Reducing the Computational Cost of UV Reactor Performance ... · Reducing the Computational Cost of...

Reducing the Computational Cost of UV Reactor Performance Calculations

by

Colin Powell

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Chemical Engineering and Applied Chemistry University of Toronto

© Copyright by Colin Powell, 2017

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Reducing the computational cost of UV reactor performance calculations

Colin Powell

Doctor of Philosophy

Department of Chemical Engineering and Applied Chemistry

University of Toronto

2017

Abstract

The performance of a UV reactor can be described using the concept of UV dose, which is the product of the

residence time of a microbe or particle and the intensity, or fluence rate, of UV light hitting that particle in a reactor.

The process to model the reactor performance of a UV reactor, hereby known as the reactor performance process

(RPP), consists of three separate models: 1) the computational fluid dynamics (CFD) model: the UV fluid flow field

modeling, typically performed using commercial software; 2) the fluence rate model: the physics of the intensity of

the UV irradiation; and 3) the reactor performance model: RED, for example, an organism-specific metric for

delivered UV dose. The combination of these three models to numerically estimate the UV reactor performance has

never been subjected to scrutiny for relevancy, efficiency, nor computational expense. The aim of this thesis was to

examine and reduce the computational cost of the RPP and as a result of this work, in certain scenarios, the cost of

calculating the reactor performance of a UV reactor was reduced from over 300 billion steps to less than 12,000.

This was accomplished by optimizing each of the models of the RPP: by focusing on four key factors in CFD

modeling (initial condition, mesh size, time step, and particle tracking), the computational cost of that step was

reduced by five times; by systematizing fluence rate model implementation, the computational cost of that step was

reduced by at least nine times; by using mixing as a proxy for reactor performance in certain lamp orientations, the

computational cost was reduced by over 99%. Further scenarios need to be examined where mixing can be used as a

proxy for reactor performance; this optimization offered the best reduction in computational cost. This optimization

would reduce the overall computational cost of the RPP for more general UV reactors but also provide a different

perspective from which to examine the performance of UV reactors.

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Acknowledgements To my family and friends who had mostly stopped asking me when I’ll be done, you can ask again now! Thank you

for your wonderful support throughout the years; I wouldn’t have finished without your constant hounding. Well, I

mean, I would have, but it’s nice to know you care!

To Mom and Dad, thanks for your constant support and encouragement and grounding and the cookies for every

occasion. To Val and Les, I appreciate the friendship and the interest anywhere we are in the world. Hopefully baby

Max can be the 2nd PhD in the family.

To Uncle Gary, Aunts Dianne and Lynette and all the cousins and cousins-in-law, thanks for the food, the wine, and

the open invitations into your homes. I think I now have some kindling for the next bonfire.

To Ari, your support and patience has been invaluable and at least now I can watch you play video games on

Saturday morning. And finally I can participate in Games Night and we can get a dog. JK. Hi Daisy!

To Yuri, your guidance, knowledge and necessary but always constructive criticism has been absolutely vital in

getting me through these past ten years as a graduate student. You have taught me so much, not only with respect to

my thesis topics, but professionalism, writing skills, idea creation and implementation, story telling, creativity, and

business acumen…I could go on, but I won’t.

To the CMTE team, what a great group of academics to be a part of. I’m honored to be included with you all and

I’m happy to call you colleagues and friends.

To the staff and Professors in the Department of Chemical Engineering and Applied Chemistry, your

professionalism and creativity has always helped me immensely.

To Ben and Kevin, who beat me, *shakes fist*. To Daniel, I promise it’ll be over soon.

Thanks.

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Table of Contents

Acknowledgements ………………………………………………………………...…………………… iii Table of Contents ……………………………………………………………………...………………... iv List of Tables………………………………………………...................................................................... vi List of Figures………………………………………………………………...………………………. …vii Nomenclature.….........……………………………………………………………………………….…... x

Chapter 1 - Introduction .............................................................................................................................. vi 1.1 Background ................................................................................................................................. 1 1.2 Problem ....................................................................................................................................... 3

1.2.1 Transient CFD Models ....................................................................................................... 3 1.2.2 Fluence Rate Models .......................................................................................................... 5 1.2.3 Reactor Performance Models ............................................................................................. 6

1.3 Hypothesis and Objectives .......................................................................................................... 7 1.3.1 Transient CFD Models ....................................................................................................... 8 1.3.2 Fluence Rate Models .......................................................................................................... 9 1.3.3 Reactor Performance Models ............................................................................................. 9

1.4 Novel Contributions .................................................................................................................... 9 1.4.1 Transient CFD Models ..................................................................................................... 10 1.4.2 Fluence Rate Models ........................................................................................................ 14 1.4.3 Reactor Performance Models ........................................................................................... 15

1.5 Thesis organization .................................................................................................................... 17 1.6 References ................................................................................................................................. 18

Chapter 2 – Transient CFD Models ............................................................................................................ 20 2.1 Key Results ................................................................................................................................ 20 2.2 Implications ............................................................................................................................... 20 2.3 Manuscript ................................................................................................................................. 21

2.3.1 Abstract ............................................................................................................................. 21 2.3.2 Introduction ....................................................................................................................... 22 2.3.3 Transient simulations for UV reactors .............................................................................. 25 2.3.4 Methodology ..................................................................................................................... 27 2.3.5 Results ............................................................................................................................... 37 2.3.6 Validation of the proposed methodology ......................................................................... 44 2.3.7 Discussion ......................................................................................................................... 50 2.3.8 Conclusion ........................................................................................................................ 51

2.4 References ................................................................................................................................. 52 Chapter 3 – Fluence Rate Models ............................................................................................................... 54

3.1 Key Results ................................................................................................................................ 54 3.2 Implications ............................................................................................................................... 54 3.3 Manuscript ................................................................................................................................. 55

3.3.1 Abstract ............................................................................................................................. 55 3.3.2 Introduction ....................................................................................................................... 55 3.3.3 Methodology ..................................................................................................................... 56 3.3.4 Effect on dose calculations ............................................................................................... 63 3.3.5 Conclusions ....................................................................................................................... 64

3.4 References ................................................................................................................................. 66 Chapter 4 – Reactor Performance Models .................................................................................................. 67

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4.1 Key Results ................................................................................................................................ 67 4.2 Implications ............................................................................................................................... 67 4.3 Manuscript ................................................................................................................................. 68

4.3.1 Abstract ............................................................................................................................. 68 4.3.2 Introduction ....................................................................................................................... 69 4.3.3 Methodology ..................................................................................................................... 73 4.3.4 Results ............................................................................................................................... 91 4.3.5 Conclusions ..................................................................................................................... 106

4.4 Appendix A ............................................................................................................................. 108 4.4.1 Additional Weighting Systems ....................................................................................... 110 4.4.2 Results ............................................................................................................................. 112 4.4.3 Conclusions ..................................................................................................................... 115

4.5 References ............................................................................................................................... 116 Chapter 5 – Conclusions and Future Work ............................................................................................... 118

5.1 Purpose .................................................................................................................................... 118 5.2 Limitations of the research ...................................................................................................... 118 5.3 Suggestions for Future Work .................................................................................................. 120 5.4 References ............................................................................................................................... 121

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List of Tables

Table 1-1 - Sampling of values for number of point or segment sources provided in the literature, with accompanying lamp length and lamp sleeve diameters……………………………………...………………………..6

Table 2-1 - Time steps used in transient simulations for G1…………………………………………………..…….37

Table 2-2 - Steady-state RED of each mesh and particle scenario for G1………………………………………..….38

Table 2-3 - Transient RED for each time step under IC1-M2 and percent change in RED from MB-TSB ………...39

Table 2-4 - Transient RED and the % difference from best-case RED for all IC2 scenarios with G1………………40

Table 2-5 - Steady-state RED of each mesh and particle scenario for G2…………………………………………...45

Table 2-6 - Overall transient RED and percent change from the smaller time step for G2………………………….46

Table 2-7 - Steady-state RED of each mesh and particle scenario for the third UV reactor………………………...48

Table 2-8 - Overall transient RED and percent change from the smaller time step for G2………………………….59

Table 3-1 - Use of MPSS and MSSS FRMs in literature………………………………………………………….....56

Table 3-2 - Technical specifications of the lamp and reactor used………………………………………………......64

Table 4-1 - Lamp and flow conditions for both UV reactors………………………………………………….……..89

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List of Figures Figure 1-1 - The general problems associated with each of the three models of the UV reactor performance calculation………………………………………………………………………………………………………....…3

Figure 1-2 - The hypotheses associated with each of the three models of the UV reactor performance process……8

Figure 1-3 - Summary of the novel contributions provided by the three manuscripts……………...…….…………10

Figure 1-4 - Transient UV reactor simulation pathway……………………………………………….…….…….....11

Figure 1-5 - Proposed methodology for transient UV reactor simulations from Powell & Lawryshyn (2016)…..…12

Figure 1-6 - New pathway for transient UV reactors…………………………………………………...…….….......13

Figure 1-7 - Summary of the computational cost reductions from the systematic review of the three models of the UV reactor performance calculation………………….…………………………………………………..…….……17

Figure 2-1 - Illustration of the proposed transient UV modeling methodology…………………………..……...….29

Figure 2-2 - Geometry of the first test UV reactor, G1, with a UV lamp oriented perpendicular to flow…………..30

Figure 2-3 - Geometry of the second test UV reactor, G2, with a UV lamp oriented perpendicular to flow and a cylinder placed 10 cm upstream of the lamp……………………………………………………………………...…31

Figure 2-4 - Geometry of the third test UV reactor, G3, with two four-lamp banks oriented parallel to the direction of flow………………………………………………………………………………………………………………..32

Figure 2-5 – Illustration of the different scenarios used to compare to the best-case solution. A dashed circle indicates a steady-state simulation was performed; a dotted circle indicates a transient simulation was performed..35 Figure 2-6 - Illustration of the mesh intensity around the lamp zone looking perpendicular to the flow in the reactor. Note this shows only a 2D picture of the mesh for illustrative purposes; the actual mesh is 3D……………………36

Figure 2-7 - Velocity magnitude at various locations downstream of the lamp for each mesh in G1……………….39

Figure 2-8 – The dose distributions of IC1-M2-Ts2, IC2-M2-TS3, and MB-TSB for G1…………………………..41

Figure 2-9 – The percent difference in the number of particles per dose bin in the dose distributions for the IC1-M2-TS2 and IC2-M2-TS3 scenarios compared to the MB-TSB dose distribution………………………………………41

Figure 2-10 - tRED (normalized by the overall RED) over a number of simulation time steps for different tRTS for G1-IC1-M2-TS2……………………………………………………………………………………………………...43

Figure 2-11 - Percent difference in RED of IC1-M2-TS2 and IC1-M2-TS3 from the best-case solution for a range of inactivation rate constants……………………………………………………………………………………………44

Figure 2-12- Overall transient dose distributions of M2 with three different time steps………………………….....46

Figure 2-13 - tRED (normalized by the overall RED) over a number of simulation time steps for different tRTS for G2 with M2 and TS2 ………………………………………………………………………………………………...47

Figure 2-14 - Overall transient dose distributions of M3 with three different time steps……………………………48

Figure 2-15 - tRED (normalized by the overall RED) over a number of simulation time steps for different tRTS for G3 with M2 and TS2……………………………………………………………………………………………........49

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Figure 3-1 – Percent error in fluence rate (from NPS = 5000) for various NPS using the MPSS without reflection and refraction. (Lamp length= 1.0m; lamp sleeve radius = 0.0250 m; UVT = 100%)………………………………58 Figure 3-2 - Error in fluence rate (from NSS = 5000) for various NSS using the MSSS with reflection and refraction. (Lamp length= 1.0m; lamp sleeve radius = 0.0250 m) ………………………………………………….60 Figure 3-3 - The NSS as a function of very small lamp lengths for the MSSS FRM………………………………..61

