}$, determine the rate of heat transfer through the window by \((a)\) natural convection and (b) radiation. Also, determine the \(R\)-value of insulation of this window such that multiplying the inverse of the \(R\)-value by the surface area and the temperature difference gives the total rate of heat transfer through the window. The effective emissivity for use in radiation calculations between two large parallel glass plates can be taken to be \(0.82\).

Short answer

Question: Calculate the rate of heat transfer through a double-pane window by both natural convection and radiation, and find the R-value of insulation for this window. Solution: 1. Calculate the temperature difference and heat transfer area: Temperature difference, ΔT = 25°F Total area of the window, A = 2.2296 m² 2. Calculate the natural convection heat transfer rate: q_conv = h A ΔT (Use given height and empirical correlation to find h) 3. Calculate the radiation heat transfer rate: q_rad = σ ε A (T1⁴ - T2⁴) 4. Calculate the total heat transfer rate: q_total = q_conv + q_rad 5. Calculate the R-value of insulation: R = (ΔT) / (q_total/A) Use the given information and empirical relationships to find the heat transfer coefficients and rates, and determine the R-value for the double-pane window.

Step-by-step solution

Calculate the temperature difference and heat transfer area

First, find the difference in glass surface temperatures and the total area of the window. Temperature difference, \(\Delta T = T_1 - T_2 = 65\,^{\circ}\mathrm{F} - 40\,^{\circ}\mathrm{F} = 25\,^{\circ}\mathrm{F}\) Convert the temperatures to Kelvin: \(T_1 = 65\,^{\circ}\mathrm{F} \times \frac{5}{9} + 273.15 = 291.48\,\mathrm{K}\) and \(T_2 = 40\,^{\circ}\mathrm{F} \times \frac{5}{9} + 273.15 = 277.59\,\mathrm{K}\) Total area of the window, \(A = 4\,\mathrm{ft} \times 6\,\mathrm{ft} = 24\,\mathrm{ft^2}\) Convert it to \(\mathrm{m^2}\) by multiplying by \(0.0929\,\frac{\mathrm{m^2}}{\mathrm{ft^2}}\): \(A = 24\,\mathrm{ft^2} \times 0.0929\,\frac{\mathrm{m^2}}{\mathrm{ft^2}} = 2.2296\,\mathrm{m^2}\)

Calculate the natural convection heat transfer rate

Using an empirical natural convection correlation to find the heat transfer coefficient (h) at the given height of the air gap (4ft). Since we are given the air gap height, use it to find the Grashof number, Gr, and Prandtl number, Pr. By combining the Gr and Pr numbers, we can calculate the Nusselt number (Nu). Next, find the convection heat transfer coefficient, h, using the following equation: \(h = \frac{\mathrm{Nu} \times k_\mathrm{air}}{L}\) Where \(k_\mathrm{air}\) is the thermal conductivity of air and \(L\) is the characteristic length, which is the height of the air gap. Finally, calculate the rate of natural convection heat transfer using the following equation: \(q_\mathrm{conv} = h A \Delta T\)

Calculate the radiation heat transfer rate

Now, calculate the radiation heat transfer rate using the following equation: \(q_\mathrm{rad} = \sigma \epsilon A \left(T_1^4 - T_2^4\right)\) Where \(\sigma\) is the Stefan-Boltzmann constant (\(5.67\times10^{-8}\,\mathrm{W/m^2K^4}\)), \(\epsilon\) is the effective emissivity given as \(0.82\), and \(T_1\) and \(T_2\) are the glass surface temperatures in Kelvin. Note: Since we are provided with the effective emissivity, we do not need to use the regular emissivity in this calculation.

Calculate the total heat transfer rate

The total heat transfer rate through the window is the sum of the natural convection heat transfer rate and the radiation heat transfer rate: \(q_\mathrm{total} = q_\mathrm{conv} + q_\mathrm{rad}\)

Calculate the R-value of insulation

Finally, calculate the R-value of insulation by using the following equation: \(R = \frac{\Delta T}{\frac{q_\mathrm{total}}{A}}\) By following these steps, you can determine the rate of heat transfer through the double-pane window by both natural convection and radiation, as well as the R-value of insulation for this window.