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Showing content from https://link.springer.com/article/10.1007/s10439-021-02819-7 below:

Active Versus Passive Flow Control in UVC FILTERs for COVID-19 Containment

We have divided the simulations into two blocks. In the first block, we treat the fluid dynamics, and in the second block, the electromagnetic radiation present in the filter column.

In addition to measuring the flow rate provided by this first device, the flow dynamics were also simulated assuming a turbulent regime. Thus, the density of velocity distributions inside the filter chamber and the pressures were determined. The simulations used the commercial software package COMSOL 5.0.2 We employ the Turbulent Flow Algebraic yPlus interface. It is used for simulating single-phase flows at high Reynolds numbers. Although the Reynolds number could indicate that we are in a laminar regime, the strong vorticity presented by the device shown in Fig.5b) and the effect of passing through the cavities where the diodes are located in the device shown in the Fig.5a) justifies the choice of a calculation or modeling under the assumption of turbulent regime. The physics interface is suitable for incompressible flows and compressible flows at low Mach number (typically less than 0.3). The equations solved by the Turbulent Flow Algebraic yPlus interface are the Reynolds averaged Navier-Stokes equations for conservation of momentum and the continuity equation for conservation of mass. Turbulence effects are included using an enhanced viscosity model based on the local wall distance. Figs. 5(a) and (b) depict the computational domain under consideration for both filters.

(a) Geometry corresponding to the boundary conditions imposed on the resolution of the flow for device one. In blue, we can see the surface type probe in which the flow rate has been calculated. (b) Geometry corresponding to the boundary conditions imposed on the resolution of the fluid dynamics in device two. In green, we can see the surface type probe in which the flow rate has been calculated. The surface type probe in blue color has been used to determine the minimum, maximum, and average normal speed to the surface in a steady state. The pattern of vertical lines surface is a symmetry boundary condition that reduces the computational domain in half

The physics interface therefore includes a wall distance equation. We solve stationary regime.

In summary it was resolved the following coupled partial differential equations5,25:

$$\begin{aligned} \rho \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}= & {} \nabla \cdot \left[ -P\mathbf {I}+\left( \mu +\mu _T \right) \left( \nabla \cdot \mathbf {u}+\left( \nabla \cdot \mathbf {u}\right) ^T\right) \right] +\mathbf {F}\end{aligned}$$

(1)

$$\begin{aligned} \rho \nabla \cdot \mathbf {u}= & {} 0\end{aligned}$$

(2)

$$\begin{aligned} (1+2\sigma _w)G^4= & {} \nabla G\cdot \nabla G+\sigma _wG\left( \nabla \cdot \nabla G\right) \end{aligned}$$

(3)

Where \(\mathbf {u}\) and P are the field of velocities and the pressure, respectively, the earth’s gravitational field is introduced by means of the force per unit of mass of fluid \(\mathbf {F}\) and \(\mathbf {I}\) is the unit tensor. In the continuity equation \(\frac{\partial \rho }{\partial t}+\nabla \cdot \left( \rho \mathbf {u} \right) =0\), which represents the conservation of mass, we assume a constant density for the air \(\frac{\partial \rho }{\partial t}=0\). In few words, we have small changes in the pressure and then in the density, so we assume an incompressible fluid where the density is constant. Consistent with the steady regime assumed there is no change in the field velocity along the time, so in the vector equation which represents conservation of momentum \(\rho \frac{\partial \mathbf {u}}{\partial t}+\rho \left( \mathbf {u}\cdot \nabla \right) \mathbf {u}=\nabla \cdot \mathbf {\sigma }+\mathbf {F}\) the term \(\rho \frac{\partial \mathbf {u}}{\partial t}\) has been omitted under the consideration of its null value. Obviously, in these equations \(\mathbf {\sigma }=-P\mathbf {I}+\mathbf {\epsilon }\) is the total stress tensor defined as a sum where the viscous stress tensor \(\mathbf {\epsilon }\) is defined through the strain rate tensor \(\nabla \cdot \mathbf {u}+\left( \nabla \cdot \mathbf {u}\right) ^T\) where we assume a general Newtonian media. In this context, \(\mu\) is the dynamics viscosity, \(\rho\) is the air density, and G the reciprocal wall distance. This last term needs an explanation. Turbulence models often use the distance to the closest wall to approximate the mixing length or for regularization purposes. One way to compute the wall distance is to solve the eikonal equation \(|\nabla D|=1\), with \(D=0\) on solid walls and \(\nabla D\cdot \hat{n}=0\) on other boundaries. COMSOL Multiphysics uses a modified eikonal equation based on the approach in 8. This modification changes the dependent variable from D to G = 1/D. Equation \(|\nabla D|=1\) then transforms to \(\nabla G\cdot \nabla G=G^4\). Additionally, the modification adds some diffusion and multiplies \(G^4\) by a factor to compensate for the diffusion. The result is the equation (3) where \(\sigma _w\) is a small constant. If \(\sigma _w\) is less than 0.5, the maximum error falls off exponentially when \(\sigma _w\) tends to zero. The default value of 0.2 is a good choice for both linear and quadratic elements.

