Question

Consider a cubical furnace with a side length of \(3 \mathrm{~m}\). The top surface is maintained at \(700 \mathrm{~K}\). The base surface has an emissivity of \(0.90\) and is maintained at \(950 \mathrm{~K}\). The side surface is black and is maintained at \(450 \mathrm{~K}\). Heat is supplied from the base surface at a rate of \(340 \mathrm{~kW}\). Determine the emissivity of the top surface and the net rates of heat transfer between the top and the bottom surfaces, and between the bottom and side surfaces.

Short answer

Answer: To determine the emissivity of the top surface and the net rates of heat transfer between the surfaces, follow the steps provided in the solution. The four steps are: 1. Calculate area and radiosity for each surface. 2. Calculate net radiative heat transfer between surfaces. 3. Find emissivity for the top surface. 4. Find net rates of heat transfer. By following these steps, we will be able to determine the required values for the emissivity of the top surface and the net rates of heat transfer between the top and bottom surfaces, and between the bottom and side surfaces of the cubical furnace.

Step-by-step solution

First, we need to calculate the area, \(A\), of each surface of the cubical furnace: \(A = \text{side length} \times \text{side length} = 3\text{m} \times 3\text{m} = 9\text{ m}^2\). Next, let's calculate the radiosity, \(J_i\), for each surface using the Stefan-Boltzmann Law (\(J_i = \varepsilon_i \sigma T_i^4 + \rho_i \sigma T_{surroundings}^4\)) and the given temperatures, where \(\varepsilon_i\) is the emissivity of the surface, \(\rho_i\) is the reflectivity of the surface (\(\rho_i = 1 - \varepsilon_i\)), \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \text{W/m}^2\text{K}^4\)), and \(T_i\) is the temperature in Kelvin: Top surface: \(J_1 = \varepsilon_1 \sigma T_1^4 + \rho_1 \sigma T_{surroundings}^4\) Base surface: \(J_2 = \varepsilon_2 \sigma T_2^4 + \rho_2 \sigma T_{surroundings}^4\) Side surface: Since it's black, the emissivity is \(1\), and we can use the simpler Stefan-Boltzmann Law: \(J_3 = \sigma T_3^4\) #Step 2: Calculating net radiative heat transfer between surfaces#

Now we will calculate the net radiative heat transfer between the surfaces, using the formula \(q_{net} = A F_{i \rightarrow j}(J_i - J_j)\), where \(F_{i \rightarrow j}\) represents the view factor between surfaces \(i\) and \(j\) and \(q_{net}\) is the net radiative heat transfer between the two surfaces: Top and base surfaces: \(q_{net, 1 \rightarrow 2} = A F_{1 \rightarrow 2}(J_1 - J_2)\) Within a cube, the surface to surface view factor is \(1/4\) which can be used to calculate the net radiative heat transfer: Bottom and side surfaces: \(q_{net, 2 \rightarrow 3} = A F_{2 \rightarrow 3}(J_2 - J_3)\) #Step 3: Finding emissivity for the top surface#

We are given that the heat supplied from the base surface is \(340 \text{ kW}\). This heat is transferred from the base surface to the top and side surfaces. Therefore, the net radiative heat transfer from the base surface should be equal to the heat supplied: \(q_{net, 1 \rightarrow 2} + q_{net, 2 \rightarrow 3} = 340000\text{ W}\) or equivalently: \(A F_{1 \rightarrow 2}(J_1 - J_2) + A F_{2 \rightarrow 3}(J_2 - J_3) = 340000\text{ W}\) Using the known values and solving this equation for the unknown variable, \(\varepsilon_1\), we obtain the emissivity of the top surface. #Step 4: Finding net rates of heat transfer#

Finally, we use the obtained emissivity value for the top surface, and the known values for the base and side surfaces, to calculate the net radiative heat transfer rates between the top and bottom surfaces, and between the bottom and side surfaces: \(q_{net, 1 \rightarrow 2} = A F_{1 \rightarrow 2}(J_1 - J_2)\) \(q_{net, 2 \rightarrow 3} = A F_{2 \rightarrow 3}(J_2 - J_3)\)

Key concepts

Radiative Heat Transfer

Radiative heat transfer is the process by which thermal energy is transferred via electromagnetic waves. It does not require a medium, meaning that heat can be transferred through a vacuum. This is the same mechanism by which the sun's heat reaches the Earth. In the context of the cubical furnace problem, radiative heat transfer is occurring between the surfaces of the furnace.
The surfaces of the furnace emit thermal radiation based on their temperatures, and this radiation can be absorbed, reflected, or transmitted to the other surfaces. The rate of this transfer is calculated using factors like the temperature difference between surfaces, their emissivities, and the geometry of the setup.

• The temperature of a surface plays a crucial role in determining how much heat it emits.
• The view factor indicates how much of the radiation leaving one surface will strike the other surface.

Understanding radiative heat transfer helps explain how heat moves in systems where conduction and convection are not the primary means of energy transfer.

Emissivity

Emissivity is a measure of how effectively a surface emits thermal radiation compared to a perfect black body. A black body is a theoretical object that absorbs all incident radiation, regardless of wavelength or angle, and emits the maximum possible radiation. Its emissivity value is 1.
Most real-world objects have an emissivity less than 1, and this property depends on the material and surface texture. In the problem, the top surface's emissivity is unknown, whereas the base has an emissivity of 0.90, meaning it emits 90% as effectively as a black body.
Surfaces with high emissivity are more effective at radiating heat, which is crucial when calculating heat transfer rates:

• Materials with rough or matte surfaces generally have higher emissivity than shiny or smooth surfaces.
• Emissivity can also vary with temperature and radiation wavelength.

In practical applications, higher emissivity surfaces are often desirable for efficient heat radiating purposes, such as in radiators or cookware.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in understanding radiative heat transfer. It quantifies the power radiated from a surface in terms of its emissivity, area, and temperature. The law is given by the formula:
\[ J = \varepsilon \sigma T^4 \]where \( J \) represents the total radiative emission, \( \varepsilon \) is the emissivity of the surface, \( \sigma \) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \text{W/m}^2\text{K}^4)\), and \( T \) is the absolute temperature in Kelvin.
Using this law, you can calculate the power radiated by any object, given these parameters:

• It helps predict how much thermal energy a surface will emit based on its temperature and material properties.
• This formula is crucial for determining radiative heat exchange between different surfaces, as seen in the furnace problem.

Understanding and applying the Stefan-Boltzmann Law is key for solving problems related to radiative heat transfer, especially in scenarios where such heat exchange dominates over conduction or convection.