Annular Ultraviolet Reactor

In this example, the fluence rate distribution is computed in a simple ultraviolet (UV) water purification reactor. The fluence rate is the amount of radiation that a tiny spherical detector would be exposed to at any point in space, divided by the cross sectional area of such a detector. Fluence rate is a key figure of merit in the evaluation of UV water purification systems because it determines the amount of radiation that bacteria and other pathogens will absorb as they are carried by water flowing past an ultraviolet lamp.

In this example, only the fluence rate will be considered. To also consider how the water flows through the reactor and how the UV dose is accumulated for particles carried by the flow, see the example Annular Ultraviolet Reactor with Particle Tracing: Ray_Optics_Module/Ultraviolet_Sterilization/annular_ultraviolet_reactor_particle.

Model Definition

The model geometry consists of the annular region between a cylindrical UV lamp and the cylindrical reactor that surrounds it. In practice, this reactor would be connected to inlet and outlet tubes so that water can flow past the UV lamp, but in this model only the fluence rate in the immediate vicinity of the lamp is considered.

Ultraviolet Lamp Model

The surface of the lamp is treated as a diffuse (Lambertian) emitter using the node. Rays are released at the lamp surface with uniform spatial density, and the initial direction of each ray is sampled according to the cosine law. The total power of the lamp is specified; in this example the power distribution over the lamp surface area is assumed to be uniform, although it is also possible to define a weighting factor.

As rays propagate through the water, away from the lamp surface, the power of each ray is attenuated based on the internal transmittance of the water, which in this example is taken to be 70% per centimeter of propagation. For distilled water, the internal transmittance of germicidal UV radiation is approximately 98%, so the value of 70% used here could represent that the water is less pure.

As rays propagate through the water, the volumetric fluence rate is computed using the dedicated node. To obtain an accurate distribution of the fluence rate, it is important to release a sufficiently large number of rays and to use a sufficiently fine mesh in the domain. In this example, 100,000 rays were released in order to balance accuracy with solution time and file size considerations, but in some publications the number of rays could be orders of magnitude greater (Ref. 1).

Rather than releasing rays diffusely at the lamp surface, a possible extension of this model would be to release rays from within the volume of the lamp, or to release them from a narrower surface inside the lamp, which could represent an arc that produces the UV radiation. At the lamp surface, then, the light could be refracted into the water domain according to Snell’s law and the Fresnel equations.

The dependence of the fluence rate on the other boundary conditions in the model is also considered. In the first study, the walls are assumed to be perfect absorbers of UV radiation, using the node. In the second study, the walls are treated as perfect reflectors using the node. Eventually, due to the attenuation within the water domain, each ray will make negligible contributions to the fluence rate even as it continues to be reflected at the outer surfaces of the reactor.

Results and Discussion

The fluence rate distribution in a slice of the reactor is shown in Figure 1. The fluence rate is greatest in the immediate vicinity of the lamp and it falls off at a greater distance.

Figure 1: Fluence rate distribution in a cross section of the annular UV reactor, assuming totally absorbing walls.

A 1D plot of the radial fluence rate distribution is shown in Figure 2. The very jagged plot is the actual computed value of the fluence rate along a dataset through the domain. The individual segments of this plot are flat with jump discontinuities between them because the node uses constant shape functions to define the fluence rate in each mesh element that the cut line intersects.

To make the plot less mesh-dependent, a finer mesh could be used, together with a significantly increased number of rays. This example uses only 100,000 rays, whereas in Ref. 1 most values are in the tens of millions of rays. In exchange for using a larger number of rays, the solution time and the memory footprint of the model would increase. Refining the mesh while keeping the number of rays fixed is not recommended because some small mesh elements will not be hit by any ray, and they will erroneously report zero fluence rate.

A low-cost alternative approach to smoothing out the fluence rate distribution is to exploit the axial symmetry of the geometry. At any point outside the lamp, the exact value of the fluence rate can be replaced with the average value over all points having the same radial and axial coordinates,

In this example the average over all azimuthal angles is performed using a coupling. In the settings for this coupling, the expressions for ρ, z, and φ are entered (in terms of the Cartesian coordinates), while in the section only ρ and z are entered. As a result, the expression genproj1(expr) integrates the expression expr along a ring centered at the z-axis.

Since the UV transmittance is 70% per centimeter in this model, and even the shortest distance from lamp to the exterior cylindrical boundaries is 4 cm, the ray power is reduced by at least 75% for rays that reach these outer boundaries (because 0.74 is approximately 0.25). Nevertheless, by applying a boundary condition, the reflected light can still notably affect the fluence rate near the outer walls. The difference in azimuthally averaged fluence rate distribution between absorbing and reflecting walls is shown in Figure 3. While the two curves appear to be very close together, at the maximum radial coordinate the difference between them is a significant fraction of the fluence rate magnitude. Therefore, the effect of reflection can significantly alter the accumulated dose for particles that flow through the reactor at the maximum possible radial coordinate.

