Gebhart factor

Equations

The Gebhart factor can be given as:

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

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

This can be rewritten as: : B_{ij} = \frac{ Q_{ij}} {\epsilon_{i} \cdot A_{i} \cdot \sigma \cdot T_{i}^{4}}

where

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

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

The 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:

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

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

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

: \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:

: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:

:B_{ij} = F_{ij} \cdot \epsilon_j + \sum_{k=1}^{N_s}((1-\epsilon_k) \cdot F_{ik} \cdot B_{kj})

where

• 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.

: \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.

References

References

1. B. Gebhart, "[https://doi.org/10.1016/0017-9310(61)90048-5 Surface temperature calculations in radiant surroundings of arbitrary complexity--for gray, diffuse radiation. International Journal of Heat and Mass Transfer]".
2. Chin, J. H., Panczak, T. D. and Fried, L. (1992), "[http://onlinelibrary.wiley.com/doi/10.1002/nme.1620350403/abstract Spacecraft thermal modeling. International Journal for Numerical Methods in Engineering]".
3. Korybalski, Michael E. Clark, John A. (John Alden), "[http://hdl.handle.net/2027.42/46657 Algebraic Methods for the Calculation of Radiation Exchange in an Enclosure]"
4. D. E. BORNSIDE, T. A. KINNEY AND R. A. BROWN, "[http://onlinelibrary.wiley.com/doi/10.1002/nme.1620300109/abstract Finite element/Newton method for the analysis of Czochralski crystal growth with diffuse-grey radiative heat transfer . International Journal for Numerical Methods in Engineering]".