Figure 3-4 - The optimal NSS as a function of UVT………………………………………………………………...62

Figure 3-5 – Dose distribution of a reactor calculated using NSS = 267 (-o-) and NSS = 1000 (-*-)……………….63

Figure 4-1 - Illustration of the division of the entrance and exit cross-sections from the single-lamp cross-flow UV reactor………………………………………………………………………………………………………………...73

Figure 4-2 - The weightings for the original (a) and modified (b) mixedness equations, respectively. Both are shown when the initial injection region is in the top left hand region……………………………………………………....78 Figure 4-3 - The weightings for the original (a) and modified (b) mixedness equations, respectively. Both are shown when the initial injection region is in the middle region…………………………………………………………….79

Figure 4-4 - Number of particles in each region after one particle moves to different regions. The local original and modified mixedness values are shown for each scenario.…………..……………………………………………….81

Figure 4-5 Number of particles in each region after 250 particles moves to different regions. The local original and modified mixedness values are shown for each scenario.…………..……………………………………………….81

Figure 4-6 - Number of particles in each region after 500 particles moves to different regions. The local original and modified mixedness values are shown for each scenario.…………..……………………………………………….82

Figure 4-7 - Number of particles in each region after the particles moves to different regions. The local original and modified mixedness values are shown for each scenario.…………..……………………………………………….82

Figure 4-8 - Perfect mixing states for each of the two local mixedness values. The left side shows the perfect mixing state for the local original mixedness whereas the right side shows the perfect mixing state for the local modified mixedness. .…………..……………………………………………………………………………………83

Figure 4-9 – Number of particles where the proportion of particles is constant for each scenario. Both the original and modified mixedness values are shown..…………..………………………….………………………………….84

Figure 4-10 - Example of one of the iterations of the local mixedness probabilities as the number of regions is increased from four (a), sixteen (b), and 64 (c). The local original mixedness values for each scenario are shown below each case for when the particles were injected from the top left corner region………………………………86

Figure 4-11 - The local original mixedness of 100 000 iterations as the number of regions increases……….……..87

Figure 4-12 - Geometry of the parallel UV reactor……………………...…………………………………………..89

Figure 4-13 – Velocity contour of cross-flow (top) and parallel (bottom) UV reactors in the exact centre of the reactors (in x and y position) and in the area around the lamp. The units are in m/s………………………...……..92 Figure 4-14 - Dose distribution and RED of the cross-flow and parallel UV reactors……………………………………………………………………………………………………………....93

Figure 4-15 – The original and modified mixedness for the cross-flow (top two squares) and the parallel (bottom two squares), respectively, at various points throughout the reactor zone…………………………………………94

ix

Figure 4-16 – Local original mixedness values using 16 regions (a) and 400 regions (c) and local modified mixedness values using 16 regions (b) and 400 regions (d) for cross-flow UV reactor…………………………….97

Figure 4-17 - Local original mixedness values using 16 regions (a) and 400 regions (c) and local modified mixedness values using 16 regions (b) and 400 regions (d) for parallel UV reactor………………………………..99

Figure 4-18 - Local RED of the cross-flow reactor with 400 regions; the RED is normalized by the maximum RED in the 400 regions……………………………………………………………………………………………….......101

Figure 4-19 - Local RED of the parallel reactor with 400 regions; the RED is normalized by the maximum RED in the 400 regions. ……………………………………………………………………………………………….........101

Figure 4-20 - Plot of local mixedness versus RED for the cross-flow reactor using the original and modified mixedness equation…...……………………………………………………………………………………….........102

Figure 4-21 - Plot of local mixedness versus RED for the parallel reactor using the original and modified mixedness equation. . …………………………………………………………….……………………………………….........103

Figure 4-22 – Dose distributions and RED (MS2) of the simulated cross-flow particle tracks using 25 (top) and 400 (bottom) regions…………………………………………………………………………………………………….105

Figure 4-23 – The local RED and mixedness for the parallel UV reactor using the three different mixedness equations……..……………………………………………………………………………………………….…….113

Figure 4-24 – The local RED and mixedness for the Cross-flow UV reactor using the three different mixedness equations……..……………………………………………………………………………………………….…….114

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Nomenclature Chapter 1 α = tolerance of the fluence rate model CFL = Courant-Freidrich-Levy Number Hi,modified = modified Shannon entropy in the i-th region of the entrance cross-section L = lamp length (m) Mi,modified = modified mixedness in the i-th region of the entrance cross-section n = number of regions in a cross-section nopt, PS = optimal number of point sources nopt, SS = optimal number of segment sources NPS = number of point sources NSS = number of segment sources p = axial distance from the centre of the lamp divided by lamp length Pij = probability that particles leaving the i-th region of the entrance region enter the j-th region of the exit cross-section r = perpendicular distance to the particle from the i–th point segment or source segment (m) rsleeve = the radius of the lamp sleeve (m) RED = reduction equivalent dose (mJ/cm2) UVT = ultraviolet transmittance (%/cm) Wij = weighting factor for that particles leaving the i-th region of the entrance region enter the j-th region of the exit cross-section Wij

* = normalized weighting factor for that particles leaving the i-th region of the entrance region enter the j-th region of the exit cross-section WT,i = sum of all weighting factors on an entrance cross-section Chapter 2 Δt = chosen time step (s) Δx = average streamwise length of an element near the lamp (m) Di = the dose of the i-th particle (mJ/cm2) D10 = the dose required to get to 1-log inactivation kµ = the inactivation rate constant (cm2/mJ) np = the number of particles per simulation ns = the number of simulations NT = total number of particles tRED = transient RED (mJ/cm2) tRTS = transient RED time step (s) U = bulk flow velocity (m/s) Chapter 3 Eo = fluence rate (W/m2) H = horizontal distance to the particle from the midpoint of the lamp (m) P = effective lamp power (W) R1 and R2 = reflection coefficients at the air-quartz and quartz-flow medium interfaces, respectively, which can be found in Liu et al. (2004) θ1 = first incident angle between the point or segment source and the particle Chapter 4 i1 and i2 = row and column indices of each region on the entrance cross-sections, respectively, and the j1 and j2 = row and column indices on each region on the exit cross-sections, respectively

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nc = total number of columns in the cross-sections Nij = number of particles leaving from region i and entering region j NTi = total number of particles that left from region i Vj = area of region j on the exit cross-section V*

j = normalized area of region j on the exit cross-section Vo = area of region of interest on the exit cross-section 𝑉"∗= total normalized area of the cross-section

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Chapter 1 - Introduction

1.1 Background

Ultraviolet (UV) disinfection is a viable alternative to chlorine disinfection technology that uses UV light (typically,

with a wavelength of 254 nm) to inactivate pathogens in water and wastewater. The benefits of using UV light for

disinfection are three-fold: 1) it has been proven to be a very effective disinfectant against a large number of

pathogens; 2) it has no chemical residual and therefore does not affect the local environment; and 3) it is

economical, in terms of both fixed and variable costs (Rudd et al., 1989).

This work will not go into significant detail on the mechanisms behind the UV disinfection process, nor any general

background on the design, operation, regulation, or applications for UV disinfection; the reader is directed towards

the the UV Disinfection Guidance Manual for more information (USEPA, 2006).

The performance of a UV reactor can be described using the concept of UV dose, which is the product of the

residence time of a microbe or particle and the intensity, or fluence rate, of UV light hitting that particle in a reactor.

An in-field test called a bioassay is used to quantify reactor performance and validate performance. Numerical

modeling of UV reactors is used to simulate UV disinfection reactors for reactor design and optimization – this still

requires the calculation of the performance of the reactor, often denoted as the reduction equivalent dose (RED),

which is an organism-specific metric for the amount of UV dose delivered to a particular organism in the UV

reactor. UV reactor modeling indeed offers the advantage of troubleshooting and optimizing the reactors before they

are constructed (Chiu et al., 1999; Lyn et al., 1999; Ducoste et al., 2005; Wols et al., 2010a) and has led to vast

improvements in the performance of UV reactors (Alpert et al., 2010; Blatchley, 1997).

The process to model the reactor performance of a UV reactor, hereby known as the reactor performance process

(RPP), consists of three separate models: 1) the computational fluid dynamics (CFD) model: the UV fluid flow field

2

modeling, typically performed using commercial software; 2) the fluence rate model: the physics of the intensity of

the UV irradiation; and 3) the reactor performance model: RED, for example, an organism-specific metric for

delivered UV dose. The process is implemented as such:

To obtain the RED, CFD modeling is first used to find the flow field and to simulate and track particles

flowing through the UV reactor. Then, a fluence rate model is used to calculate the fluence rate at each

particle’s position as it flows through the reactor. The fluence rate at every position for each particle is

integrated over the time each particle spends in the reactor to obtain the dose of each particle. Then, the dose

of each particle is used to calculate the RED.

The above three types of models have been examined thoroughly in the literature, but have yet to be examined

holistically as a process, the RPP, as is the central problem discussed in the next section.

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1.2 Problem

The combination of these three models to numerically estimate the UV reactor performance has never been

subjected to scrutiny for relevancy, efficiency, nor computational expense. The problems associated with each of the

models are summarized in Figure 1-1 and further expanded upon in this section.

Figure 1-1 – The general problems associated with each of the three models of the UV reactor performance calculation.

1.2.1 Transient CFD Models

Typically, steady-state CFD models have been used to simulate UV wastewater reactors, because they are easy to

implement and understand (Lyn, 2004). However, a number of studies have shown that transient modeling captures

more of the transient features in a UV reactor that can potentially alter dose distribution (Lyn, 2004; Munoz et al.,

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2007; Younis & Yang, 2010). While a host of studies have used transient UV reactor simulations previously, such as

Lyn (2004), there has never been an examination of the key factors associated with the modeling of transient flow in

UV reactors, which we identified in Powell & Lawryshyn (2016) as: (1) initial conditions, (2) mesh size, (3) CFD

simulation time step, and (4) a particle-tracking strategy. The systematic review of the transient simulation

methodology for UV reactors is examined in terms of these four factors; each of these approaches are discussed

briefly below.

To find the velocity field for a transient simulation, an initial condition must be imposed. Georgiadis et al. (2010)

recommended that a recycle condition be utilized, namely, using the velocity field at the outlet of the reactor as the

initial velocity field.

Ensuring mesh independence for a transient simulation is often quite difficult because the transient velocity fields

and high computational expense make it challenging to compare results from different mesh sizes (Wols et al.,

2010b; Breuer, 2000; Georgiadis et al., 2010). A consistent approach to transient mesh independence is not clear in

the broader CFD literature, but the modeler must be aware of the balance between accuracy and computational

expense (Wols et al., 2010b). For CFD simulations, the mesh, which overlays the geometry of the body being

simulated in order to facilitate calculation of the relevant physical equations and parameters, must be independent of

the solution. For a steady-state solution, this mesh independence study is typically done using three meshes of

increasing density, with the working mesh being the mesh with the lowest number of elements that can calculate a

factor relevant to the simulation within a certain tolerance of the mesh with a greater number of elements. For UV

reactors, that factor is typically the RED.

The choice of the CFD simulation time step is also critical for transient simulations. Typically, at least three time

steps are chosen. Then, the time step that produces a factor relevant to the simulation within a certain tolerance of

that calculated using a smaller time step is chosen as the working time step. Wols et al. (2010b) and Younis & Yang

(2010) chose a time step based on the Courant-Friedrich-Levy (CFL) number, a dimensionless number based on

how long a particle takes to pass through one mesh element (Courant et al., 1928). Wols et al. (2010b) chose a time

step that was intended to produce a CFL number less than one, which ensures that a particle will move the distance

5

of less than one mesh element per time step. Regardless, there has been little agreement on a consistent

methodology.

A particle-tracking strategy for a transient UV reactor simulation has yet to be addressed in the literature. Wols et al.