To solve this system of coupled partial differential equations, several boundary conditions have been imposed. The walls have been imposed non-sliding: \(\mathbf {u}|_{\ell _w}=\bar{0}\). At the upper outlet of the filter, the pressure (relative to the atmospheric pressure) is free. That is, in the following equations,

$$\begin{aligned}&\hat{n}^T\left[ -P\mathbf {I}+\left( \mu +\mu _T \right) \left( \nabla \cdot \mathbf {u}+\left( \nabla \cdot \mathbf {u}\right) ^T\right) \right] \hat{n}=-\hat{p}\end{aligned}$$

(4)

$$\begin{aligned}&\quad \hat{p}\ge p_0\end{aligned}$$

(5)

$$\begin{aligned}&\quad \mathbf {u}\cdot \mathbf {t}=0\end{aligned}$$

(6)

$$\begin{aligned}&\quad \nabla G\cdot \hat{n}=0 \end{aligned}$$

(7)

\(p_0\) is the difference in pressure which is measured by the sensors. In summary, at the filter inlet, on the fan side, the boundary condition has also been the relative pressure measured by the BMP280 probes. Besides this, in both cases, normal flow to the surfaces is assumed. Figure 6(a) shows a velocity density map for various longitudinal cuts of the device.

In the second device the approach was analogous. However, the vorticity considerably lengthens the convergence (see Fig.7c)). It was for this reason that we used a symmetry plane, which is sketched with vertical pattern lines in Fig. 5(b), that reduced the size of the computational domain by half. As the calculations are scaled cubically with the size of the grid, this reduction facilitated and reduced the simulation time. The Symmetry boundary condition prescribes no penetration and vanishing shear stresses. The boundary condition is a combination of a Dirichlet condition and a Neumann condition for the compressible and incompressible formulations:

$$\begin{aligned} \mathbf {u} \cdot \hat{n}=\bar{0} \left( -P\mathbf {I}+\left( \mu \left( \nabla \cdot \mathbf {u}+\left( \nabla \cdot \mathbf {u}\right) ^T\right) -\frac{2}{3}\mu \left( \nabla \cdot \mathbf {u} \right) \right) \right) \cdot \hat{n}=\bar{0} \end{aligned}$$

The Dirichlet condition takes precedence over the Neumann condition. The sense that this has is relative to the number of boundary conditions with respect to the partial derivative equations that we have. When the number of boundary conditions or constraints exceed the number of equations, the Dirichlet conditions will prevail over the Neumann ones. Therefore, the above equations are equivalent to the following equation for the incompressible formulation:

$$\begin{aligned} \mathbf {u} \cdot \hat{n}=0 \mathbf {K}-\left( \mathbf {K}\cdot \hat{n} \right) \hat{n}=\bar{0} \end{aligned}$$

Where employing the strain rate tensor, we can define \(\mathbf {K}\) as,

$$\begin{aligned} \mathbf {K}=\mu \left( \nabla \cdot \mathbf {u}+\left( \nabla \cdot \mathbf {u}\right) ^T\right) \hat{n} \end{aligned}$$

