Question

Consider a \(10-\mathrm{ft} \times 10-\mathrm{ft} \times 10-\mathrm{ft}\) cubical furnace whose top and side surfaces closely approximate black surfaces and whose base surface has an emissivity \(\varepsilon=0.4\). The base, top, and side surfaces of the furnace are maintained at uniform temperatures of $800 \mathrm{R}, 1600 \mathrm{R}\(, and \)2400 \mathrm{R}$, respectively. Determine the net rate of radiation heat transfer between \((a)\) the base and the side surfaces and \((b)\) the base and the top surfaces. Also, determine the net rate of radiation heat transfer to the base surface.

Short answer

Answer: To determine the net rate of radiation heat transfer, follow these steps: 1. Calculate effective emissivities for each interaction between surface pairs (1 to 2 and 1 to 3) using the formula provided. 2. Calculate the net radiation heat transfer between Surface 1 and Surface 2 using the Stefan-Boltzmann Law formula and the values of effective emissivities, areas, and temperatures for the base and side surfaces. 3. Calculate the net radiation heat transfer between Surface 1 and Surface 3 using the Stefan-Boltzmann Law formula and the values of effective emissivities, areas, and temperatures for the base and top surfaces. 4. Determine the net rate of radiation heat transfer to the base surface (Surface 1) by adding the heat transfer from the side surfaces (Surface 2) and the top surface (Surface 3). By following these steps, the net rate of radiation heat transfer for each interaction and to the base surface can be calculated.

Step-by-step solution

Compute effective emissivities for each interaction

First, we need to find the effective emissivities for each pair of surfaces: base to side surfaces (1 to 2), and base to top surface (1 to 3). In general, the effective emissivity between surfaces is given by the following formula: Effective Surface Emissivity = \(\frac{1}{\frac{1}{\varepsilon_{1}}+\frac{1}{\varepsilon_{2}}-1}\) For Surface 1 (Base - emissivity \(\varepsilon_1 = 0.4\)) and Surface 2 (Side - emissivity \(\varepsilon_2 \approx 1\), as it is approximate black body): Effective Emissivity (1 to 2) = \(\frac{1}{\frac{1}{0.4}+\frac{1}{1}-1}\) Similarly, For Surface 1 (Base) and Surface 3 (Top - emissivity \(\varepsilon_3 \approx 1\)): Effective Emissivity (1 to 3) = \(\frac{1}{\frac{1}{0.4}+\frac{1}{1}-1}\) These values of effective emissivities will be used for further calculations.

Calculate radiation heat transfer between base and side surfaces

For this, we will use the Stefan-Boltzmann Law formula for radiation heat transfer: \(q = F \cdot A_1 \cdot(\varepsilon_1 \cdot \sigma \cdot T_1^4 - \varepsilon_2 \cdot \sigma \cdot T_2^4)\) Since the base surface (Surface 1) and side surfaces (Surface 2) have the effective emissivity calculated in the previous step and we know their respective areas (\(10\,\text{ft}^2\)) and temperatures (\(800\,\mathrm{R}\) and \(2400\,\mathrm{R}\)), we can substitute the values and calculate the net radiation heat transfer between Surface 1 and Surface 2.

Calculate radiation heat transfer between base and top surfaces

Again we will use the Stefan-Boltzmann Law formula for radiation heat transfer: \(q = F \cdot A_1 \cdot (\varepsilon_1 \cdot \sigma \cdot T_1^4 - \varepsilon_3 \cdot \sigma \cdot T_3^4)\) As before, we will use the effective emissivity calculated in step 1 for the interaction between base surface (Surface 1) and top surface (Surface 3) and their respective areas (\(10\,\text{ft}^2\)) and temperatures (\(800\,\mathrm{R}\) and \(1600\,\mathrm{R}\)). Substituting the values, we can calculate the net radiation heat transfer between Surface 1 and Surface 3.

Determine the net rate of radiation heat transfer to the base surface

Now, we can determine the net rate of radiation heat transfer to the base surface (Surface 1) by adding the heat transfer from the side surfaces (Surface 2) and top surface (Surface 3): Net Radiation Heat Transfer to Surface 1 = \(q_{1 \to 2} + q_{1 \to 3}\) By plugging in the values calculated in Step 2 and Step 3, we can determine the net rate of radiation heat transfer to the base surface.