(2010b) injected 50,000 particles over 1,000 time steps to “obtain more statistically reliable results” using LES, but

it is unclear whether this gradual approach to particle injection was used for the transient k-epsilon simulation.

By considering each of the above four factors together, it means that for transient simulations, at least nine transient

simulations must be performed to ensure mesh and time step independence (three meshes, with three time steps

tested for each). Even still, as noted above, this independence may not be found (Georgiadis et al., 2010). This is a

computationally expensive exercise that is examined systematically in this thesis to reduce computational cost.

1.2.2 Fluence Rate Models

For fluence rate models (FRMs) that model a UV lamp using a number of point or segment sources, such as the

multiple point source summation (MPSS) or multiple segment source summation (MSSS) FRMs, there has never

been agreement on an optimal number of point or segment sources (NPS or NSS, respectively) to discretize that

lamp. The calculation is dependent on the summation of the fluence rate over all the point sources on the lamp.

Thus, as the number of point sources increases, the calculation becomes more computationally expensive with no

gains in accuracy.

As shown in Powell & Lawryshyn (2015), and below in Table 1-1, there is a wide range in the number of point or

segment sources used in the literature to implement fluence rate models, with no apparent evidence for those

choices.

6

Table 1-1 Sampling of values for number of point or segment sources provided in the literature, with accompanying lamp length and lamp sleeve diameters.

FRM used Reference Lamp length (m)

Sleeve diameter (mm)

NPS/ NSS1

nopt, PS nopt, SS

MPSS Bolton (2000) 0.645 100 1001 23 36 MSSS Quan et al.

(2004) 0.762 25.5 1E6 101 177

UVCalc®3D3 and MSSS

Liu et al. (2004) 0.28 37 1000 27 276

UVCalc®3D Jin et al. (2005) 0.375 n/p2 1000 - - MPSS Rahn et al.

(2006) 0.15 64 10014 9 66

UVCalc®3D Liu et al. (2007) <0.4775 n/p 1000 - - UVCalc®3D Munoz et al.

(2007) 1.1971 68 1000 60 104

MPSS Wols et al. (2010a)

<0.15 n/p 200 - -

MSSS Wols et al. (2010b)

<0.46 50 n/p 28 36

UVCalc®3D Li et al. (2012) 0.297 23 1000 44 49 1NPS = number of point sources; NSS = number of segment sources 2n/p = not provided; the information was not provided in the paper or difficult to infer. 3UVCalc®3D (Bolton Photosciences, Inc: Edmonton, Canada) is a commercially available software version that implements the MSSS FRM with 1000 segment sources. 4This particular value was inferred based upon references in the text to Bolton (2000). 5This particular value was inferred based on diagrams provided in the text; the length could not have been longer than the value given above.

6This particular value is not valid, however, because the methodology proposed here is not valid for lamp lengths less than nine times the lamp sleeve radius.

Table 1-1 also shows the optimal point and segment sources, nopt, that were calculated using the models developed

in Powell & Lawryshyn (2015), with the nopt up to 20 times lower than those used by some authors. As a result, the

reduction in computational cost that could be achieved by using the method developed in Powel & Lawryshyn

(2015) is clear for each study: For Bolton (2000), the computational cost could be reduced by over forty times and

for Rahn et al. (2006), the computational cost could be reduced by over 150 times.

1.2.3 Reactor Performance Models

RED has long been used as a suitable reactor performance metric for UV reactors. There have been other attempts to

model the reactor performance of the UV reactor, such as Blatchley (1997), with the use of spherical balls, but these

efforts have not been widely adopted. Regardless, an alternative reactor performance metric with a lower

computational cost may be necessary with the use of higher complexity reactors, but perhaps more importantly, with

higher complexity turbulence models.

7

Mixing of particles within the flow domain has long been thought to correlate directly with reactor performance, but

the impact has yet to be quantified in the literature. The majority of work discussing the effect of mixing on UV

reactor performance has been qualitative and experimental (Haas & Sakellaropoulos, 1979; Severin et al., 1984;

White et al., 1986; Qualls & Johnson, 1985; Thampi & Sorber, 1987; Gandhi et al., 2011); there was clear consensus

that maximizing radial mixing increased UV reactor performance and that greater axial mixing was undesirable

(Qualls & Johnson, 1985). It is perceived that greater radial mixing reduces the variation in the average distance

between the organisms and the lamp (White et al., 1986) which produces a narrow dose distribution and a more

efficient reactor.

A metric that can quantify the UV reactor particle mixing process is necessary because it can not only provide a

better description of mixing compared to qualitative methods, but could also replace RED as the measurement for

reactor performance. The Shannon entropy (Shannon, 1948) can provide a local mixing quantity and accurately

describes the mixing in the radial direction in a UV reactor, so it is well suited to describe mixing in UV reactors.

Ogawa & Ito (1973) were the first to use the Shannon entropy to determine the mixedness of a general chemical

reactor system. They used the example of a solute spreading from one region to another in a chemical reactor to

calculate Shannon entropy. The Shannon entropy was more recently applied to Lagrangian mixedness quantification

for general fluid flow in a channel by Camesasca et al. (2006); this application is the most relevant to quantifying

mixing in UV reactors.

One problem associated with the application of the original Shannon entropy and the mixedness equation to UV

reactors is that it treats particles that do move and particles that do not move equally – they both add to mixing. This

criterion is not logical as a particle that stays within the same region should be considered to be not-mixed. For this

reason, as part of the work in this thesis, the original Shannon entropy was modified to better describe mixing in UV

reactors but also to improve the correlation between mixedness and RED, thus negating the need for calculating

RED altogether in some scenarios and dramatically reducing the computational cost of the RPP.

1.3 Hypothesis and Objectives

8

The general hypothesis of this thesis is that there is potential to reduce the computational cost of reactor

performance calculations through a systematic review of the three models associated with the RPP. Thus, the overall

objective of this thesis is to reduce the computational cost associated with the RPP by meeting a number of sub-

objectives connected to each of the three models of the RPP. These sub-objectives are summarized in Figure 1-2

and discussed in more detail below.

Figure 1-2 The hypotheses associated with each of the three models of the UV reactor performance process.


To reduce the computational cost of the implementation of the CFD models, a systematic review of each of the four

factors involved in deciding on solution independence discussed earlier (initial condition, mesh size, simulation time

step and particle tracking). The specific objective is, using the above factors, produce a standard methodology for

transient UV reactor simulations. Meeting this objective will enhance the systemization of the implementation of the

CFD models and thus lead to a reduced computational cost for the RPP.

9


To reduce the computational cost of the fluence rate models, a process needs to be developed that can find the

optimal number of point or segment sources to calculate the fluence rate that is computationally expensive but

maintains accuracy. Thus, the objective within the fluence rate models is to use the MPSS and MSSS equations to

find that optimal NPS/NSS in terms of the relevant factors, such as lamp length, lamp sleeve radii, UVT, and others.

A methodical way to find the optimal discretization is necessary to reduce computational cost.


To reduce the computational cost of the reactor performance models, a new reactor performance metric must be

used. The objective here is to find a new reactor performance metric that reduces computational cost, but also

describes reactor performance in a way that is strongly positively correlated with RED to ensure we know the

reactor performance is consistent with the RED.

1.4 Novel Contributions

By completing the objectives as listed in the previous section, a series of novel contributions have been developed.

A summary of the contributions is depicted in Figure 1-3; each of these are explained briefly in this section and in

much more detail in the manuscripts provided later in this thesis. For each of the three models of the RPP, an

estimate of the reduction in computational cost of each major contribution is provided.

10

Figure 1-3 – Summary of the novel contributions provided by the three manuscripts


The typical method for obtaining a suitable solution for a transient UV reactor simulation is shown visually in

Figure 1-4, where each mesh is represented by an M (M1, M2 and M3) and each time step for each mesh is

represented by a TS (TS1, TS2, TS3). As can be seen in Figure 1-4, there are a total of nine simulations, three

simulations per mesh density based on the three different time steps for each mesh density. Three time steps are

chosen for each mesh and thus, at least nine simulations were required to ensure mesh and time step independence.

11

Figure 1-4 – Transient UV reactor Simulation Pathway

The four factors that affect solution suitability were identified in Powell & Lawryshyn (2016) as: (1) initial

conditions; (2) mesh; (3) time step; and (4) particle-tracking strategy. By performing a systematic review of each of

these strategies, a standard methodology was developed in Powell & Lawryshyn (2016) that can be used to

dramatically reduce the computational cost of the RPP. The methodology is provided below in Figure 1-5.

12

Figure 1-5 – Proposed methodology for transient UV reactor simulations from Powell & Lawryshyn (2016).

13

Implementing this methodology reduces the computational cost simply by reducing the number of transient

simulations that need to be completed. The new process, as developed in Powell & Lawryshyn (2016), to obtain a

converged solution is shown in Figure 1-6.

Figure 1-6 – New Pathway for Transient UV Reactors

In the new pathway, M1, M2, and M3 are all steady-state solutions and M2 is the “converged” steady-state solution.

While there may be more than three meshes required, the process of finding that converged steady-state solution is

far less computationally expensive than finding this converged mesh and time step using transient simulations. The

guidelines in Powell & Lawryshyn (2016), especially with respect to initial choices for mesh size and time step

should be used with caution and with some forethought with respect to what makes sense in the context of the

reactor, however the guidelines prescribed are often a good first step. Furthermore, there are some situations where

using the steady state solution as the initial condition may not provide you a better picture of the flow field to start,

compared to other methods such as a bulk flow velocity; for most situations, however, the guidelines prescribed are

applicable.

Using the new pathway in Figure 1-6, only the converged steady-state solution is brought forward to test time-step

independence. Since the time steps are chosen using the Courant Number as a guide, it is highly unlikely that more

than three transient simulations would be needed to determine solution convergence. This results in a large reduction

in computational cost.

14

Previously, the number of computations to achieve the converged flow field to find the reactor performance was

over 1.9 million due to the need to conduct at least nine transient simulations. By using the methodology prescribed

in Figure 1-5 and further visualized in Figure 1-6, the number of computations has been now reduced four-fold to

under 500,000.


Whereas previously the NPS/NSS chosen to implement the FRMs was either arbitrary or very high, a novel

contribution of this thesis was the development of an equation for the optimal NPS where reflection and refraction

associated with the lamp sleeve/water interface are not considered. The implicit equation to find the NPS without

reflection and refraction must be solved numerically to find nopt and is shown below:

(1-1)

where r is the perpendicular distance to the particle from the i–th point segment or source segment, p is the axial

distance from the centre of the lamp divided by lamp length, L is the lamp length, and α is the tolerance of the

fluence rate model.

Through numerical experiments it was found that a higher NPS was required for (1) locations closer to the lamp

sleeve; (2) lamps with larger lamp sleeve radii; and (3) lamps that have a longer length. Most importantly, it was

found that there was a specific location near the lamp to calculate the optimal NPS. The radial location is just

outside the lamp sleeve and the axial location (the location along the orientation of the lamp) is given by the

following equation, which is in terms of the distance from the centre of the lamp:

15

𝑝 = 0.5 + +.,-./011213

(1-2)

These equations were used to find the optimal NPS for an actual UV reactor RED calculation. It was found that the

reduction equivalent dose (RED) and dose distribution were virtually identical when using 267 segment sources (as

prescribed by the equations developed) and 1000 segment sources, resulting in a five-fold decrease in computational

cost.

To put it in perspective, the number of computations required to calculate the RED for a single-lamp UV reactor

with one million particles was estimated at 300 billion. After the application of the models developed in Powell &

Lawryshyn (2015), the number of computations was reduced to 8 billion. As shown in Table 1-1 and discussed in

Powell & Lawryshyn (2015), the reduction in computational cost depends on the lamp length and lamp sleeve radii

and far greater reductions are possible. The manuscript also examined the fluence rate models when reflection and

refraction associated with the lamp sleeve/water interface was considered. An implicit equation was also developed

to find the optimal discretization and while it does require slightly more computational time than the equation

without reflection and refraction, the reduction in computational cost is still comparable, especially because the

fluence rate models with reflection and refraction take more computational time in general: reducing the

computational cost when implementing those fluence rate models is even more important.


