# Modeling spherical irradiance optimizes designs for UV-C air disinfection (MAGAZINE)

We have been using ultraviolet radiation as a means of disinfection for over a century. However, the novel coronavirus pandemic has led to an explosion of interest in ultraviolet germicidal irradiation (UVGI) systems. From mercury vapor and pulsed xenon lamps to excimer lamps and UV LEDs, there have been, and continue to be, remarkable advances in UV radiation technology.

What we have yet to see, however, are comparable advances in our ability to model — and more importantly, analyze — the performance of UVGI systems.

Architectural lighting designers have had the ability to model and analyze their designs for more than fifty years. Using measured luminous intensity distributions provided by the luminaire manufacturers and architectural CAD models provided by their clients, they can quickly model the distribution of surface illuminance, taking into account both direct illuminance from the light sources and indirect illuminance due to interreflections from environmental surfaces.

Now, it is a straightforward process to adapt architectural lighting design software to UVGI applications. After all, both ultraviolet radiation and visible light are optical radiation that obey the same physical laws of optics. One of the advantages of doing this is that it becomes possible to predict the irradiances of surfaces that are hidden from direct view of the radiation sources. Fig. 1 shows a mobile UV-C platform being used to disinfect an operating theater — which would have many surfaces along vertical and horizontal planes in the intended application environment. The developer of such a product would need to consider how surface reflections factor into the radiation levels and time needed to sanitize the environment (dependent upon the pathogen of concern).

### UVGI design software

It is possible to model the UV-C dose, or fluence, of surfaces by considering multiple positions and dwell times for the mobile platform. Intelligent UV-C robots can be programmed to determine an optimal path for room disinfection. Using UVGI design software, it is possible to identify surfaces that may not receive a sufficient dose and flag them for enhanced terminal cleaning.

At the beginning of the COVID-19 pandemic when we were being advised to disinfect every surface, UVGI design software like this made perfect sense. And a multi-modal approach to disinfection will have its place in some applications, regardless of the current pandemic. However, we have known for more than 80 years that aerosols are the primary transmission vector of respiratory diseases such as tuberculosis, measles, and influenza. The focus is now on the disinfection of air for interior spaces. This opens the door for the widespread adoption of UVGI systems using low-pressure mercury vapor lamps, UV-emitting LEDs, and far-UV excimer lamps.

We have and are continuing to develop the technology, but how can we model its performance? In particular, how can we model the spatial distribution of UV radiation in a volume of air?

### Spherical irradiance

We begin with a few definitions. Surface irradiance refers to the radiant power per unit area incident on a surface, regardless of the direction the radiation is coming from (Fig. 2). For UVGI applications, it is measured in microwatts per square centimeter (μW/cm^{2}).

For aerosols and liquid droplets suspended in air, spherical irradiance refers to the radiant power per unit area incident on the aerosol droplet from all directions. For any given direction, the area of interest is the cross-sectional area of the spherical droplet. Integrating over all possible directions, we have the spherical irradiance, or fluence rate, of the droplet, also measured in μW/cm^{2} (Fig. 3).

It is important to note that these definitions consider radiation received both directly from radiation sources and indirectly from surface reflections. While many common materials have low UV-C reflectances, some materials — for instance, sheet aluminum — can have reflectances as high as 70%.

For UVGI applications, we are interested in the ultraviolet dose, or fluence, of the droplet, measured in millijoules per square centimeter (mJ/cm^{2}). Unfortunately, there are no practical instruments to accurately measure UV fluence, apart from spherical actinometers — hollow quartz spheres filled with photoreactive chemicals such as ferrioxalate in solution.

We are, however, interested not in measuring but modeling spherical irradiance in a volume of air, and so we need a virtual spherical irradiance (SI) meter.

### Virtual SI meters

From a mathematical perspective, it is easy enough to model the performance of spherical actinometers. However, the mathematics of spherical geometry are complex and difficult to solve in a reasonable amount of time. Moreover, we will need hundreds to thousands of SI meters to model large volumes of air in complex architectural environments.

The solution is to replace the sphere with a six-sided dual cubic tetrahedron — imagine two three-sided pyramids mounted base-to-base (Fig. 4, left). Like the sphere, each face will receive direct and indirect irradiance from its surrounding environment. Fortunately, the planar faces result in a much simpler mathematical model. This will allow us to position a three-dimensional (3-D) array of meters in the virtual space in order to sample its UV radiation field.

