Space-time Fourier transform

2013-04-13T00:33:52Z

129.105.207.87:

The '''''Gebhart factors''''' are used in [[radiative heat transfer]], it is a means to describe the ratio of radiation absorbed by any other surface versus the total emitted radiation from given surface. As such, it becomes the radiation exchange factor between a number of surfaces. The Gebhart factors calculation method is supported in several radiation heat transfer tools, such as TMG <ref>{{cite web
| url = http://www.apic.com.tw/support/ideas_data/ESCTMG/TrainingData/Tmg/TMG_Radiation.pdf
| title = Radiation
| work = SDRC
| date = 2000-01-01
| accessdate = 2010-11-26
| publisher = SDRC/APIC
}}</ref> and TRNSYS.

The method was introduced by Benjamin Gebhart in 1957.<ref name="Gebhart">B. Gebhart, "[http://www.sciencedirect.com/science/article/B6V3H-481MR6M-3D/2/45002ab06cfbe37d6ba2dbf9658d8c67 Surface temperature calculations in radiant surroundings of arbitrary complexity--for gray, diffuse radiation. International Journal of Heat and Mass Transfer]".</ref> Although a requirement is the calculation of the [[view factor]]s beforehand, it requires less computational power, compared to using ray tracing with the [[Monte Carlo Method]] (MCM).<ref name="Chin">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]".</ref> Alternative methods are to look at the [[radiosity (heat transfer)|radiosity]], which Hottel <ref name="Korybalski">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]"</ref> and others build upon.

== Equations ==

The Gebhart factor can be given as:

:<math>B_{ij} = \frac{\mbox{Energy absorbed at }A_{j}\mbox{ originating as emission at } A_{i}}{\mbox{Total radiation emitted from }A_{i}}</math>
.<ref name="Korybalski" />

The Gebhart factor approach assumes that the surfaces are gray and emits and are illuminated diffusely and uniformly.<ref name="Chin" />

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

where

* <math>B_{ij}</math> is the Gebhart factor

* <math>Q_{ij}</math> is the heat transfer from surface i to j

* <math>\epsilon</math> is the [[emissivity]] of the surface

* <math>A</math> is the surface area

* <math>T</math> is the temperature

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

The <math>B_{ij}</math> factor can then be used to calculate the net energy transferred from one surface to all other, for an opaque surface given as:<ref name="Gebhart" />

<math>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 </math>

where

* <math>q_{i}</math> is the net heat transfer for surface i

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

:<math> \epsilon_{i} \cdot A_{i} \cdot B_{ij} = \epsilon_{j} \cdot A_{j} \cdot B_{ji}</math>

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

:<math>q_{1-2} = A_{1} \cdot \epsilon_{1} \cdot B_{12} \cdot \sigma \cdot (T_{1}^4-T_{2}^4)</math>

Realizing that this can be used to find the heat transferred (Q), which was used in the definition, and using the [[view factor]]s as auxiliary equation, it can be shown that the Gebhart factors are:<ref>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]".</ref>

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

where
* <math>F_{ij}</math> 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.

:<math> \sum_{j=1}^{N_s}(B_{ij}) = 1</math>

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.

==See also==
* [[Radiosity (heat transfer)|Radiosity]]
* [[Thermal radiation]]
* [[Black body]]

== References ==
{{Reflist}}

[[Category:Heat transfer]]

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Space-time Fourier transform