Anecdotally, higher mixing in a UV reactor was thought to bring better disinfection. Mixing was thought to reduce

the variance in the dose distribution, resulting in a more efficient use of the supplied UV light energy. Thus, it is

critical that we measure mixing in order to quantify this impact; however, it should be recognized that the mixing

part of hydrodynamics is not the only factor in reactor performance. The RED, as a reactor performance metric, is a

function of hydrodynamics (mixing) and fluence rate distribution. Regardless, mixing was used here as a novel way

to examine reactor performance. In this thesis, a new mixing metric based on the Shannon entropy was developed.

This Shannon-based mixing metric quantifies particle mixing based on the path it takes over time, just like the

calculation of UV dose; numerical experiments show that, in UV reactors, the Shannon-based mixedness also

correlates strongly with RED in certain scenarios. This means that in these scenarios, the mixing calculation can be

16

used as a proxy for RED; now, there is no need to calculate fluence rate and thus using the mixing metric to define

reactor performance dramatically reduces the computational cost of the reactor performance calculation.

The modified mixing metric developed in Powell & Lawryshyn (2017), described here

𝑀5,7895:5;9 = <=,>?@=A=1@

<=,>?@=A=1@,>BC=

D E=FG8HIJ=FK=F∗

LFMN

G8HIOP,= (1-3)

is different than the original mixedness metric, first popularized by Ogawa & Ito (1973), in that the modified

mixedness allows a different particle distribution to describe perfect mixing.

For a UV reactor with a lamp oriented parallel to the direction of flow, the R2 between the local RED and modified

mixedness was approximately 0.98; that is, mixing and reactor performance are very strongly correlated. Further

modifications of the weighting system used in the modified mixedness are also discussed later in this thesis as an

additional short communication.

The application of this mixing metric to define UV reactor performance has dramatically reduced the computational

cost of the calculation. Essentially, fluence rate calculations are now unnecessary if the performance of two or more

reactors needs to be compared. While the number of computations to find the RED of a particular reactor, including

the CFD and fluence rate calculations was reduced to 8 billion using the models produced in Powell & Lawryshyn

(2015), the number of computations to calculate reactor performance using the modified mixedness metrics is now

only 12,000.

A summary of the computational cost savings of each of the three models is provided below in Figure 1-7; when

using the methodology developed for this thesis to implement the RPP, using transient CFD models to find a

converged solution and then using the modified mixedness to calculate reactor performance (instead of fluence rate

models and the RED), the computational cost is reduced from 300 billion to just under 500,000.

17

Figure 1-7 – Summary of the computational cost reductions from the systematic review of the three models of the UV reactor performance calculation.

1.5 Thesis organization

This thesis is organized into chapters that correspond to each of the three published manuscripts and then a

conclusion chapter with future work. Chapter 2 includes the paper “Standard Methodology for Transient Simulations

of UV Disinfection Reactors” (Powell & Lawryshyn, 2016). Chapter 3 includes the paper “A method for

determining the optimal discretization of UV lamps for emission-based fluence rate models” (Powell & Lawryshyn,

2015). Chapter 4 includes the paper “A modification of the entropy-based mixing to quantify mixing in UV

reactors” (Powell & Lawryshyn, 2017), as well as a short communication with additional options for modification of

the mixedness metric to fit the reactor performance calculation. In each chapter, a brief synopsis of the manuscript,

as well as the key results and the calculation of the reduction in the computational cost produced by the novel

contributions is shown. Chapter 5 presents the conclusions of the paper, including a summary of the contributions

and how the objectives were met, as well as future work that will improve the work completed herein.

18

1.6 References

Alpert, S.M., Knappe, D.R.U., and Ducoste, J.J. (2010) Modeling the UV/hydrogen peroxide advanced oxidation process using computational fluid dynamics. Water Research 44(6) 1797-808. Blatchley, E. R., III (1997) Numerical modeling of UV intensity: Application to collimated-beam reactors and continuous-flow systems. Water Research 31(9), 2205-18. Bolton, J.R. (2000) Calculation of ultraviolet fluence rate distributions in an annual reactor: significance of refraction and reflection. Water Res. 34(13), 3315-24. Breuer, M. (2000). “A challenging test case for large eddy simulation: High Reynolds number circular cylinder flow.” Int. J. Heat Fluid Flow, 21(5), 648–654. Camesasca, M., I. Manas-Zloczower, M. Kaufman (2006) Entropic characterization of mixing in microchannels. Journal of Micromechanics and Microengineering 15(11) 2038-2044 Chiu, K., Lyn, D.A., Savoye, P. and Blatchley, E.R., III (1999) Integrated UV disinfection model based on particle tracking. J. Environ. Eng. 125(1), 7-16. Courant, R., Friedrichs, K., and Lewy, H. (1928). “Über die partiellen Differenzengleichungen der mathematischen Physik.” Mathematische Annalen, 100(1), 32–74 (in German). detector. Water Res. 46 (11), 3595–3602. Ducoste, J.J., Liu, D. and Linden, K. (2005) Alternative approaches to modeling dose distribution and microbial inactivation in ultraviolet reactors: Lagrangian and Eulerian. J. Environ. Engineering, ASCE, 1310(10) 1393-403. Gandhi, V., Roberts, P.J.W., Stoesser, T., Wright, H., Kim, J.-H. (2011) UV reactor flow visualization and mixing quantification using three-dimensional laser-induced fluorescence. Water Research 45, 3855-3862. Georgiadis, N. J., Rizzetta, D. P., and Fureby, C. (2010). “Large-eddy simulation: Current capabilities, recommended practices, and future research.” AIAA J., 48(8), 1772–1784. Haas, C.N., and G.P. Sakellaropoulos (1979) Rational analysis of UV disinfection reactors. Proceedings of ASCE National Conference on Environmental Engineers, ASCE, New York. 540-547 Jin, S., Linden, K. G., Ducoste, J. & Liu, D. (2005) Impact of lamp shadowing and reflection on the fluence rate distribution in a multiple low-pressure UV lamp array. Water Res. 39 (12), 2711–2721. Li, M., Qiang, Z., Bolton, J. & Weiwei, B. (2012) Impact of reflection on the fluence rate distribution in a UV reactor with various inner walls as measured using a micro-fluorescent silica detector. Water Res. 46 (11), 3595–3602. Liu, D., Ducoste, J., Jin, S., and Linden, K. (2004) Evaluation of alternative fluence rate distribution models. J. Water Supply Res. Technol. 53(6) 319-408. Liu, D., Ducoste, J., Jin, S. and Linden, K. (2007) Numerical simulation of UV disinfection reactors: evaluation of alternative turbulence models. Appl. Math. Model. 31, 1753-69. Lyn, D. A., Chiu, K. and Blatchley, E.R., III (1999) Numerical modeling of flow and disinfection in UV disinfection channels. J. Environ. Eng. 125(1) 17-26. Lyn, D.A (2004) Steady and unsteady simulations of turbulent flow and transport in ultraviolet

19

disinfection channels. Journal of Hydraulic Engineering 130(8), 762-70. Munoz, A., Craik, S. & Kresta, S. Computational fluid dynamics for predicting performance of ultraviolet disinfection – sensitivity to particle tracking inputs. J. Environ. Eng. Sci. 6 (3), 285–301. Ogawa, K. & Ito, S. (1973) A definition of quality of mixedness. J. Chem. Eng. Jpn. 8 148–151 Powell, C. & Y. Lawryshyn (2015) A method for determining the optimal discretization of UV lamps for emission-based fluence rate models. Water Science and Technology 71(12) 1768-1774 Powell, C. & Y. Lawryshyn (2016) Standard methodology for transient simulations of UV disinfection reactors. Journal of Environmental Engineering doi:10.1061/(ASCE)EE.1943-7870.0001153 Powell, C. & Y. Lawryshyn (2017) A modification of the entropy-based mixing to quantify mixing in UV reactors. Journal of Environmental Engineering (in-press) Qualls, R.G., and J.D. Johnson (1985) Modelling and efficiency of ultraviolet disinfection systems. Wat. Res. 19, 1039-1046. Quan, Y., Pehkonen, S. & Ray, M. (2004) Evaluation of three different lamp emission models using novel application of potassium ferrioxalate techniques. Ind. Eng. Chem. Res. 43 (4), 948–955. Rahn, R., Bolton, J. & Stefan, M. (2006) The iodide/iodate actinometer in UV disinfection: determination of the fluence rate distribution in UV reactors. Photochem. Photobiol. 82 (2), 611–615. Rudd, T. and Hopkinson, L.M. (1989) Comparison of disinfection technologies for sewage and sewage effluents. Water and Environment Journal 3(6) 612-8. Severin, B.F., Suidan, M. T., Engelbrecht, R. S. (1984) Mixing Effects in UV Disinfection. Journal (Water Pollution Control Federation) 56:7 881-888 Shannon, C.E. (1948) A mathematical theory of communication Bell System 1051 Technical Journal. 27:379-425, 623-656. Thampi, M.V. and C.A. Sorber (1987) A method for evaluating the mixing characteristics of UV reactors with short retention times Water Research 21(7):765-771 USEPA (2006): Ultraviolet Disinfection Guidance Manual for the Final Long Term 2 Enhanced Surface Water Treatment Rule. O!ce of Water (4601), EPA 815-R-06-007, November 2006. . White, S. C., et al. (1986) A study of operational ultraviolet disinfection equipment at secondary treatment plants. J. Water Pollut. Control Fed. 58, 181. Wols, B.A., Shao, L., Uijttewaal W.S.J., Hofman, J.A.M.H., Rietveld, L.C., and van Dijk, J.C. (2010a) Evaluation of experimental techniques to validate numerical computations of the hydraulics inside a UV bench-scale reactor. Chemical Engineering Science 65(15), 4491-502. Wols, B.A., Uijttewaal W.S.J., Hofman, J.A.M.H., Rietveld, L.C., and van Dijk, J.C. (2010b) The weaknesses of a k–ɛ model compared to a large eddy simulation for the prediction of UV dose distributions and disinfection. Chemical Engineering Journal 162(2):528-536. Younis B.A. and Yang, T.-H. (2010) Computational modeling of ultraviolet disinfection. Water Sci. & Tech. 62(8), 1872-78.

20

Chapter 2 – Transient CFD Models

2 Introduction

This section of the thesis provides a standard methodology for describing the performance of UV wastewater

treatment reactors with transient simulations. The purpose of this work was to address inconsistencies in the way

transient simulations ere used to model UV reactors, including issues with mesh size, time steps, number of

particles, and initial conditions, but also to demonstrate the amount of variability in the RED in a UV reactor over

time. This chapter includes the manuscript “Standard Methodology for Transient Simulations of UV Disinfection

Reactors” (Powell & Lawryshyn, 2016) as published.

2.1 Key Results

The manuscript shows that:

1) A standard methodology for transient UV reactors can help reduce overall computational cost;

2) when using the transient k-epsilon model to simulate a simple UV reactor, there is less than a 1%

variation in the transient RED (tRED) and a higher transient RED time step (tRTS) reduces the

variation in RED;

3) more complex UV reactors produce a much higher variation in tRED, up to 20% for the most complex

reactor examined.

2.2 Implications

This manuscript further implies that the number of transient simulations that were needed to be performed to ensure

a converged solution was higher than used in previous studies. By using the steady state solution for the initial

conditions of the transient simulation, as well as the other guidance provided for mesh, time step, and particle

tracking, the number of transient simulations drops from at least 9 to 3 plus three steady state solutions, thus

requiring far fewer computations.