The calculation procedure is both simple and fast. Each dual cubic tetrahedron face is subdivided into a triangular array of elements, much like a computer display consists of a rectangular array of pixels (Fig. 4, center). If we imagine being at the center of the meter, we have a 360° spherical view of the environment. In theory, we cannot tell whether we are viewing the actual environment or looking at six triangular computer displays surrounding us.

With this, we can project each visible object (or portion thereof for partially hidden objects) onto the faces (Fig. 4, right). This allows us to calculate the irradiance received from each visible object, and to sum the results to obtain the spherical irradiance.

This is only a conceptual outline of the virtual SI meter design. The design itself is based on radiative transfer theory, referred to in computer graphics research as radiosity. Developed in the 1980s for architectural visualization, it is fully applicable to UVGI modeling. (Today, the radiosity method forms the basis of Lighting Analysts’ AGi32 and ElumTools architectural lighting design software.)

The key point here is not the implementation of the meter design but its speed. Using a desktop computer, it takes only seconds to minutes to model thousands of SI meters in complex environments.

Virtual SI meters have a variety of applications, including the modeling of air disinfection systems. In Fig. 5, air flows through the serpentine channel of a sample recessed T-grid air sanitizer while being exposed to UV radiation emitted by 72 LEDs. From a design perspective, the average spherical irradiance within the channel is strongly dependent on the reflectance of channel walls. Modeling the system allows us to optimize the design.

Using CAD modeling, it is easy to represent even the most complex air-handling unit geometries with their UV disinfection sources. Again, the spherical irradiance distribution within the units can be sampled with arbitrary density using a 3-D array of SI meters.

However, the problem is that the air flow within most air-handling units is turbulent. It is not the fluence rate that is important in deactivating pathogens but the fluence itself. Turbulence affects how long the aerosols remain in the irradiance field, so it must be accounted for in predicting the effectiveness of air disinfection.

### Computational fluid dynamics (CFD)

To address this issue, it may therefore be necessary to employ computational fluid dynamics techniques to predict the time-averaged air flow through the air-handling unit. The predicted static irradiance field provides an input to any commercial or open-source CFD program.

Unfortunately, mechanical engineers rarely employ CFD techniques when designing air-handling systems for entire rooms, let alone complex architectural spaces such as hotel lobbies and theaters. Instead, building codes simply specify minimum air changes per hour.

This will undoubtedly change in response to the COVID-19 pandemic. Studies of infections in restaurants and other situations have highlighted the importance of differential air flow within enclosed spaces, and building codes will need to be revised to address this issue.

To summarize, it is possible to adapt existing architectural lighting design software to predict the UV irradiance of surfaces in complex environments, including both direct and indirect radiation. However, modeling air disinfection requires virtual spherical irradiance meters to be implemented within this software.

This article has outlined a novel approach that enables thousands of SI meters to densely sample large volumes of air in these environments. The calculation times range from seconds to minutes, making the approach ideal for modeling air-disinfection systems, air-handling units, upper-room applications with UV-C, and whole-room applications with far-UV.

Spherical irradiance meters are part of the solution in that they can predict fluence rates within air volumes. However, due to air-flow turbulence in complex environments, computational fluid dynamics calculations will be needed to model the fluence that any aerosols may be exposed to.

Regardless, virtual SI meters enable the modeling of UVGI systems. With them, we can design safe and effective air-quality systems for the post-pandemic era.

### Get to know our expert

IAN ASHDOWN*, P. Eng. (Ret.), FIES, is currently senior scientist at SunTracker Technologies. A senior member of the Optical Society of America (OSA) and a recipient of the Illuminating Engineering Society (IES) Gold medal, Ashdown has multiple patents and publications to his credit. With experience in electrical and software engineering, he has specialized knowledge of photometrics, radiometry, and colorimetry, and has worked with companies such as the former TIR Systems Ltd., Cooledge Lighting, and Lighting Analysts. Ashdown is a frequent presenter at LED and SSL industry events, including Strategies in Light and the HortiCann Light + Tech Conference. He holds BAppSc and MSc degrees from the University of British Columbia.*

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