Classical thermodynamics, heat transfer Definition

View factor

Definition

Let $A_i$ and $A_j$ be two surfaces in space, with areas also written $A_i$ and $A_j$ . For points $p \in A_i$ and $q \in A_j$ , let $S$ be the distance from $p$ to $q$ . Let $\theta_i$ and $\theta_j$ be the angles between the segment $pq$ and the normals of the surfaces at $p$ and at $q$ . The integrand below is set to $0$ when the segment $pq$ is blocked or when a surface faces away from the other point. The view factor from $A_i$ to $A_j$ is $F_{i \to j} = \frac{1}{A_i} \int_{A_i} \int_{A_j} \frac{\cos\theta_i \cos\theta_j}{\pi S^2} \, dA_j \, dA_i.$ If $A_i$ emits and reflects diffusely with uniform radiosity, then $F_{i \to j}$ is the fraction of the thermal radiation leaving $A_i$ that strikes $A_j$ directly.

Remarks

Reciprocity $A_i F_{i \to j} = A_j F_{j \to i}$ follows from the symmetry of the integrand. If $A_1, \dots, A_n$ form a closed enclosure, then $\sum_{j=1}^n F_{i \to j} = 1$ for each $i$ , and $F_{i \to i} = 0$ when $A_i$ is plane or convex.

Example: a hemispherical shell $A_1$ of radius $R$ closed by its base disk $A_2$ . The disk sees only the shell, so $F_{2 \to 1} = 1$ . Reciprocity gives $F_{1 \to 2} = (\pi R^2)(1) / (2\pi R^2) = 1/2$ , and summation gives $F_{1 \to 1} = 1/2$ : the concave shell sees itself.

Non-example: if $A_i$ is a mirror, reflecting specularly, the reflected part of the radiation leaving $A_i$ is not distributed according to $F_{i \to j}$ , since the diffuse assumption fails.

Other names are configuration factor, angle factor and shape factor.

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Radiosity concept
Thermal radiation concept
View factor definition

Tag	`RFP1CXYYZ3`
URI	`m2://RFP1CXYYZ3`
Kind	definition
Writing	W0 stub
MSC	80A21, 78A40
Defines	View factor
Uses	Thermal radiation, Radiosity
Changed	3 November 2026

View factor

Definition

Remarks

Reading path

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