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Posted on 05 May, 2023

Theory of View Factor

Introduction of Radiation View Factor

View factor is purely geometric quantity and is independent of the surface properties and temperature. It is also known as shape factor, configuration factor, and angle factor. View factor assumes that the surfaces are diffuse emitters and diffuse reflectors (diffuse view factor) or specular reflectors (specular view factor). View factor ranges between 0 and 1.

Radiation heat exchanged between surfaces depends on the orientation of the surfaces relative to each other, and this dependence on orientation is accounted for by the view factor. The net radiation flow from a surface through its surface resistance is equal to the sum of the radiation flows from that surface to all the other surfaces through the corresponding space resistance.

View Factor Relations

The Reciprocity Relation (Rule)

$F*(j→i)= F*(i→j) A_j= A_i$

The Summation Rule

$∑\_(j=i )^N▒〖=1〗 , N^2-[N+1/2N(N-1) ]=1/2N(N-1)$

The sum of the view factors from surface i of an enclosure to all surfaces of the enclosure must equal unity.

The Superposition Rule

$F*(1→(2,3))= F*(1→2)+ F\_(1→3)$

Apply the reciprocity relation: {{ latex(equation="F*((2,3)→1)=(A_2 F*(2→1 )+ A1 F(1→3))/(A_2+A_3 )"))}}

The view factor from a surface to a surface is equal to the sum of the view factors from the surface to the parts of the surface.

The Symmetry Rule

$F*(i→j)= F*(i→k) and F*(j→i)= F*(k→i)$

Two (or more) surfaces that possess symmetry about a third surface will have identical view factors from that surface.

Computation of the view factors

In the radiation analysis of an enclosure, either the temperature or the net rate of heat transfer must be given for each of the surfaces to obtain a unique solution for the unknown surface temperatures and heat transfer rates.

In this report, we explore the following methods:

Analytical methods- Particular configurations & The Monte Carlo Method.

Finite area to finite area

A.(Contour Integration-) this method is significantly more accurate than the area integration method. The view factor results based on the analytical expressions derived for these finite length geometries are compared with that of the exact expressions available in the literature for infinite length. It is observed that the use of infinite length approximations in finite length cases can lead to significant errors.

B. View factors between infinitely long surfaces; the crossed-strings methods. Channels and ducts that are very long in one direction relative to the other direction (surface) can be considered to be 2D. These geometries can be modeled as being infinitely long.

C. Algebraic rule and matrix formulation method

Monte Carlo Method

The Monte Carlo method is a statistical method for calculating view factors. It involves randomly generating rays of radiation and tracking the paths between the surfaces. The view factor is calculated as the ratio of the number of rays that reach one surface to the total number of rays generated.

The steps for the method are as follows:

Generate a large number of random rays from the source surface.
Determine the intersection points of the rays with the target surface.
Calculate the solid angle subtended by the target surface at each intersection point.
Calculate the fraction of rays that intersect the target surface.
Calculate the view factor between the source and target surfaces as the average of the solid angles multiplied by the fraction of rays that intersect the target surface.
Repeat the above steps for all pairs of surfaces in the enclosure.

The user increases the number of simulations to reduce the statistical scatter of the estimated view factor. While there is currently no existing test to verify the convergence of the results to an algebraically proven solution for any arbitrary configuration, the Monte Carlo method is one of the most flexible and efficient techniques for complex surface geometries but can be computationally expensive for larger surfaces.

Summary

The Reciprocity rule, Summation rule, Superposition rule, and Symmetry rule are important relations in the calculation of view factors. Methods for calculating view factors include Direct Integration, Contour Integral, Unit Sphere, Cross-String, and Algebraic Analytical [5]. Radiative properties of surfaces may vary depending on factors such as temperature, wavelength, and surface orientation, and the presence of other gases or materials in the enclosure can also affect radiation heat transfer. The choice of method depends on the system geometry and the required result accuracy.

Radiation heat transfer in two-surface and three-surface enclosures can be calculated using the Stefan-Boltzmann law and view factor calculation methods, while a system of energy balance equations can be used to analyse more complex multi-surface situations.

Analytical solutions are ideal for simple radiation heat transfer problems but lack flexibility for complicated geometries. Hence, we focus on the Monte Carlo method in its stead. Cross-validation of the Monte Carlo method algorithm is then achieved through the algebraic analytical method for simpler geometries.