CFD is slow, difficult to interpret and has a lot of free parameters. To first order, it seems to replicate the exponential decay for a single zone with a given deactivation constant. But, we know that fan dynamics, ventilation and air mixing matter, which suggests the need for another zone. ## Single zone model $\text{UVPD} = P \times N_{lamp} / A_{floor} $ $\text{eACH} = \alpha k \text{UVPD}$ _α_ = 3.23 W s mW–1 h–1 ## Multizone models [A Multi-Zone Model Evaluation of the Efficacy of Upper-Room Air Ultraviolet Germicidal Irradiation](https://www.tandfonline.com/doi/abs/10.1080/104732299302909)gives a zonal modal for upper room UVC, by partitioning it into three zones. ![[Screenshot 2025-08-14 at 22.24.53.png]] Here we have - $G$ 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} $ 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 ) $ Now, we can also do this for Far UVC, which is what I try to do here. ![[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 ) $ We can rewrite this system of equations as $ \frac{dC_U}{dt} = \left( -\frac{\beta}{V_U} - \frac{\alpha}{V_U} - (k_1 + k_2) \right) C_U(t) + \frac{\beta}{V_U} C_O(t) + \frac{\alpha}{V_U} C_{NF}(t) $ $ \frac{dC_O}{dt} = \frac{\beta}{V_O} C_U(t) + \left( -\frac{Q}{V_O} - \frac{\beta}{V_O} - k_1 \right) C_O(t) $ $ \frac{dC_{NL}}{dt} = \frac{\alpha}{V_{NL}} C_U(t) + \left( -\frac{\alpha}{V_{NL}} - (k_1 + k_2) \right) C_{NL}(t) + \frac{G}{V_{NL}} $ $ \frac{ dG }{ dt } = 0 $ We can then rewrite this as a matrix second order differential equation, with the coefficients below:> $ \mathbf{x}(t)= \begin{bmatrix} C_U(t)\\ C_O(t)\\ C_{NL}(t)\\ G(t) \end{bmatrix}, \qquad \dot{\mathbf{x}}(t)=A \mathbf x ( t ) , $ $ A= \begin{bmatrix} -\dfrac{\beta}{V_U}-\dfrac{\alpha}{V_U}-(k_1+k_2) & \dfrac{\beta}{V_U} & 0 & 0\\[8pt] \dfrac{\beta}{V_O} & -\dfrac{Q}{V_O}-\dfrac{\beta}{V_O}-k_1 & 0 & 0\\[8pt] \dfrac{\alpha}{V_{NL}} & 0 & -\dfrac{\alpha}{V_{NL}}-(k_1+k_2) & \dfrac{1}{V_{NL}}\\[8pt] 0 & 0 & 0 & 0 \end{bmatrix} $ In the expression above, - $\beta$ is meant to be the airflow give through half of the interzonal surface area. Same with $\alpha$. This is in units of volume / s. This is given by random airspeed in the air (m / s) and then multiplying by the surface area. The units here should be m^3 / s - In the work of Miller, 1999 - air speed is 3 m/min. - What is the CFM of a regular fan? - 1m^3 / s - Not sure how to translate one fan to the rest of the room - $Q$ is meant to be the natural ventilation rate. - $k_1$ is meant to be the natural die-off rate - $k_2$ is the susceptibility from Far UVC Then the plots would look like this.