Figure 2: Comparison of the computed radial fluence rate distribution with the azimuthally averaged value and a simplified analytic solution.

Figure 3: Comparison of the fluence rate distribution for absorbing and reflecting walls.

Reference

1. Y.M. Ahmed, M. Jongewaard, M. Li, and E.R. Blatchley III, “Ray Tracing for Fluence Rate Simulations in Ultraviolet Photoreactors,” Environ. Sci. Technol., vol. 52, no. 8, pp. 4738–4745, 2018.

Ray_Optics_Module/Ultraviolet_Sterilization/annular_ultraviolet_reactor

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
r_lamp	1[cm]	0.01 m	Lamp radius
r_reac	5[cm]	0.05 m	Reactor radius
L_reac	100[cm]	1 m	Reactor length
L_lamp	80[cm]	0.8 m	Lamp length
d_lamp	L_reac-L_lamp	0.2 m	Lamp displacement
mid_lamp	d_lamp+L_lamp/2	0.6 m	Lamp midplane location
P	40[W]	40 W	Total source power

Geometry 1

Reactor

In the toolbar, click

In the window for , type Reactor in the text field.

Locate the section. In the text field, type r_reac.

In the text field, type L_reac.

Lamp

In the toolbar, click

In the window for , type Lamp in the text field.

Locate the section. In the text field, type r_lamp.

In the text field, type L_lamp.

Locate the section. In the text field, type d_lamp.

Difference 1 (dif1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

Find the subsection. Click to select the

toggle button.

Select the object only.

In the toolbar, click

Add a coupling to be used in results processing.

Definitions

General Projection 1 (genproj1)

In the toolbar, click

and choose .

Select Domain 1 only.

In the window for , locate the section.

In the text field, type sqrt(x^2+y^2).

In the text field, type z.

In the text field, type atan2(y,x).

Locate the section. In the text field, type sqrt(x^2+y^2).

In the text field, type z.

The above expressions represent a conversion from Cartesian to cylindrical polar coordinates. For any point (r, z) in the geometry, the input argument to this projection coupling will be integrated for all azimuthal angles φ from 0 to π.

Geometrical Optics (gop)

In the window, under click .

In the window for , locate the section.

In the text field, type 0.

Select the check box.

Locate the section. From the list, choose .

Ray Properties 1

In the window, under click .

In the window for , locate the section.

In the λ0 text field, type 254[nm]. A low-pressure mercury vapor lamp would emit most of its UV radiation at this wavelength (Ref. 1).

Medium Properties 1

In the window, click .

In the window for , locate the section.

From the n list, choose . In the associated text field, type 1.38. This is approximately the refractive index of water at 254 nm (Ref. 1).

From the list, choose .

From the τi,10 list, choose . In the associated text field, type 0.7. Different values of the internal transmittance of water can be used here, depending on the clarity of the water. For pure water, the internal transmittance of germicidal UV radiation is about 0.98 per centimeter (Ref. 1). The value of 0.7 shown here indicates that the water is less clear.

Release from Boundary 1

In the toolbar, click

and choose .

Select Boundaries 5, 6, 9, and 10 only. These are the curved surfaces of the inner cylinder. Use the middle mouse wheel to select an interior boundary in the Graphics window. You can also make it easier to view the interior boundaries by enabling wireframe rendering.

In the window for , locate the section.

From the list, choose .

In the N text field, type 100000.

Locate the section. From the list, choose .

Select the check box.

In the Nw text field, type 1.

Specify the r vector as


0	t1
0	t2
1	n

From the list, choose .

These settings will cause rays to be released from the lamp surface with uniform spatial density, with a distribution of initial directions following the cosine law with respect to the surface normal at each release position. To ensure that the surface normal points in the outward direction rather than the inward direction, look for the arrows in the Graphics window.

Locate the section. In the Psrc text field, type P.

Fluence Rate Calculation 1

In the toolbar, click

and choose .

Select Domain 1 only.

Wall 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Add a boundary condition. This will be disabled in the first study, so that all of the exterior boundaries absorb the outgoing UV radiation. Then the boundary condition will be enabled in a second study to predict the effects of reflection on the fluence rate distribution.

Mirror 1

In the toolbar, click

and choose .

Select Boundaries 1–4, 8, and 11 only.

In this example a mesh is used. To get higher resolution of the radial dependence of the fluence rate, a structured mesh could be created, although this may require the addition of mesh control surfaces to the geometry sequence.

Mesh 1