Question

What are the summation rule and the superposition rule for view factors?

Short answer

Question: Explain the summation rule and superposition rule for view factors and provide examples for each rule. Answer: The summation rule for view factors states that the sum of view factors from one surface (i) to all the other surfaces (j) in an enclosure is equal to 1, ensuring conservation of energy. For example, in an enclosure with three surfaces, the summation rule for each surface would have the following equations: For surface 1: F₁₂ + F₁₃ + F₁₁ = 1 For surface 2: F₂₁ + F₂₃ + F₂₂ = 1 For surface 3: F₃₁ + F₃₂ + F₃₃ = 1 The superposition rule states that the view factor from a surface (i) to a composite surface (k), which is subdivided into smaller surfaces (j), is equal to the sum of view factors from surface i to the individual smaller surfaces (j). This rule helps in solving complex geometries by breaking the problem into smaller components. For example, if surface i is emitting energy towards a composite surface k, which is subdivided into three smaller surfaces, the superposition rule would be represented as: Fᵢₖ = Fᵢ₁ + Fᵢ₂ + Fᵢ₃

Step-by-step solution

Summation Rule

The summation rule states that the sum of view factors from one surface (i) to all the other surfaces (j) in an enclosure is equal to 1, mathematically represented as: \[ \sum_{j=1}^n F_{i,j} = 1 \] This rule implies that all the energy leaving surface i must be received by any of the other surfaces in the enclosure. It ensures conservation of energy. Example: Consider an enclosure with three surfaces (n=3). The summation rule for each surface is given by: For surface 1: \[F_{1,2} + F_{1,3} + F_{1,1} = 1\] For surface 2: \[F_{2,1} + F_{2,3} + F_{2,2} = 1\] For surface 3: \[F_{3,1} + F_{3,2} + F_{3,3} = 1\]

Superposition Rule

The superposition rule states that the view factor from a surface (i) to a composite surface (k), which is subdivided into smaller surfaces (j), is equal to the sum of view factors from surface i to the individual smaller surfaces (j). Mathematically represented as: \[F_{i,k} = \sum_{j=1}^m F_{i,j}\] where m is the number of smaller surfaces into which the composite surface k is divided. This rule helps in solving complex geometries by breaking the problem into smaller and simpler components. Example: Consider a surface i emitting energy towards a composite surface k, which is subdivided into three smaller surfaces (j=1, 2, 3). The superposition rule for the given scenario is given by: \[F_{i,k} = F_{i,1} + F_{i,2} + F_{i,3}\]