Gebhart factor

The Gebhart factor can be given as:

${\displaystyle B_{ij}={\frac {{\mbox{Energy absorbed at }}A_{j}{\mbox{ originating as emission at }}A_{i}}{{\mbox{Total radiation emitted from }}A_{i}}}}$

.[4]

The Gebhart factor approach assumes that the surfaces are gray and emits and are illuminated diffusely and uniformly.[3]

This can be rewritten as:

${\displaystyle B_{ij}={\frac {Q_{ij}}{\epsilon _{i}\cdot A_{i}\cdot \sigma \cdot T_{i}^{4}}}}$

where

• ${\displaystyle B_{ij}}$ is the Gebhart factor
• ${\displaystyle Q_{ij}}$ is the heat transfer from surface i to j
• ${\displaystyle \epsilon }$ is the emissivity of the surface
• ${\displaystyle A}$ is the surface area
• ${\displaystyle T}$ is the temperature

The denominator can also be recognized from the Stefan–Boltzmann law.

The ${\displaystyle B_{ij}}$ factor can then be used to calculate the net energy transferred from one surface to all other, for an opaque surface given as:[2]

${\displaystyle q_{i}=A_{i}\cdot \epsilon _{i}\cdot \sigma \cdot T_{i}^{4}-\sum _{j=1}^{N_{s}}A_{j}\cdot \epsilon _{j}\cdot \sigma \cdot B_{ji}\cdot T_{j}^{4}}$

where

• ${\displaystyle q_{i}}$ is the net heat transfer for surface i

Looking at the geometric relation, it can be seen that:

${\displaystyle \epsilon _{i}\cdot A_{i}\cdot B_{ij}=\epsilon _{j}\cdot A_{j}\cdot B_{ji}}$

This can be used to write the net energy transfer from one surface to another, here for 1 to 2:

${\displaystyle q_{1-2}=A_{1}\cdot \epsilon _{1}\cdot B_{12}\cdot \sigma \cdot (T_{1}^{4}-T_{2}^{4})}$

Realizing that this can be used to find the heat transferred (Q), which was used in the definition, and using the view factors as auxiliary equation, it can be shown that the Gebhart factors are:[5]

${\displaystyle B_{ij}=F_{ij}\cdot \epsilon _{j}+\sum _{k=1}^{N_{s}}((1-\epsilon _{k})\cdot F_{ik}\cdot B_{kj})}$

where

• ${\displaystyle F_{ij}}$ is the view factor for surface i to j

And also, from the definition we see that the sum of the Gebhart factors must be equal to 1.

${\displaystyle \sum _{j=1}^{N_{s}}(B_{ij})=1}$

Several approaches exists to describe this as a system of linear equations that can be solved by Gaussian elimination or similar methods. For simpler cases it can also be formulated as a single expression.

• Radiosity
• Thermal radiation
• Black body