View Factor Models

View factor models are used for radiative transfer calculations of the amount of radiation leaving one surface (A) that reaches a second surface (B). The view factor $$F_{A\rightarrow B}$$ is the fraction of the radiation from surface A that hits surface B. In the context of bifacial PV arrays, the surface B is the back side of the bifacial module or row of modules and surface A is a collection of surfaces near the array such as the unshaded ground, the shaded ground, the front of the PV modules behind the module of interest, etc.). View factor models rely on the fact that all radiation is conserved and thus the sum of all view factors from any surface A equals 1. View factors, also termed shape factors, configuration factors and angle factors, implicitly assume that all radiation is scattered isotropically from any reflecting surface. View factor models estimate the radiant flux (W) reaching the backside of a bifacial module as:

$$E_{B}=\sum F_{A\rightarrow B}\times E_{A}$$

where $$E_{B}$$ is the radiant flux (W) on the back-side of the bifacial module and $$E_{A}$$ is the radiant flux (W) reflected or scattered from each surface A.  Note that radiant flux (W) on surface B is equal to irradiance (W/m^2) on surface B multiplied by the area of B.

View factors can be computed by numerically integration. The view factor from a general surface A1 to another general surface A2 is given by:

$$F_{1\rightarrow 2}=\frac{1}{A_{1}}\int_{A_{1}}^{ }\int_{A_{2}}^{ }\frac{\cos \theta _{1}\cos \theta _{2} }{\pi s^{2}}dA_{2}dA_{1}$$

where $$\theta_{1}$$ and $$\theta _{2}$$ are the angles between the surface normals and a ray between the two differential areas.

Image of view_factor_variables
Figure 1. Calculation of view factors between planar differential elements.

Source : Dov Grobgeld <>, CC BY-SA 3.0,

Catalogs are available that list view factor formulas for many simple geometries, e.g.,

Calculation of the radiant flux leaving a surface A depends on the surface reflectivity or albedo and the amount of light reaching that surface from both direct and diffuse sources. For example, a simple view factor model may assume that the irradiance reaching any illuminated surface is GHI, and the irradiance reaching any shaded surface is DHI.

Computationally efficient implementation of a view factor model enables rapid exploration of the relationships between back-side irradiance and sun position, diffuse/direct ratio, array geometry, albedo and/or near-field structures. Back-side irradiance can be estimated on a cell-by-cell basis leading to an understanding of the spatial variability of backside irradiance as a function of other variables and design parameters. Spatial variability may lead to potential mismatch losses within the bifacial module or string.

Our project is developing view factor models to be used to predict back side irradiance for bifacial PV systems: