The net radiant heat exchange between the opening (fire door) and the wall when the fire door is open is approximately 0.4373 watts.
To calculate the net radiant heat exchange between the opening (fire door) and the wall when the fire door is open, we can use the Stefan-Boltzmann Law and the view factor method. The Stefan-Boltzmann Law relates the radiant heat exchange between two objects to their temperatures and the emissivity of their surfaces.
Given:
- Temperature inside the kiln (T1) = 1395°C = 1668 K
- Temperature of the wall (T2) = 60°C = 333 K
- Emissivity of the wall (E) = 0.92
- Distance from the door to the wall (d) = 3 m
- Door dimensions: 30 cm x 30 cm
- The normal from the center of the door strikes the wall midway from the ends and 1 m above the floor.
First, we need to calculate the view factor (F) between the door and the wall. The view factor represents the fraction of radiation emitted by one surface that reaches the other surface.
1. Calculate the areas involved:
- Area of the fire door (A1) = (30 cm x 30 cm) = 0.09 m²
- Area of the wall (A2) = 10 m x 6 m = 60 m²
2. Calculate the distance between the center of the door and the wall (r). This can be done using the Pythagorean theorem, as the normal strikes the wall midway from the ends:
r² = (3 m)² + (1 m)²
r² = 9 m² + 1 m²
r² = 10 m²
r = 10 m
3. Now, calculate the view factor (F) using the formula for a small, finite area:
F = (A1 / π) * [(1 / r²) * (arctan(B/A) - (B / (A² + r²))) + (1 / (A * r)) - (1 / (π * r))]
Where:
- A = Area of the door (A1)
- B = Area of the wall (A2)
- r = Distance between the door and the wall
Plug in the values:
F = (0.09 m² / π) * [(1 / (10 m)²) * (arctan(60 m² / 0.09 m²) - (60 m² / (0.09 m² + 10 m²))) + (1 / (0.09 m² * 10 m)) - (1 / (π * 10 m))]
F ≈ 0.000143
Now, we can calculate the radiant heat exchange using the Stefan-Boltzmann Law:
4. Calculate the radiant heat exchange (Q) between the opening and the wall using the Stefan-Boltzmann Law:
Q = F * E * σ * (T1^4 - T2^4)
Where:
- σ (Stefan-Boltzmann constant) ≈ 5.67 x 10^-8 W/(m²·K⁴)
Plug in the values:
Q ≈ 0.000143 * 0.92 * (5.67 x 10^-8) * ((1668 K)^4 - (333 K)^4)
Calculate this expression:
Q ≈ 0.000143 * 0.92 * (5.67 x 10^-8) * (8.5153 x 10^16 - 3.1175 x 10^11)
Q ≈ 0.4373 W