21

In Powell & Lawryshyn (2016), a steady-state solution was found within 400 iterations for the UV reactor in

question. For the transient solutions, approximately 8000 time steps were used, and each time step required

approximately 20 iterations to converge. Using the previous methodology, which employed nine transient solutions

to find a converged solution, the total number of iterations required by Fluent was 1.92 M. For the new

methodology, using three steady-state solutions and three transient simulations, the number of iterations was only

481,200. It should also be recognised that the procedure provided in this work has not been tesed in all cases, and

could be unnecessary for certain scenarios. For example, if a very complicated geometry has a steady-state solution

very different than its transient pseudo-steady state solution, then using the steady state as an initial condition may

not reduce computational time. Regardless, this method is prescribed for very general UV reactor design. Part of the

future work here will examine more test cases for applicability.

2.3 Manuscript

This section showcases the manuscript “Standard Methodology for Transient Simulations of UV Disinfection

Reactors” (Powell & Lawryshyn, 2016) as published.

2.3.1 Abstract

Using ultraviolet (UV) light to disinfect water has recently become more commonplace and using computational

fluid dynamics (CFD) is increasingly being used to model, optimize, and test UV disinfection reactors. The vast

majority of CFD based UV reactor models assume the steady-state because it was thought that transient phenomena

will not lead to significant differences in performance estimates. Recent studies have shown that this assumption

may not be valid under certain circumstances, especially when large eddy simulation (LES) is utilized. However,

there is no standard methodology to conduct a transient simulation of a UV reactor. The objective of this paper is to

propose a standard methodology to model the performance of UV reactors using transient CFD. A simple single-

lamp cross-flow UV reactor was used to perform a series of computational experiments to develop a proposed

standard methodology for transient simulations and this method was then verified using two more complex reactors.

The paper examined mesh, time step, initial conditions, and particle tracking to determine model convergence using

transient turbulent k-epsilon models. In addition, transient RED (tRED) was introduced as a means to determine the

22

variability in RED over time in a UV reactor. The results showed that although the tRED does not vary significantly

for the single-lamp reactor, there is upwards of 10% of variability in the RED for more complex reactors. The

proposed methodology developed in this paper leads to a structured and efficient approach to model UV reactor

performance based on transient CFD.

2.3.2 Introduction

Computational fluid dynamics (CFD) has been used extensively in determining the disinfection performance of

ultraviolet (UV) reactors (Chiu et al., 1999; Lyn et al., 1999; Wright & Hargreaves, 2001; Lyn, 2004; Sozzi &

Taghipour, 2006; Liu et al., 2007; Younis & Yang, 2011; Wols et al., 2010; Wols et al., 2011). CFD requires the

development of a UV geometry and associated mesh to solve the resultant flow field. The modeler must also decide

whether to treat the flow field as steady-state or transient. Most of the published work in UV reactor modeling has

been done using steady-state as opposed to transient models, as the steady-state models are simple to implement

(Lyn et al., 1999; Liu et al., 2007). Munoz et al. (2007) mention using transient flow modeling in UV reactor design

and deem it unnecessary because their transient models did not produce results different from steady-state; however,

they did not provide sufficient methodology to replicate their transient work. Conversely, Lyn (2004) and Younis &

Yang (2011) stated that transient simulations were necessary to resolve transient flow phenomena but they warned

of the added computational expense. Using a 3D Laser Induced Fluorescence (LIF) technique, Gandhi et al. (2012)

showed that the experimental disinfection was actually transient and varied with Reynolds number. Younis & Yang

(2011) also developed a new transient Reynolds-averaged Navier-Stokes (RANS) model to examine vortex shedding

downstream of lamps; their work showed that dose delivery depends on Reynolds number and that steady state

standard RANS models were not properly capturing transient phenomena that affect dose delivery. Thus, there is

ample evidence to suggest that transient simulations of UV reactors can be used to provide more accurate results.

To date, no attempt has been made to provide a standardized methodology for transient CFD modeling of UV

reactors. Thus, the purpose of this paper is to propose a methodology for transient UV reactor simulations in order to

make transient simulations more comparable.

This paper is organized as follows. First, a review of the methodology of steady-state and transient modeling

23

techniques for UV reactors is discussed. Following this, the major issues that are critical to the successful transient

simulation of a UV reactor are identified. These issues are tested using a simple UV reactor and, using these results,

a standardized methodology for transient UV reactor modeling is introduced and then validated using two, more

complex, UV wastewater reactors.

2.3.2.1 Steady-state solutions for UV reactors

Modeling a UV reactor with CFD combines three techniques: 1) fluid flow modeling; 2) fluence rate modeling; and

3) microbial kinetics modeling (Chen et al., 2011). The latter two techniques are well documented in the published

literature and will not be discussed at length here because they have a standardized methodology that does not

impact the way a transient simulation is performed. The reader is directed to Liu et al. (2004) for the state of the art

on fluence rate modeling and Ducoste et al. (2005) for the state of the art on microbial kinetics modeling for UV

reactors. This section will focus on the state of the art for fluid flow modeling in UV reactors, namely mesh and

particle independence studies for steady-state solutions.

Regardless of the type of solution (steady-state or transient), modeling a UV reactor first involves designing a

geometry and choosing a mesh that divides the geometry into elements or volumes in which the Navier-Stokes

equations are solved. The next step is to solve the Navier-Stokes equations with a turbulence model, such as the k-

epsilon model (Sozzi & Taghipour, 2006; Liu et al., 2007), typically utilizing the finite volume model method

(Greene et al., 2006; Sozzi & Taghipour, 2006; Munoz et al., 2007; Liu et al., 2007; Wols et al., 2010). To determine

the disinfection provided by a UV reactor, there are two approaches: Lagrangian and Eulerian (Sozzi & Taghipour,

2006; Munoz et al., 2007). In the Lagrangian framework, often called the particle tracking method, microbes are

considered as discrete parts of the dispersed phase and the UV dose and inactivation are found by numerically

integrating the fluence rate field over the time spent in the reactor (Sozzi & Taghipour, 2006; Munoz et al., 2007). In

the Eulerian framework, microbes are treated like a tracer in a reactive system and the inactivation is determined by

using an advection-diffusion equation that includes a reaction term (Lyn et al., 1999; Munoz et al., 2007). Sozzi &

Taghipour (2006) evaluated these two frameworks for calculating inactivation in UV reactors and found them to be

comparable. The Lagrangian framework, however, is used more often in the literature (Wols et al., 2010; Wols et al.,

2011; Younis & Yang; 2011) presumably due to the ease of implementing fluence rate models.

24

Modeling a UV reactor using the steady-state Navier-Stokes equations provides a “snapshot” of the velocity flow

field. A steady-state simulation typically takes a short time to converge and is easy to implement and, as a result, is

used extensively in UV reactor modeling (Blatchley, 1997; Tartera et al., 1998; Liu et al., 2007). There are two main

issues which must be examined to ensure a valid steady-state solution (Munoz et al., 2007): the solution must be

independent of 1) the mesh and 2) the number of particles that are injected in the computational flow domain.

Standard approaches in the literature for dealing with both are discussed next.

The mesh must be independent of the solution; a velocity field is independent of the mesh of a UV reactor when

increasing the number of elements into which the geometry is divided does not change the solution. The process to

choose a mesh and then achieve mesh independence typically involves using a percentage of the diameter of the

lamp to determine an average element size, like Nishino et al. (2008), then reducing that size to find three meshes

with progressively higher concentrations of elements or volumes in the area where transient phenomena occur (i.e.

around the lamp(s)) and then using one or more performance metrics to find the coarsest mesh that produces no

change in the metric(s). For UV reactor modeling, the metrics used for mesh independence include the velocity

magnitude along a line (Li et al., 2010), the coefficient of determination for the velocity magnitude in all elements

(Sozzi & Taghipour, 2006), microbial inactivation (Munoz et al., 2007; Wols et al., 2010; Wols et al., 2011), and

reduction equivalent dose (RED) (Powell & Lawryshyn, 2015). There are advantages and disadvantages with using

these particular metrics, however. While using the velocity magnitude at a point or line to determine mesh

independence is straightforward, the points chosen are often arbitrary. Conversely, requiring high correlation among

the velocity magnitudes in every element for successive meshes provides a strict measure, but is often prohibitive in

practice. Indeed, due to the high accuracy required with this methodology, the chosen mesh may be much finer than

necessary; using a metric more closely associated with the performance of the UV reactor, like RED, to ensure mesh

independence may not require as fine a mesh.

The steady-state solution for a UV reactor must also be independent of the number of particles injected in the

reactor. Particle independence is achieved when the mean dose or RED does not change when the number of

particles captured in the reactor is increased. Graham & Moyeed (2002) standardized this procedure and it was

25

subsequently used by Munoz et al. (2007). Both papers found that the variability in RED with multiple simulations

of a UV reactor was inversely proportional to the product of two variables: np, the number of particles per

simulation, and ns, the number of simulations, and that this variability stabilized as this product increased. Munoz et

al. (2007) logically note that there are two ways to increase this product: increase the np or increase the ns. They also

note that it is easier to run one simulation with more particles than many simulations with fewer particles. Wols et

al. (2010) used a more ad hoc particle independence approach when performing a transient simulation, but it was

similar to the recommended approach by Munoz et al. (2007). Wols et al. (2010) increased the number of particles

injected in a UV reactor until the disinfection varied less than a specified tolerance. The particles were injected at

different times, but then grouped together regardless of the time they exited the reactor. They performed this

procedure without reference to past transient particle tracking approaches, namely because there are no published

references, but it is unclear whether this is the correct procedure.

These issues with mesh and particle independence for steady-state solutions are examples of the need for a

standardized methodology for transient UV reactor simulations.

2.3.3 Transient simulations for UV reactors

A methodology for a transient simulation of a UV reactor has not yet been “standardized” because there are so few

studies dealing specifically with UV reactors. This may be due to the computational expense, as transient

simulations require more iterations for parameter convergence, and/or because of the reported comparable results

between steady-state and transient simulations. But developing a standardized methodology is still important

because there may be a requirement for higher resolution of solutions, like results given by large eddy simulation

(LES), or a desire to examine the variability in reactor performance over time, which only a transient simulation can

provide. This section will focus on the extent of the published work in transient UV reactor modeling.

The same three techniques that are used in steady-state UV reactor modeling are used for transient UV reactor

simulations: fluid flow modeling, fluence rate modeling, and microbial kinetics modeling. Again, the latter two will

not change if the reactor is modeled using steady-state or transient flow. For transient fluid flow modeling of a UV

reactor, in the absence of previous studies, four important issues have been identified: 1) initial conditions; 2) mesh;

26

3) CFD simulation time step and 4) a particle tracking strategy.

To find the velocity field for a transient simulation, an initial condition must be imposed. Georgiadis et al. (2010)

recommended that a “recycle” condition be utilized: that is, using the velocity field at the outlet of the reactor as the

initial velocity field. Another technique that is possibly less computationally expensive is to use a steady-state

simulation as the initial condition (ANSYS, 2012), but this presumes that the turbulence model used has a steady-

state analogue, which is not true for LES. Wols et al. (2010) imposed a uniform velocity at the inlet for the k-ε

simulation and a logarithmic velocity at the inlet for LES, but it is not clear whether sensitivity to this assumption

was tested.

Ensuring mesh independence for a transient simulation is often quite difficult because the transient velocity fields

and high computational expense make it challenging to compare results from different meshes (Wols et al., 2010;

Breuer, 2000; and Georgiadis et al., 2010). A consistent approach to transient mesh independence is not clear in the

broader CFD literature, but the modeler must be aware of the balance between accuracy and computational expense

(Wols et al., 2010). Performing numerous mesh independence studies for transient simulations that are

computationally expensive may not provide a greater accuracy benefit. Alternative approaches, like using the

steady-state solution to find mesh independence, are discussed later in this paper.

