Question

Two very large parallel plates are maintained at uniform temperatures of \(T_{1}=600 \mathrm{~K}\) and \(T_{2}=400 \mathrm{~K}\) and have emissivities \(\varepsilon_{1}=0.5\) and \(\varepsilon_{2}=0.9\), respectively. Determine the net rate of radiation heat transfer between the two surfaces per unit area of the plates.

Short answer

Solution: Using the Stefan-Boltzmann law and the view factor, the net rate of radiation heat transfer per unit area (q_net) can be calculated as follows: q_net = (1 / (1/0.5 + 1/0.9 - 1)) * (5.67 * 10^-8) * (600^4 - 400^4) Compute the value of q_net to obtain the net rate of radiation heat transfer per unit area of the plates.

Step-by-step solution

Understand the concept of radiation heat transfer

Radiation heat transfer occurs when energy is emitted by one body and absorbed by another. For two large parallel plates, the radiation heat transfer can be determined using the Stefan-Boltzmann law and the view factors.

Apply the Stefan-Boltzmann law

The Stefan-Boltzmann law states that the rate of radiation heat transfer per unit area (q) is directly proportional to the temperature difference between the two surfaces raised to the power of 4: q = σ * (T1^4 - T2^4) Where σ is the Stefan-Boltzmann constant (5.67 * 10^-8 W/m²K⁴), T1 = 600 K, and T2 = 400 K.

Apply view factors

The net heat transfer between the two surfaces can be found by multiplying the heat transfer rate per unit area by the view factor between the surfaces. The view factor F₁₂ for two parallel plates with emissivities ε₁ and ε₂ is given by: F₁₂ = 1 / (1/ε₁ + 1/ε₂ - 1) We are given ε₁ = 0.5, and ε₂ = 0.9. Now let's calculate F₁₂: F₁₂ = 1 / (1/0.5 + 1/0.9 - 1)

Calculate the net rate of radiation heat transfer

Using the Stefan-Boltzmann law and the view factor, we can now calculate the net rate of radiation heat transfer per unit area between the two plates: q_net = F₁₂ * q Substitute the values of σ, T1, and T2 from step 2, as well as the value of F₁₂ obtained in step 3: q_net = (1 / (1/0.5 + 1/0.9 - 1)) * (5.67 * 10^-8) * (600^4 - 400^4) Finally, calculate q_net to get the net rate of radiation heat transfer per unit area of the plates.