Fig. 6(a) shows that, for device one working around 30% of its capacity, the average speed is around 3 meters per second and that it is higher in the cylinder axis, reaching more than 3.5 meters per second close to the outlet surface. In Fig. 5(a), we can see a surface probe that corresponds to the plane shown in blue. In this plane perpendicular to the air circulation, there is a flow rate of \(0.084\frac{L}{s}\). Figures 7(a) and 7(b) present the most relevant simulated results regarding fluid dynamics for the second device. In particular, we are interested in speeds/flow rate, and most especially, in the average speed in the region where the LEDs illuminate the flowing air. In Fig. 5(b), we can see a surface probe that corresponds to the plane shown in blue. In this plane perpendicular to the air current, the maximum value (1.2\(\frac{m}{s}\)), the minimum value (0.3\(\frac{m}{s}\)), and the average value (0.42\(\frac{m}{s}\)) of the fluid velocity that crosses it when the fan runs at 100% of its capacity are calculated. On average, the normal speed that crosses the plane probe is seven times slower than the one determined for the first device. In addition, we calculated the flow rate and velocity through the surface probe plotted in green color in Fig. 5(b). The result is \(2.8\frac{L}{s}\) flow rate and \(2.5\frac{m}{s}\) average velocity. Therefore we observe a flow rate more than thirty times the value obtained in device one.

a) Speed density map in device one. A zoom of the speed map of the region illuminated by the UVC LEDs is also shown. b) Pressure density and pressure isoline map in device one. A zoom of the pressure map of the region illuminated by the UVC LEDs is also shown. c) Velocity field and streamlines

(a) Speed density map in device two. A zoom of the speed map of the region illuminated by the UVC LEDs is also shown. (b) Pressure density and pressure isoline map in device two. A zoom of the pressure map of the region illuminated by the UVC LEDs is also shown. (c) Velocity field and streamlines.

In 17 we explain how to deal with the Helmholtz equation,

$$\begin{aligned} \nabla \wedge \mu _r^{-1}\left( \nabla \wedge \mathbf {E} \right) -k^2_0\left( \epsilon _r-\frac{j\sigma }{\omega \epsilon _0} \right) \mathbf {E}=\bar{0} \end{aligned}$$

(8)

in the frequency domain by using COMSOL Multiphysics. In COMSOL the package Frequency Domain Study is used to compute the response of a linear or linearized model subjected to harmonic excitation for one or several frequencies. In this particular case, we assume the diode frequency as the main harmonic frequency of the source. The source was introduced as a boundary condition which is called impedance boundary condition (IBC). This is a kind of absorbing boundary condition16:

$$\begin{aligned}&\sqrt{\frac{\mu _0\mu _r}{\epsilon _0\epsilon _r-j\frac{\sigma }{\omega }}}\hat{n}\wedge \mathbf {H}+\mathbf {E}+\mathbf {E}_{inc}=\end{aligned}$$

(9)

$$\begin{aligned}&\quad \left( \hat{n}\cdot \mathbf {E}_{inc}+\hat{n}\cdot \mathbf {E} \right) \end{aligned}$$

(10)

In 17 we explain how to feed/illuminate a computational domain with this kind of source. We consider two kinds of boundary conditions that partially enclose the computational domain and fix a unique solution, the physical one. These are the perfect electric conductor (PEC) surface \(\hat{n}\wedge \mathbf {E}=\bar{0}\) and the perfect magnetic conductor (PMC) surface \(\hat{n}\wedge \mathbf {H}=\bar{0}\). All these boundary conditions are portrayed in Fig. 8(c). So as to reduce the computational domain, we study a simple diode in each geometry. Figure 8a) depicts the computational domain simulated for device one. In an identical form Fig. 8(b) illustrates the computational domain simulated for device two.

(a) The region where the electromagnetic simulation is performed for device one. (b) Region where the electromagnetic simulation is performed for device two. (c) Types and locations of the boundary conditions used in the resolution of the differential equation in partial derivatives of Helmholtz wave.

For device one, Fig. 9(a), on the left, shows the map of power density per unit area that is distributed in the filtering region. To the right and as a complement, we can see the electric field density. Similarly, in the Fig. 9(b), on the left, illustrates for device two, the distribution of power per unit area. As before, we complement this figure with the one seen on the right, where we have the electric field density map.

In the next subsection, we draw some conclusions from these results, which combined with the flow rates, and therefore the exposure time give us the doses in units of energy per unit area.

(a) On the left, the figure shows a two-dimensional density map or slice of the power per unit area for device one. On the right, the cut corresponding to the electric field density map is shown, also for device one. (b) On the left, the figure shows a two-dimensional density map or slice of the power per unit area for device two. On the right, the cut corresponding to the electric field density map is shown, also for device two.

