🔲 Multi-Surface Enclosure (N surfaces)

Compute temperature profiles and net heat rates within a closed gray enclosure containing up to 10 surfaces using the matrix radiosity method.

Matrix Radiosity Method

Radiation heat exchange within multi-surface enclosures is resolved by solving linear systems of radiosities ($J_i$). The model supports:

  • Temperature Imposed ($T_i$): Computes the net heat transfer rate ($Q_i$) required to maintain it.
  • Flux Imposed ($q_{net, i}$): Computes the unknown surface temperature ($T_i$). Set $q = 0$ for adiabatic surfaces.
  • View Factors Validation: Automatically checks that summation ($\sum F_{ij} = 1$) and reciprocity ($A_i F_{ij} = A_j F_{ji}$) rules are respected.

📝 Configuration

1. Surface Boundary Properties

Surface Area [m²] Emissivity Boundary Condition Imposed Value (T [°C] or q [W/m²])

2. View Factor Matrix ($F_{i\to j}$)

📊 Results & Verification

Configure enclosure and click "Solve Radiation Enclosure" to run matrix solver.

📘 Calculation Methodology & Analytical Presets

Matrix Radiosity Solver

Thermal radiation within gray, diffuse enclosures is solved using a net radiation balance for each surface node:

Radiosity Equation ($J_i$):
$$J_i - (1-\varepsilon_i) \sum_{j=1}^N F_{i\to j} J_j = \varepsilon_i \sigma T_i^4 \quad (\text{if Temp imposed})$$ $$J_i - \sum_{j=1}^N F_{i\to j} J_j = q_{net, i} \quad (\text{if Net Flux imposed})$$

After solving the system $\mathbf{A} \cdot \mathbf{J} = \mathbf{C}$ for radiosities, unknown variables are derived:

Net Heat Rate ($Q_{net, i}$):
$$Q_{net, i} = A_i \cdot (J_i - G_i) \quad \text{where } G_i = \sum_{j=1}^N F_{i\to j} J_j$$ Unknown Temperature ($T_i$, if Flux imposed):
$$T_i = \left(\frac{J_i + \frac{1-\varepsilon_i}{\varepsilon_i} q_{net, i}}{\sigma}\right)^{0.25} - 273.15$$

Concentric Shells Preset ($N=3$)

For nested concentric cylinders or spheres where Surface 2 has two active radiating sides (inside facing 1, outside facing 3):

Concentric Cylinders ($A_2 = 4\pi r_2 L$):
$$F_{12} = 1.0, \quad F_{21} = \frac{r_1}{2 r_2}, \quad F_{23} = 0.5, \quad F_{22} = 0.5\left(1 - \frac{r_1}{r_2}\right)$$ $$F_{32} = \frac{r_2}{r_3}, \quad F_{33} = 1.0 - \frac{r_2}{r_3}$$
Concentric Spheres ($A_2 = 8\pi r_2^2$):
$$F_{12} = 1.0, \quad F_{21} = \frac{r_1^2}{2 r_2^2}, \quad F_{23} = 0.5, \quad F_{22} = 0.5\left(1 - \frac{r_1^2}{r_2^2}\right)$$ $$F_{32} = \left(\frac{r_2}{r_3}\right)^2, \quad F_{33} = 1.0 - \left(\frac{r_2}{r_3}\right)^2$$