Question

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) \(66.4 \mathrm{~kW}\) (e) \(75.5 \mathrm{~kW}\) (For water, use $k=0.644 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad \operatorname{Pr}=3.55\(, \)\nu=0.554 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \beta=0.451 \times 10^{-3} \mathrm{~K}^{-1}$. Also, the applicable correlation is \(\mathrm{Nu}=0.069 \mathrm{Ra}^{1 / 3} \mathrm{Pr}^{0.074}\).)

Short answer

Answer: To find the rate of heat transfer, follow the steps of calculating the temperature difference, Rayleigh number, Nusselt number, heat transfer coefficient, and heat transfer rate. Once you have calculated the heat transfer rate (Q), compare it with the given options to find the correct answer.

Step-by-step solution

Calculating temperature difference

First, calculate the temperature difference between the bottom and top of the window (\(ΔT\)): \(ΔT = 60^{\circ}\mathrm{C} - 40^{\circ}\mathrm{C} = 20^{\circ}\mathrm{C} \)

Calculating the Rayleigh number

Calculate the Rayleigh number (Ra) using the formula: \(Ra = \frac{g\beta ΔTL^{3}}{να}\) First, we need to find α. α is the thermal diffusivity which equals to: \(α = \frac{k}{ρc_p}\) We know that for water, \(k = 0.644 \frac{\mathrm{W}}{\mathrm{m}⋅\mathrm{K}}\), \(ν = 0.554×10^{-6} \frac{\mathrm{m}^{2}}{\mathrm{s}}\), and \(\beta = 0.451×10^{-3} K^{-1}\). Also, Prandtl number (\(Pr\)) \(= 3.55\). We will start by finding α using Prandtl number: \(Pr = \frac{ν}{α}\) => \(α = \frac{ν}{Pr} = \frac{0.554×10^{-6}\frac{\mathrm{m}^2}{\mathrm{s}}}{3.55}\) Now we can calculate Ra: \(Ra = \frac{9.81 \times 0.451×10^{-3} K^{-1} \times 20 K \times(3.5×10^{-2} m)^3}{0.554×10^{-6} \frac{\mathrm{m}^{2}}{\mathrm{s}} \times \frac{0.554×10^{-6}\frac{\mathrm{m}^2}{\mathrm{s}}}{3.55}}\)

Calculating the Nusselt number

Calculate the Nusselt number (Nu) using the given correlation: \(Nu = 0.069 \times Ra^{1/3} \times Pr^{0.074}\) By inserting the previously calculated values, we get: \(Nu = 0.069 \times Ra^{1/3} \times 3.55^{0.074}\)

Calculating heat transfer coefficient

Find the heat transfer coefficient (\(h\)) using Nusselt number: \(h = \frac{Nu \times k}{L}\) Inserting the calculated Nusselt number: \(h = \frac{Nu \times 0.644 \frac{\mathrm{W}}{\mathrm{m}⋅\mathrm{K}}}{3.5×10^{-2} m}\)

Calculating heat transfer rate

Finally, we calculate the heat transfer rate (\(Q\)) using the formula: \(Q = h \times A \times ΔT\) where \(A\) is the area of the window, and it is given by: \(A = 1.5\,\mathrm{m} \times 4.5\,\mathrm{m}\) By inserting the relevant values, we get: \(Q = h \times (1.5\,\mathrm{m} \times 4.5\,\mathrm{m}) \times 20\,K\) Now, compare the calculated value of Q with the given options to find the correct answer.