The choice of the CFD simulation time step is also critical for transient simulations. Wols et al. (2010) and Younis

& Yang (2011) chose a time step based on the Courant-Friedrich-Levy (CFL) number, a dimensionless number

based on how long a particle takes to pass through one streamwise volume (Courant et al., 1928). Wols et al. (2010)

chose a time step that was intended to produce a CFL number less than 1, which ensures that a particle will move the

distance of less than one mesh element per time step. An alternative approach is to choose at least three time steps

that ensure the CFL tolerance is met and to choose the lowest time that produces an unchanged performance metric.

For example, Nishino et al. (2008) modeled a bluff body in a channel with a number of time steps and evaluated

time step independence by using the Strouhal number, a measure of the vortex shedding past a bluff body, and the

drag coefficient. A useful performance metric for UV reactors would be the RED, as is done in this paper.

27

A particle tracking strategy for a transient UV reactor simulation has yet to be addressed in the literature. Wols et al.

(2010) injected 50000 particles over 1000 time steps to “obtain more statistically reliable results” using LES, but it

is unclear whether this gradual approach to particle injection was used for the transient k-epsilon simulation. The

authors then combined all particles into one dose distribution to calculate the overall disinfection performance, but

there was no reason given for this approach. It is clear that particles must be injected gradually for a transient

simulation, especially when using Fluent for particle tracking, but it is still not clear how many particles to inject at

each injection and how many particles to inject overall. Another approach, introduced later in this paper, would be

to inject particles gradually over a series of time steps, just as Wols et al. (2010), but also collect them gradually

over a series of time steps at a particular location: this would give a “transient” dose distribution and RED, and

potentially a more accurate picture of the variability in reactor performance – a phenomena frequently encountered

in field bioassay studies (Petri et al., 2006).

The purpose of this paper is to provide a standardized methodology, that is systematic, reproducible, and

computationally less expensive than current methods, for simulating the transient performance of UV reactors. This

methodology is determined by performing a number of numerical experiments examining the four critical issues

identified above (initial condition, mesh size, CFD simulation time step, and a particle tracking strategy) and

comparing them to a “best-case” solution.

2.3.4 Methodology

The methodology is divided into three parts. First, the proposed methodology for a transient UV reactor simulation

is described; this was found from the results of the numerical experiments conducted here. Following that, the

scenarios that were used to determine the proposed methodology are described, as well as the geometries, reactor

conditions, and flow models that were used to determine and verify the proposed methodology.

2.3.4.1 Proposed methodology

The major steps of the proposed standardized methodology for transient simulations of UV reactors are illustrated in

Figure 2-1. The methodology has eleven distinct steps that are described in more detail below.

28

1. Choose three meshes of varying coarseness. The coarsest mesh should have elements with an average

streamwise length at most 5% of the diameter of the UV lamp sleeve and closer to the lamp, at most 0.5%

of the diameter of the UV lamp sleeve. (this follows the work by Nishino et al. (2008) in transient

RANS/detatched eddy simulation (DES) modeling). To obtain the two finer meshes, choose average

streamwise lengths lower than 5% of the diameter of the lamp sleeve.

2. Solve for the steady-state solution for each of the meshes.

3. Inject a progressively increasing number of particles into each flow domain.

4. Calculate the dose received by each particle using an appropriate fluence rate model and calculate the RED

of each injection set.

5. Determine the coarsest mesh and the minimum number of particles that produces an RED that changes less

than a chosen tolerance.

6. Using the chosen mesh, determine three time steps using the CFL equation,

∆𝑡 = 𝐶𝐹𝐿∆𝑥𝑈

(2-1)

where U is the bulk flow velocity, Δt is the chosen time step, and Δx is the average streamwise length of an

element near the lamp.

7. Solve for the transient solution by running each time step scenario for at least three hydraulic residence

times (HRTs).

8. After 3 HRTs, begin to inject the number of particles determined in Step 5, at each CFD simulation time

step, for as long as transient RED (tRED,, which is described later in this paper) data is required. For

example, if one HRT of tRED data is required then particle injections should be made every time step for

one HRT and the CFD simulation should be continued for another 1.5 HRTs to ensure all particles have left

the reactor (in this case, a total of 5.5 HRTs would need to be simulated).

9. Choose a time interval by which to sample the reactor: bin the particles by the time they exit the reactor in

between each time interval.

10. Calculate the dose and RED of each bin of particles.

11. The RED of each bin shows the tRED over the time period desired.

This methodology produced the most computationally inexpensive, yet converged solution; more on this is

discussed later in the results section.

29

Figure 2-1 - Illustration of the proposed transient UV modeling methodology

30

2.3.4.2 UV Reactor Geometries

Three different UV reactor geometries were used in this paper to determine and apply a proposed methodology for

transient simulations. The first geometry, a single-lamp cross-flow reactor, was used to perform various numerical

experiments to determine the proposed methodology. Two other reactors, a single-lamp cross flow reactor with a

cylinder placed perpendicularly and upstream of the lamp, and an 8-lamp, two bank UV wastewater treatment

reactor, were used to apply the proposed methodology that resulted from those numerical experiments.

The first UV reactor, shown in Figure 2-2 and known as G1, has a 0.1 m by 0.1 m square cross-section, is 5.5 m

long in the direction of flow; the single cylindrical UV lamp sleeve has a length of 0.1 m, a 0.05 m sleeve diameter,

300 W effective lamp power and is orientated perpendicular to the direction of flow. The entrance and exit length

are sufficient to ensure fully developed flow. The lamp length and diameter was chosen to be typical of a particular

commercial UV disinfection apparatus.

Figure 2-2 - Geometry of the first test UV reactor, G1, with a UV lamp oriented perpendicular to flow.

31

The second UV reactor, known as G2, is shown in Figure 2-3. The lamp and operating parameters used in this

geometry are the exact same as the first, except for the addition of a vertically-oriented 0.1 m long, 0.05 m diameter

cylinder placed 10 cm upstream of the UV lamp and a proportional reduction in the entrance and exit lengths to

reduce computational expense compared to the first geometry. This geometry was chosen to induce some transient

behaviour upstream and downstream of the lamp. The flow rate used for the first two UV reactors was 0.00397 m3/s

with a hydraulic diameter Reynolds number of 25000.

Figure 2-3 - Geometry of the second test UV reactor, G2, with a UV lamp oriented perpendicular to flow and a cylinder placed 10 cm upstream of the lamp.

The third UV reactor that was tested more closely matches that of a practical UV wastewater treatment reactor; it is

shown in Figure 2-4 and is known as G3. The reactor is 7 m long, with a 0.2 m by 0.4 m cross-section; there are

eight lamps arranged in two side-by-side four-lamp banks oriented parallel to the flow. The lamps are 1.5 m long

with a sleeve diameter of 0.05 m and an effective lamp power of 100 W. The flow rate used is 0.0051 m3/s with a

hydraulic diameter Reynolds number of 17000. Sufficient entrance and exit length were provided.

32

Figure 2-4 - Geometry of the third test UV reactor, G3, with two four-lamp banks oriented parallel to the direction of flow.

2.3.4.3 Flow Model

To solve for the flow field to conduct the numerical experiments and find the proposed methodology, the realizable

k-ε model was implemented with FLUENT v. 14.5 (Ansys, 2012). This turbulence model, a modification and

improvement of the standard k-ε model, solves two additional equations over the standard Navier-Stokes equations,

one for k, the turbulent kinetic energy, and one for ε, the dissipation rate of the turbulent kinetic energy. It also

contains a new formulation for resolving the turbulent viscosity and a different equation for solving for ε. The two

transport equations are not provided here for brevity, but are shown in Wilcox (1998). The limitations of the

standard k-epsilon model in accurately predicting vortex shedding has been shown by Franke & Rodi (1991) and the

realizable k-epsilon model has been shown to be superior in predicting flow near walls (Shih et al., 1995), which is

why the realizable model was chosen.

33

The above flow equations must be accompanied by boundary conditions. The water enters the reactors only in the

direction of flow and is uniform across the inlet surface. On all solid boundaries, including the lamp sleeve, no-slip

conditions were imposed.

Particle tracking in FLUENT (Ansys, 2012) was used to determine particle trajectories within the flow domain. The

discrete random walk model in Fluent was used to introduce randomness associated with turbulence to the particle

trajectories. It was assumed that the particles were massless and smaller than the Kolmogorov scales so they moved

with the flow, without disturbing it. In this paper, the fluence rate was calculated using the radial model as described

in Liu et al. (2004). While the radial model is not the most accurate (Liu et al., 2004), it is the fastest to implement

and is sufficient for the purposes of the demonstrating the proposed transient methodology. To determine the UV

dose for the i-th particle, Di, the fluence rate was integrated over the time spent in the reactor using the trapezoidal

method. The disinfection was found by calculating the RED (US EPA, 2006):

𝑅𝐸𝐷 = −1𝑘^ln

𝑒Dbcd=eP.fg𝑁"

(2-2)

where n is the number of particles in the reactor and the inactivation rate constant, kµ, is related to the D10 value, the

dose required for 1-log inactivation, by the expression:

𝑘^ =ln 10𝐷g+

(2-3)

As discussed previously, a new particle tracking parameter, the transient RED, or tRED, is introduced in this paper.

The tRED is defined as the RED of a group of particles that are collected at a given location in a UV reactor over a

particular time interval. The particular interval is arbitrary, but should be a regular interval that is greater than the

numerical simulation time step. This interval is called the transient RED time step (tRTS) for the remainder of this

paper. The tRED can be thought of as a “numerical bioassay” where a sample is collected at a given location over a

fixed period of time. The equation to find the tRED at a particular time step, j, is:

𝑡𝑅𝐸𝐷(𝑗) = −1𝑘^ln

𝑒Dbcd=eP,F.fg𝑁",l

(2-4)

34

where NT,j represents the total number of particles collected in that particular tRTS. It differs from the RED

presented in equation (2-2) because it only calculates the RED of particles that were collected during that particular

time interval.

2.3.4.4 Tests used to develop the proposed methodology

The goal of this paper is to provide a standardized methodology for transient UV reactor simulations that is

systematic and computationally inexpensive compared to other methods. To determine the proposed methodology, a

“best-case” model was chosen as the baseline to compare the results of numerical experiments focused on the four

issues identified previously: initial condition, mesh, time step and particle tracking. The best-case RED was

calculated using the initial bulk velocity as the initial condition, a very fine mesh with an element size of

approximately 1.875% of the lamp sleeve diameter around the lamp region, and a very small time step ensuring a

CFL less than 0.1.

The proposed methodology was determined by comparing the best-case RED to the RED of a number of UV reactor

scenarios, which are shown in Figure 2-5. In this paper, a difference of less than 1% between the best-case RED and

the RED of the other scenarios tested in Figure 2-5 was considered to be “converged”1.

Figure 5 shows the various scenarios that are compared to the best-case RED. The scenarios are divided into two

main sections: IC1 and IC2 for the first and second initial conditions.

1The 1% threshold was used because it is typically chosen as a convergence criterion, for example, Wols et al. (2010) and Bolton (2000).

35

Figure 2-5 – Illustration of the different scenarios used to compare to the best-case solution. A dashed circle indicates a steady-state simulation was performed; a dotted circle indicates a transient simulation was performed.

Under IC1, there were seven total simulations (four steady-state simulations in dashed circles and three transient

simulations in dotted circles). These scenarios were examined as follows:

1) The steady-state solutions were found for the four meshes for G1, known as G1-M1, G1-M2, G1-M3, and

G1-MB (best-case), shown in Figure 2-6. These meshes were chosen to ensure an average element size that

is a function of the UV lamp sleeve diameter; for G1-M1, G1-M2, G1-M3, and G1-MB, an average

element size that was 20%, 5%, 2.5% and 1.875% of the UV lamp sleeve diameter, respectively, were used.

2) A progressively larger number of particles were injected in each steady-state solution.

3) Each particle dose and the RED for each scenario was calculated

4) The mesh and number of particles required for a “converged” solution were chosen as the working mesh

and the required particles.

