In analyses of collimated beam apparatus (CBA), it is common practice to treat the water factor (WF) and divergence factor (DF) as independent correction terms and to combine them multiplicatively, as described by Bolton and Linden (2003). While this approach has been widely adopted in the literature, its mathematical basis merits further examination.

When the water factor is considered independently, it can be expressed as,

Similarly, when the divergence factor is considered independently, it can be written as,

Since water absorption and beam divergence act simultaneously, the combined effect should be expressed as,

In mathematics, 

As commonly adopted (Bolton and Linden, 2003),

The relative error, , is calculated and presented in Figure 1.

Fig.1 Analysis of water factor and divergence factor

Equation (4) indicates that, in a strict mathematical sense, Equation (3) is not exact. Nevertheless, for most engineering applications, the relative error introduced by treating WF and DF multiplicatively remains negligible. Only under certain conditions, such as in close proximity to the lamp, does this deviation warrant further consideration.

Nomenclature

Reference

Bolton, James R., and Karl G. Linden. 2003. “Standardization of Methods for Fluence (UV Dose) Determination in Bench-Scale UV Experiments.” Journal of Environmental Engineering 129 (3): 209–215. https://doi.org/10.1061/(ASCE)0733-9372(2003)129:3(209).

Submitted by Lianfeng Zhang, Research Institute of Tsinghua University in Shenzhen