View Factors: What they are and how to calculate them

In my first post, I described the goals of my and Mukund’s project. We are looking to create thermal models describing the heat processes going on in regolith, the loose small pieces of rock that cover the surface of asteroids like Bennu. These heat processes are due to conduction and radiative transfers. (No atmosphere on asteroids means there is no medium for convection to occur)

My portion of the project is centered on answering the question “When a body radiates heat, where exactly does it go?”. Answering this question requires calculating the view factors of the surfaces.

Let’s consider two surfaces 1 & 2, with temperatures T1 and T2, areas A1 and A2. The rate of heat transfer from 1 to 2, derived from the Stefan Boltzmann constant is equal to R1->2 =  σA1F1->2(T41 – T42).  σ is the Stefan-Boltzmann constant.

That F1->2 there is the view factor from 1 to 2. Also known as the form factor or configuration factor, it describes the portion of heat that is emitted by 1 that is received by 2. My project centres on finding for each surface, as described by a triangular mesh, what other surfaces can it see, and calculating the view factor between them.

Double Area Integration

The formula defining the factor is:

for the surfaces seen here to the right. While the calculation isn’t complicated, a double area integral is going to be expensive to compute accurately when thousands of surfaces are being used. Thus other methods exist to solve this problem that I will outline

Monte Carlo Method

One alternative to the double area integral is the Monte Carlo method. Readers will likely be aware of the principle of Monte Carlo methods – that random sampling can be used to calculate numerical results, sometimes faster than deterministic methods. In this case, one can, when calculating the view factor from a particular point, dA1, choose a random angle at which to project a ray from that point, and find what surface it hits. If one repeats this process N times, and find that a surface A2 is hit x times, at N increases we know -> FdA1 -> A2.

This can have the advantage that it can be done to find the factor from dA1 to multiple surfaces at the same time. However, the efficiency and convergence of this method can be variable based on the situation, and we have not chosen to use this in this project.

These are just two examples of methods that can be exercised to find the view factor. Others exist, including the Nusselt Sphere, the Crossed-Strings method and others. A good review of the matter can be found here

In my next article, I will recount my experiences implementing view factor calculations for my project

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