Question

The side surfaces of a 3 -m-high cubic industrial furnace burning natural gas are not insulated, and the temperature at the outer surface of this section is measured to be \(110^{\circ} \mathrm{C}\). The temperature of the furnace room, including its surfaces, is \(30^{\circ} \mathrm{C}\), and the emissivity of the outer surface of the furnace is \(0.7\). It is proposed that this section of the furnace wall be insulated with glass wool insulation $(k=0.038 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ wrapped by a reflective sheet \((\varepsilon=0.2)\) in order to reduce the heat loss by 90 percent. Assuming the outer surface temperature of the metal section still remains at about \(110^{\circ} \mathrm{C}\), determine the thickness of the insulation that needs to be used. The furnace operates continuously throughout the year and has an efficiency of 78 percent. The price of the natural gas is \(\$ 1.10 /\) therm ( 1 therm \(=105,500 \mathrm{~kJ}\) of energy content). If the installation of the insulation will cost \(\$ 550\) for materials and labor, determine how long it will take for the insulation to pay for itself from the energy it saves.

Short answer

Answer: To calculate the required thickness of insulation and the time it takes for the insulation to pay for itself, we need to follow the steps discussed in the solution, such as calculating the current heat loss, determining the heat loss after insulation, finding the required thickness, calculating the total cost and energy saved, and finally determining the time it takes for the insulation to pay for itself. The specific thickness and time will depend on the given values and calculations.

Step-by-step solution

Calculate the current heat loss from the furnace

We can use the Stefan-Boltzmann Law to calculate the heat loss from the furnace's outer surface. \(q = \varepsilon \sigma A(T_h^4 - T_c^4)\) Where: \(q\) is the heat loss from the furnace's outer surface, \(\varepsilon\) is the emissivity of the outer surface, \(\sigma\) is the Stefan-Boltzmann constant (\(\approx 5.67 \times 10^{-8} \mathrm{W} / \mathrm{m}^2\mathrm{K}^4\)), \(A\) is the area of the outer surface, \(T_h\) is the higher temperature (\(110^{\circ}\mathrm{C}+273.15\mathrm{K}=383.15\mathrm{K}\)), \(T_c\) is the lower temperature (\(30^{\circ}\mathrm{C}+273.15\mathrm{K}=303.15\mathrm{K}\)). \(A = 6s^{2}\) (There are 6 faces in a cube, with side \(s\)) We are given that the height of the cube is \(3 m\), therefore \(s=3\). \(A = 6(3)^{2} = 54 \mathrm{m}^2\) Now we can calculate the heat loss from the furnace: \(q = 0.7\times (5.67 \times 10^{-8})(54)(383.15^4-303.15^4)\)

Determine the heat loss after insulation is applied

We are tasked to reduce the heat loss by 90%, therefore, the desired heat loss after the insulation is applied is: \(q_{insulated} = 0.1 \times q\)

Use the glass wool insulation properties to find the required thickness of insulation

We can use Fourier's Law to calculate the heat loss through the insulation: \(q = k \frac{A (T_h - T_c)}{d}\) Where: \(k\) is the thermal conductivity of the glass wool insulation \((0.038 \mathrm{W} / \mathrm{m} \cdot \mathrm{K})\), \(d\) is the thickness of the insulation. Now we can find the required thickness of the insulation: \(d = k \frac{A (T_h - T_c)}{q_{insulated}}\)

Calculate the total cost of insulation and the difference in heat loss before and after insulation

The total cost of insulation is given as \(\$ 550\). Heat loss difference before and after insulation: \(Δq = q - q_{insulated}\)

Determine the energy saved and the time it takes for the insulation to pay for itself

Energy saved per unit time: \(E = Δq \times \mathrm{efficiency}\) Energy cost saving per unit time: \(C = \frac{E}{105,500} \times 1.10\) Calculate the time it takes for the insulation to pay for itself: \(time = \frac{550}{C}\)