For the first device, the exposure time is directly proportional to the number of rings of LEDs located in the cylindrical exit region and inversely proportional to the speed of the fluid. We could approach the problem by assuming that there is a continuous length of illumination and that this length of illumination is the radiated region. Then, the exposure time is proportional to that length, and therefore the dose received will be directly proportional to this radiated length, which is nothing other than the length of the irradiated cylinder. By considering a mean velocity of 3\(\frac{m}{s}\) and an irradiated cylindrical length of 7.5cm, the exposure time for this device is 25ms. In Fig. 9(a) we can see a peak of irradiance around 1.5\(\cdot 10^5\frac{W}{m^2}\). However, we should consider the average irradiance that is calculated in 4.5\(\cdot 10^4\frac{W}{m^2}\). Hence, the fluence or doses in the filtering region for the first device is 112\(\frac{mJ}{cm^2}\).

To determine the exposure time in the case of device two, the air in the illuminated region is assumed to have an exposure path of one centimeter, which implies an exposure time of 23.8ms for an average speed of 0.42\(\frac{m}{s}\). In Fig. 9(b) we can see this time a peak of irradiance around 1.3\(\cdot 10^5\frac{W}{m^2}\). However, we should consider again the average irradiance that due to the volume under consideration is higher, 6.2\(\cdot 10^4\frac{W}{m^2}\). Hence, the fluence or doses in the filtering region for the second device is 148\(\frac{mJ}{cm^2}\).

In the Fig. 3(a) we can see that device two has been printed in PLA with the appropriate geometry. Subsequently, the piece through which the air flows has been painted. It can be seen that red paint has been used. This painting is not merely decorative but serves two functions. The first and most important is to caulk, in the sense of waterproofing, the pores of the printed parts. This ensures that flow takes place inside the printed geometry and not through it. The second function is to shield and act as a resonance box for the lighting provided by the diodes. The paint that is basically roof waterproofing/sealing has been mixed with aluminum filings.

The air velocity has been measured with an anemometer https://www.pce-instruments.com/english/api/getartfile?fnr=1279276&dsp=inlinePCE-009 which has a measurement range [0.2,20.0] ± 0.1\(\frac{m}{s}\). The measured values, 2.9±0.1\(\frac{m}{s}\) and 2.4±0.1\(\frac{m}{s}\) for device one and two respectively, are extremely consistent with the simulations carried out.

Combining the table in Fig. 10(c) and the radiation doses obtained in the previous section, we can infer that both devices are capable of deactivating pathogens with an efficiency greater than 99%. In 10 we can read:

“Results: The available data reveals large variations, which are apparently not caused by the coronaviruses but by the experimental conditions selected. If these are excluded as far as possible, it appears that coronaviruses are very UV sensitive. The upper limit determined for the log-reduction dose (90% reduction) is approximately 10.6 mJ/cm² (median), while the true value is probably only 3.7 mJ/cm² (median).

Since coronaviruses do not differ structurally to any great exent, the SARS-CoV-2 virus -as well as possible future mutations- will very likely be highly UV sensitive, so that common UV disinfection procedures will inactivate the new SARS-CoV-2 virus without any further modification.” From this sentence, we know that our criteria for coronaviruses, including the one responsible for COVID-19, deactivation provided by Fig. 10c) is realistic. Other sources as 21,7 are in agreement with.10

However, as is clear from the summary in Table 1, device two is substantially better than one in all the valuations considered (including energy consumption). Device one has been designed to be able to deactivate pathogens, despite penalizing its filtering ratio in terms of flow. We conclude three things from this work. First of all, in a UVC filter, it is necessary to pay attention not only to the radiated power but to the dose that a pathogen would receive. In this sense, the devices must consider the radiation exposure time. The second conclusion is that complexity and electronics do not always provide the best performance. The third conclusion follows from the second and is that there is not or there is not always proportionality between the performance and the cost of a device. What is more, it is desirable that this relation does not exist. The reason, we would obtain the same service with fewer material resources and fewer work units.

(a) The figure outlines the process by which type C ultraviolet radiation is capable of breaking down a DNA or RNA molecule and inactivating a pathogen. b) The illustration shows how the chosen diode has its emission maximum located at the maximum absorption peak of the DNA molecule. (c) The table synthesizes for some bacteria, viruses, and fungi the radioactive doses necessary to inactivate them with the safety of 90%, 99%, and 99.9%. the dose is established as a function of the energy per unit area that is applied to the regions where these pathogens are present

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