Calculating UV radiation dose in a moving airstream
I am trying to calculate the UVC irradiation dose absorbed by an airstream passing around a set of multiple (3) UVC lightbulbs.
Here is a brief description of the scenario: 3000 CFM of 55F air is passing through a 24"x24", 24" long device with three UVC bulbs installed in a cross flow orientation (equally spaced installation at 4", 12", 20" heights). The air is traveling approximately 750ft/min. The bulbs are rated for 36W total and 11.2W UVC radiation at 100hr. Bulb length is 415mm.
If the distance from the lamp is assumed to be 4", I can use the following equations to calculate the irradiance:
$$E=\frac{\phi}{(2\pi^2ld)} (2\alpha+\sin(2\alpha))$$
Where $\alpha = \arctan \frac{L}{2D}$, in radians -> $\alpha =arctan \frac{0.415m}{2*0.1016m}$ = 1.115 rad.
Using the irradiance formula, I calculate total irradiance E=5108µW/cm2. 565
(Multiplying a single bulb irradiance by 3, since there are three bulbs.)
Going back to 750fpm air velocity in the device, it takes 0.16 seconds for the air to pass through the 24" device. Using the formula to calculate UV dose, $D_{UV}=It$ -> $D_{UV}=\frac{817.22µW*s}{cm^2}$
Here is where I run into a fundamental question that I have not had been able to find an answer for. Are those formulas assuming an exposure distance of 0.1016m from the lamp for the entire 0.16s? Obviously, the airstream will begin receiving UV dose as soon as it enters the device, albeit at a much lower level, with intensity peaking for the split second the air passes directly adjacent to the UVC bulbs.
Are there any differential equations or integrals that can be established to sum the total UV dosage received by a particle across the entire device length? Would the radiation at the boundaries be so insignificant that it can be ignored? Additionally, I have ignored factors such as reflectivity, view factor, and wind chill as I have not been able to work them into my formulas yet. Any guidance on this would be appreciated!
I appreciate any assistance on this problem! I am new to this site so forgive me if I posted anything incorrectly. Thank you 565