CFD is slow, and seems to first order replicate models for a single zone with deactivation. Unfortunately, this doesn't take into account ventilation and fan mixing, which means that we will probably need to do multi-zonal models. Here's an excrutiatingly dumb question regarding papers on cfd for modelling. The ceiling mounted is the best, but what is unclear to me is a) if it's a _major_ increase (since the absolute values of the concentrations of every placement is small relative to no uvc) b) why it is the best. it is because this particular geometry just happens to maximise the intensity-weighed surface area for the space? In that case, im not sure how well this is generalisable to different spaces.sorry if the questions are so dumb!!! c) and finally - the dumbest question of all. here's a reduced order model I was thinking of: the graph above looks suspiciously like the below, where C_0 and C_infty are functions of a) ventilation rate b) fluence in the breathing zone and c) size of room?. do we expect k in the equation below to be the same k / susceptbility parameter in the papers? if so wouldn't this be a candidate for a first order model (edited I also learn that fan mixing seems to affect room UVC a lot. Which means that somehow more than one zone is likely to be appropriate. In this case, I think this model is slightly more likely to work, but adapted ![[Screenshot 2025-08-14 at 22.24.53.png]] I found the above paper in [[A Multi-Zone Model Evaluation of the Efficacy of Upper-Room Air Ultraviolet Germicidal Irradiation]] The above paper does the following model. - $G$ is the emission rate - this is given as particles / voume - The constants $\beta_1$ represents the flow of air going between the upper zone and lower zone. It s in the units of volume, and so the concentration entry should be $\beta_1 \times C_{VL }$. - 31.5 m³ for the second and third test days. To assign interzonal air exchange values, assume that the average random air speed in the test room was 3 m/min (10 fpm), which represents relatively still air. (27) In a multi-zone model, it is assumed that air moves into a given zone through half the interzonal surface area, and moves out of the zone through the other half. Let's look at each region one by one, and scrutinize where these euquations are coming from in detail. In the upper region the total number of pathogens per unit time is given by - The volume of air transferred from the lower room, multiplied by the concentration of the lower room. - Then there is the inactivation from the light itself, the rate of which is proportional to the number of pathograns in that room. - Then, there is inactivation from the natural die-off, which is again proportional to the current number of pathogens in that room. Finally, we have room outflow, which reduced the number of pathogens by the concentration in the upper room multiplied by the outflow are volume. $ V_U \frac{dC_U}{dt} = \beta_I \, C_I(t) - \Big[ (k_1 + k_2)\, V_U + \beta_I \Big] \, C_U(t) \tag{8'} $ Ok, now lets do the lower room. In the lower room, we have - Pathogens coming from the near field zone, the number of which is the concentration multiplied by the airflow $\beta_2$ - Similarly, we have entry and exit from the upper room zone. - We have entry and exit from the near field zone - We have ventilation in and out $ \frac{dC_L}{dt} = \frac{\beta_2}{V_L} \, C_{NF}(t) + \frac{\beta_1}{V_L} \, C_U(t) - \frac{(k_1 \cdot V_L + \beta_1 + \beta_2 + Q)}{V_L} \, C_L(t) \tag{9} $ In the near field zone, we have - constant emission - airflow in and out from the lower zone - natural die off $ \frac{dC_{NF}}{dt} = \frac{G}{V_{NF}} + \frac{\beta_2}{V_{NF}} \, C_L(t) - \frac{(k_1 \cdot V_{NF} + \beta_2)}{V_{NF}} \, C_{NF}(t) \tag{10} $ The interzonal air field is given by This is a linear system of differential equations, which can be solved by taking the matrix exponential at t. $ \dot { \mathbf C } ( t ) = A \mathbf C \implies \mathbf C ( t ) = \exp( A t ) \mathbf C ( 0 ) $ ![[Screenshot 2025-08-19 at 18.59.08.png]] In this case, let - $C_ U$ be concentration of pathogens in the field enclosed by the Far UVC light - which is a cone. - $C_ { NL}$ be concentration of pathogens in the near field around people. - $C_O$ be concentration of pathogens the region outside the Far UVC light. Let $V_X$ be their respective volumes of each reason Then let - $G$ be the emission rate of someone who is infected - $\alpha$ the rate of air flow between the near field area and the UVC cone - $\beta$ the rate of air flow between the near field area and outside the room - Q be the natural ventilation rate outside For deactivation coefficients - Let $k_1$ be natural die off rate of pathogen - Let $k_2$ be susceptibility of pathogen to far UVC Then: $ \frac{ dC _ U} { dt } = \frac{ \beta } { V _ U } \left[ C _ O ( t ) - C _ U ( t ) \right] + \frac{ \alpha } { V_ U } \left[ C _ {NF}( t) - C _U ( t ) \right] -(k_1 + k_ 2 ) \cdot C_ U ( t ) $ $ \frac{ d C _ O } { dt } = -\frac{ Q } { V_O } C _ O ( t )+ \frac{ \beta} { V_ 0 } \left[ C_ U ( t ) - C_ O ( t ) \right] - k_1 C_O ( t ) $ $ \frac{ d C_ {NL}} { dt } = \frac{ G } { V_{ NL } }+ \frac{ \alpha }{ V _ {NL} }\left[ C_ U ( t ) - C_{ NL } ( t ) \right] - ( k _ 1 + k _ 2 ) \cdot C_ { NL } ( t ) $