5) The transient simulations were run with the working mesh, the converged velocity field in that mesh for the

36

initial condition, and the required particles, with three time steps shown in Table 2-1, ensuring a CFL of at

least 5, 1 and 0.5, respectively (for the best-case scenario, CFL = 0.1).

6) Each particle dose and RED for each time step scenario was calculated.

7) Each scenario was compared with the G1-MB-TSB transient solution.

Figure 2-6 - Illustration of the mesh intensity around the lamp zone looking perpendicular to the flow in the reactor. Note this shows only a 2D picture of the mesh for illustrative purposes; the actual mesh is 3D.

Under IC2, there were twelve transient simulation scenarios. The scenarios were examined as follows:

1) Using each mesh, G1-M1, G1-M2, G1-M3, and G1-MB, with the bulk flow velocity as the initial

condition, run the transient simulations with the required time steps shown in Table 1.

2) Calculate each particle dose and RED for each mesh and time step scenario.

3) Compare with the G1-MB-TSB transient solution.

37

Table 2-1 - Time steps used in transient simulations for G1

M1 M2 M3 MB TS1

(CFL = 5) 0.252 s 0.126 s 0.09 s TS1 (CFL = 5) 0.063 s

TS2 (CFL = 1) 0.0504 s 0.0252 s 0.018 s TS2

(CFL = 1) 0.0126 s

TS3 (CFL = 0.5) 0.0252 s 0.0126 s 0.0090 s TSB

(CFL = 0.1) 0.00126 s

The above scenarios were used to arrive at the proposed methodology that reduces computational cost for transient

simulations for UV reactors, a barrier to many researchers. It should be evident, however, that the IC1 scenarios in

Figure 2-5 would require far less computational time than the IC2 scenarios because fewer transient simulations are

required. This assumption is validated in the results, which are shown in the next section.

2.3.5 Results

The results are presented here in two sections. The first section shows the results of the scenarios identified in Figure

2-5. The second section shows the performance of the two other UV reactors using the proposed methodology. All

scenarios shown were run at 70% UVT; other scenarios at 50% and 90% UVT were run but did not change the

results so were not included in the paper.

2.3.5.1 Determining the proposed methodology

2.3.5.1.1 Initial Condition 1 – IC1

The first scenarios in Figure 2-5 are the steady-state simulations of G1-M1, G1-M2, G1-M3 and G1-MB. These

simulations were used to determine the working mesh, the initial condition and the required particles for the

transient simulations under IC1. There are two types of convergence sought here: 1) “particle” convergence, where a

solution is independent of the number of particles injected in the flow field, and 2) “mesh” convergence, where a

solution is independent of the mesh used to solve the flow field.

Table 2-2 shows the RED for each steady-state scenario in Figure 2-5 under IC1: there is an RED provided for each

mesh and number of particles injected.

38

Table 2-2 – Steady-state RED of each mesh and particle scenario for G1

Number of Particles RED for each mesh (mJ/cm2)

G1-M1 G1-M2 G1-M3 G1-MB 575 12.39 11.75 11.73 11.82

2500 12.17 11.85 11.66 11.51 5600 12.06 11.96 11.90 11.51

10000 12.14 11.96 11.82 11.60 22500 12.19 11.89 11.83 11.66 40000 12.19 11.89 11.86 11.68 62500 12.16 11.94 11.88 11.64

The particle convergence is examined by looking at the change in RED for each mesh vertically in Table 2-2. For all

meshes, the change in the RED as the number of particles decreased was only less than 1% for at least 5600

particles; this number of particles ensures particle convergence and is thus known as the “required particles”. The

mesh convergence is examined by looking at the change in RED for each number of particles horizontally in Table

2-2. For each number of particles, the change in the RED as the mesh was changed to one with fewer elements (i.e.

from G1-M3 to G1-M2) was only less than 1% between G1-M2 and G1-M3 when the number of particles was

22500; thus, M2 and 22500 particles ensured mesh convergence. However, before concluding that G1-M2 should be

the working mesh, the velocity magnitude at various positions in the flow domain for each mesh, G1-M1, G1-M2,

G1-M3 and G1-MB, was examined and are shown in Figure 2-7.

39

Figure 2-7 - Velocity magnitude at various locations downstream of the lamp for each mesh in G1.

The velocity magnitudes do not visibly differ between G1-M2, G1-M3, or G1-MB. This further confirms that G1-

M2 should be chosen as the working mesh and the steady-state solution using G1-M2 should be used as the initial

condition for the transient simulations.

Next, using G1-M2 as the mesh and initial condition and 22500 particles as the required particles, the transient

simulation was performed using M2-TS1, M2-TS2, and M2-TS3. The RED for each time step is shown in Table 2-

3, along with that of G1-MB-TSB.

Table 2-3 – Transient RED for each time step under IC1-M2 and percent change in RED from MB-TSB

Time Step RED (mJ/cm2) % difference in RED from MB-TSB

TS1 11.49 1.1% TS2 11.42 0.53% TS3 11.43 0.62% MB-TSB 11.36 -

The two smaller time steps, M2-TS2 and TS3, were able to match the transient best-case RED within 1%. It should

also be noted that the lower time step (TS3) produced a slightly higher difference in RED than the higher time step

40

(TS2): this was due to transient variations, and regardless, the difference in RED for both was still less than 1%.

Also, M2-TS2 required less computational time than TS3 because fewer time steps were necessary to ensure all

particles left the reactor. The IC1 scenario successfully modelled the best-case RED within the acceptable tolerance

and with low computational time.

2.3.5.1.2 Initial Condition 2 - IC2

The other scenarios on the right side of Figure 2-5, under IC2, used the same meshes as above but with an initial

condition of the bulk flow velocity and with three different time steps, TS1, TS2 and TS3. The RED and the percent

difference in the RED from the best-case solution are shown in Table 2-4.

Table 2-4 – Transient RED and the % difference from best-case RED for all IC2 scenarios with G1

Mesh and Time Step

RED (mJ/cm2)

% difference in RED from transient best-case solution

M1 – TS1 15.03 32.3% M1 – TS2 12.45 9.6% M1 – TS3 12.18 7.2% M2 – TS1 12.24 7.7% M2 – TS2 11.64 2.5% M2 – TS3 11.44 0.70% M3 – TS1 12.17 7.0% M3 – TS2 11.59 2.0% M3 – TS3 11.44 0.70% MB – TS1 11.71 3.1% MB – TS2 11.39 0.26% MB – TSB 11.36 -

It is evident that G1-M1, the coarsest mesh, was not sufficient to model the best-case RED according to the

convergence tolerance that was chosen for this paper. Only the smallest time step, TS3, for G1-M2 and G1-M3,

matched the best-case RED within the tolerance chosen. All of the scenarios clearly required less computational

time, though, than the MB-TSB scenario; however, G1-M2-TS3 (with a CFL less than 0.5) required the lowest

computational time for this geometry and thus was chosen as the converged solution for this transient solution.

41

2.3.5.2 IC1 and IC2

The dose distributions for G1-IC1-M2-TS2, G1-IC2-M2-TS3, and G1-MB-TSB are shown in Figure 2-8; they are

virtually identical.

Figure 2-8 – The dose distributions of IC1-M2-Ts2, IC2-M2-TS3, and MB-TSB for G1.

To better show the difference between the distributions, Figure 2-9 shows the percent difference in the number of

particles in each bin between G1-IC1-M2-TS2 and G1-MB-TSB and G1-IC2-M2-TS3, and G1-MB-TSB.

Figure 2-9 – The percent difference in the number of particles per dose bin in the dose distributions for the IC1-M2-TS2 and IC2-M2-TS3 scenarios compared to the MB-TSB dose distribution.

42

The difference between the two scenarios is more clear in Figure 2-9: for each bin, the percent difference in particle

frequency per bin is generally lower for the IC1-M2-TS2 scenario. This contributes to a lower overall difference

(albeit not significant) in RED for that scenario.

It is fair to assume that the choice of initial condition, either the steady state flow field or the bulk flow velocity,

does not produce a significantly different dose distribution or RED, but the former is superior as an initial condition

for a number of reasons: 1) it required far less computational time, as predicted previously, because fewer transient

simulations were needed; 2) it prescribes a number of particles that must be injected and collected from the reactor

to ensure a converged RED, which is impossible to know when using the other initial condition; and 3) it facilitates

the use of the tRED and ensures that each collection of particles over time will have a converged RED. Thus, it is

recommended that the steady-state solution be used as the initial condition for all transient simulations; this is

reflected in the proposed methodology in Figure 2-1. As well, G1-IC1-M2-TS2 is chosen as the preferred solution.

2.3.5.3 Transient RED

The tRED was introduced in this paper to examine the transient nature of the performance of the UV reactor. Figure

2-10 shows the tRED normalized by the overall RED for the G1-IC1-M2-TS2 (the chosen mesh and time step using

the proposed methodology) solution when the tRTS is two, four, and eight times the simulation time step and at least

22500 particles were collected for each tRTS.

43

Figure 2-10 - tRED (normalized by the overall RED) over a number of simulation time steps for different tRTS for G1-IC1-M2-TS2.

Figure 2-10 demonstrates that a lower tRTS provides more frequent “sampling” of the reactor, leading to slightly

higher variance in the tRED compared to a higher tRTS. However, no variation of the tRED exceeds 1% of the

overall RED. As well, the maximum and minimum transient RED is generally consistent for all tRTS. As a result, it

does not necessarily matter what the tRTS is, as long as it is higher than the simulation time step. Essentially, the

choice of tRTS is dependent on the scale of transient phenomena the modeler wants to examine. However, choosing

a very large tRTS would require a much longer simulation time to ensure there is a suitable time period for which to

collect particles. To the extreme, a very large tRTS would eventually just calculate the overall RED.

It is also prudent to examine the sensitivity of using the proposed methodology to the test microbe being used to

determine RED. The RED to this point was calculated using MS2 as the test microbe. Figure 2-11 shows the

percent difference in RED between the best-case RED and the scenarios IC1-M2-TS2 and IC2-M2-TS3 as the the

test microbe changes.

44

Figure 2-11 - Percent difference in RED of IC1-M2-TS2 and IC1-M2-TS3 from the best-case solution for a range of inactivation rate constants

It is evident that the percent difference in RED from the best-case solution for IC1-M2-TS2 is lower than 1% for the

full range of test microbes, including that of MS2 (k = 0.14 cm2/mJ) and for T1 (k = 0.46 cm2/mJ, (NWRI, 2012)).

The IC2-M2-TS3 scenario performs essentially the same, except for microbes with a low inactivation rate constant,

i.e. with a high UV resistance. With a high-UV resistance microbe, the RED tends towards the mean dose; thus,

using the IC2 scenario, the mean dose is not being predicted as well. Regardless, this shows that the convergence

process is indeed dependent on the test microbe, but if the microbe is kept consistent, it does not affect the validity

of the procedure outlined here.

2.3.6 Validation of the proposed methodology

In this section, the UV reactor geometries shown, G2 and G3 in Figures 2-3 and 2-4 are used to validate the

proposed methodology. For each, the relevant results from using the proposed methodology, outlined in Figure 2-1,

are shown here. For each geometry, three meshes, five cases of different number of particles, and three time steps

were chosen.

45

2.3.6.1 Geometry 2

The three meshes for G2 were designed using the same methods as G1, with an aim to ensure a specific ratio of

element size to lamp sleeve diameter near the lamp zone with a mesh that gets finer near the lamp sleeve and a

hexahedral sweep mesh in the entrance and exit lengths; the meshes are not shown here for brevity and similarity to

the mesh around the lamp for G1. G2-M1, G2-M2 and G2-M3 contain 460000, 1.1 M, and 4.0 M elements,

respectively, each corresponding to elements near the lamp having maximum streamwise lengths of 5%, 3.75% and

2.4% the lamp sleeve diameter, respectively. The meshes are not shown here for brevity because they match closely

with those from G1. Pursuant to the first five steps in the proposed methodology, the steady-state RED for each

mesh and number of particles injected into the second UV reactor is shown in Table 2-5.

Table 2-5 – Steady-state RED of each mesh and particle scenario for G2

Number of Particles

RED for each mesh (mJ/cm2) G2-M1 (5%)

G2-M2 (3.75%)

G2-M3 (2.4%)

2500 11.80 11.97 11.88 10000 11.89 12.02 11.96 22500 11.88 12.05 11.95 40000 11.90 12.03 11.95 62500 11.86 12.03 11.94

For G2, at least 10000 particles were required for particle convergence. However, at least 22500 particles and M2

were required to ensure mesh convergence. Thus, G2-M2 and at least 22500 particles were chosen as the working

mesh and required particles for the transient simulations, respectively.

The dose distribution for the transient simulations using the three time steps for G3-M2 are shown in Figure 2-12. A

total of 2 million particles were injected and collected over the time of the simulation.

46

Figure 2-12 - Overall transient dose distributions of M2 with three different time steps

It is evident that the dose distributions for each time step are virtually identical and the REDs for each time step

confirm this; they are shown in Table 2-6, as well as the percent difference from the smaller time step and the

corresponding CFL for each time step.

Table 2-6 – Overall transient RED and percent change from the smaller time step for G2

Time Step RED (mJ/cm2)

% difference from smaller time step

TS1 = 0.016 s (CFL = 5) 11.96 0.25% TS2 = 0.0032 s (CFL = 1) 11.98 0.67% TS3 = 0.0016 s (CFL = 0.5) 11.90 -

Time step convergence was achieved for all time steps but to avoid performing another simulation to determine if

G2-TS1 or G2-TS2 achieved time step convergence, G2-TS2 was chosen as the working time step.

Figure 2-13 shows the tRED normalized by the RED (i.e. the RED for all particles collected, regardless of the time

they exited the reactor) for G2-M2-TS2 when the tRTS is two, four, and eight times the simulation time step and at

least 22500 particles were collected for each tRTS.

47

Figure 2-13 - tRED (normalized by the overall RED) over a number of simulation time steps for different tRTS for G2 with M2 and TS2.

As was seen for G1, there is not a significant difference in the tRED when the tRTS changes. However, there is

certainly variability within the tRED. While the tRED varies little within the first 50-100 time steps, there is

upwards of 10% variation in the tRED after that period. This is contrast to the tRED for G1, where there was little

variation throughout. The variability could potentially be due to changes in flow past the cylinder that is upstream

of the UV lamp: the Coanda effect could force the vortex street downstream of that cylinder to change direction thus

causing particles to follow lower dose paths.

2.3.6.2 Geometry 3

The three meshes for G3 were designed using the same methods as G1 and G2, with an aim to ensure a specific ratio

of element size to lamp sleeve diameter near the lamp zone with a mesh that gets finer near the lamp sleeve and a

hexahedral sweep mesh in the entrance and exit lengths. G3-M1, G3-M2 and G3-M3 have 669000, 2.2 M, and 4.0

M elements, respectively, each corresponding to individual elements near the lamp having streamwise lengths of

8%, 4% and 2% the lamp sleeve diameter, respectively. Pursuant to the first five steps in the proposed methodology,

the steady-state RED for each mesh and number of particles injected into G3 is shown in Table 2-7.

48

Table 2-7 - Steady-state RED of each mesh and particle scenario for the third UV reactor

Number of Particles

RED for each mesh (mJ/cm2) G3-M1 (8%)

G3-M2 (4%)

G3-M3 (2%)

575 234.91 290.31 277.86 2500 234.73 285.09 282.27 5600 236.39 285.71 283.79

10000 234.84 284.34 283.52 22000 236.34 285.33 283.38 40000 236.63 285.00 283.43

For G3, at least 10000 particles for each individual mesh were required to achieve particle convergence. To achieve

mesh convergence, the same number of particles and M2 was sufficient. Thus, G3-M2 and 10000 particles were

chosen as the working mesh and the number of particles required to be injected for the transient simulations.

The dose distribution for the transient simulations using three time steps for G3-M2 are shown in Figure 2-14. A

total of 2 million particles were injected and collected over the time of the simulation.

Figure 2-14 - Overall transient dose distributions of M3 with three different time steps

The RED for each time step are shown in Table 2-8, as well as the percent difference from the smaller time step and

49

the corresponding CFL for each time step.

Table 2-8 – Overall transient RED and percent change from the smaller time step for G3

Time Step RED (mJ/cm2) % difference from smaller time step

TS1 = 0.2133 s (CFL = 5) 220.45 6.0% TS2 = 0.05866 s (CFL = 1) 234.60 0.78% TS3 = 0.02933 s (CFL = 0.5) 233.12 -

Time step convergence was achieved between TS2 and TS3 as the change in the RED to a smaller time step was just

less than 1%.

Figure 2-15 shows the tRED normalized by the RED (i.e. the RED for all particles collected, regardless of the time

they exited the reactor) for the solution with the chosen mesh and time step when the tRTS is two, four, and eight

times the simulation time step and at least 10000 particles were collected for each tRTS.

Figure 2-15 - tRED (normalized by the overall RED) over a number of simulation time steps for different tRTS for G3 with M2 and TS2.

As was seen with G2, there is not a significant difference in the tRED when the tRTS changes. However, there is

variability within the tRED. The tRED behaves different than the tRED for G1 and G2 as it rises to a maximum and

then falls back down to the overall value. This could be due to variability in the flow through the banks and this

50

geometry requiring more time to reach a “pseudo” steady-state. Regardless, it is interesting to note that the

variability is as high as 10% but does seem to converge after a period of time.

2.3.7 Discussion

This paper proposed a standard methodology to determine the RED and tRED for a transient UV reactor simulation.

By using a series of numerical experiments, new standards for resolving issues with mesh, initial conditions, time

step and particle tracking were introduced.

These standards now contained in the proposed methodology include:

1) using the UV lamp sleeve diameter as a guide for determining mesh size and ensuring an average element

size of less than 5% of the lamp sleeve diameter;

2) using the steady-state solution as the initial condition;

3) using the CFL as a guide for determining time step and ensuring the CFL is less than one;

4) using the steady-state solution to find the required number of particles per injection;

5) using the tRED to determine the transient variation in RED in the reactor.

While the main goal of this work was to propose a methodology for transient simulations of UV reactors, the

application of the methodology to UV reactors is also important. Although the tRED changed less than 1% for the

simplest geometry chosen here, the variability was much higher for more complex reactors: up to 10% for the

single-lamp reactor with a cylindrical obstruction and up to 20% for the multi-lamp reactor; this may increase for

more complex modeling techniques, as well.

It is important to have a standard methodology for a transient UV reactor simulation because of the need for

reproducible solutions, which has been a problem in the literature so far. As well, by standardizing the methodology,

it is now possible to codify the process by which the transient behavior of UV reactors is evaluated. The

methodology proposed requires lower computational time than other methods, ensures a converged solution, and,

most important, is systematic.

51

2.3.8 Conclusion

This purpose of this paper was to provide a standardized methodology for a transient simulation of a UV reactor.

Past work has shown disagreement in the usefulness of transient simulations for UV reactors. One side notes that

the simulations are computationally expensive and do not produce different results than steady-state solutions. The

other side has shown that there is a different in disinfection for transient simulations. However, for those works,

there was not a standardized methodology for performing these transient simulations so it is difficult to compare the

results.

This paper examined three parameters, initial condition, mesh and time step, as well as the particle tracking strategy

that will affect a transient simulation for UV reactors. This paper also introduced a new concept in UV reactor

simulation: the tRED. This parameter allows the modeler to produce an RED that gives a picture of the disinfection

performance in the reactor over time as the transient phenomena in the reactor develop. This tRED was shown to

vary no more than 1% from the overall RED for a single-lamp cross flow UV reactor when particles were collected

well downstream of the lamp, but this variation was much for more complex reactors. The next step in this work is

to apply the methodology proposed here for a more complex transient model, such as LES.

52

2.4 References

ANSYS Fluent (2012). Ansys Fluent User Manual v. 14.5. Blatchley, E. R. (1997) Numerical modeling of UV intensity: Application to collimated-beam reactors and continuous-flow systems. Water Research. 31(9), 2205-2218. Bolton, J. (2000) Calculation of ultraviolet fluence rate distributions in an annular reactor: significance of refraction and reflection. Water Research 34(13) 3315-3324 Breuer, M. (2000) A challenging test case for large eddy simulation: high Reynolds number circular cylinder flow. International Journal of Heat and Fluid Flow 21:5 648-54 Chen, J., Deng, B., Kim, C.N. (2011) Computational fluid dynamics (CFD) modeling of UV disinfection in a closed-conduit reactor. Chemical Engineering Science 66(21):4983-4990 Chiu, K., Lyn, D.A., Savoye, P., Blatchley III, E.R., 1999. Integrated UV disinfection model based on particle tracking. Journal of Environmental Engineering 125 (1), 7–16. Courant, R., Friedrichs, K., Lewy, H. (1928), Über die partiellen Differenzengleichungen der mathematischen Physik" Mathematische Annalen (in German) 100 (1): 32–74 Gandhi, V., Roberts, P. J.W, Stoesser, T., Wright, H., Kim, J.-H. (2011) UV reactor flow visualization and mixing quantification using three-dimensional laser-induced fluorescence. Water Research 45:13 3855-62 Gandhi, V., Roberts, P.J.W., Kim, J.H. (2012) Visualizing and quantifying dose distribution in a UV reactor using three-dimensional laser-induced fluorescence. Environmental Science and Technology 46, 13220-13226 Georgiadis, N.J., Rizzetta, D.P., Fureby, C. (2010) Large-eddy simulation: current capabilities, recommended practices, and future research. AIAA Journal 48(8), 1772-1784 Graham, D. and Moyeed, R. (2002) How many particles for my Lagrangian simulations? Powder Technology 125(2):179-186 Kim S.-E. (2004) Large eddy simulation using unstructured meshes and dynamic subgrid-scale turbulence models. Technical Report AIAA-2004-2548. 34th Fluid Dynamics Conference and Exhibit: American Institute of Aeronautics and Astronautics. Franke, R., W. Rodi.(1991) Calculation of vortex shedding past a square cylinder with various turbulence models, in: Proc. Of the 8th Symp. On Turbulent Shear Flows, Munich, Germany, p.189. Lawryshyn, Y. & Hoffman, R. (2015) Theoretical evaluation of UV reactors in series. Journal of Environmental Engineering 141(10):04015023 Li, C., Deng, B., Kim, C.N. (2010) A numerical prediction on the reduction of microorganisms with UV disinfection. Journal of Mechanical Science and Technology 24 (7) 1465-1472. Liu, D., Ducoste, J., Jin, S., Linden, K. (2004) Evaluation of alternative fluence rate distribution models. J. Water Supply Res. Technol. 53(6), 319-408. Liu D., Wu, C., Linden, K., and Ducoste, J. (2007) Numerical simulation of UV disinfection reactors: evaluation of alternative turbulence models. Applied Mathematical Modelling 31(9), 1753–1769.

53

Lyn, D.A., Chiu, K., Blatchley III, E.R. (1999). Numerical modeling of flow and disinfection in UV disinfection channels. Journal of Environmental Engineering 125 (1), 17–26. Lyn, D.A. (2004) Steady and transient simulations of turbulent flow and transport in ultraviolet disinfection channels. Journal of Hydraulic Engineering 130(8), 762-770. Maka, P. & Lawryshyn, Y. (2011) An assessment of the checkpoint bioassay concept for full scale wastewater UV reactor validation. Water Science and Technology 64(1):43-9 Munoz, A., S. Craik, and S. Kresta (2007). Computational fluid dynamics for predicting performance of ultraviolet disinfection - sensitivity to particle tracking inputs. Journal of Environmental Engineering and Science 6(3), 285–301 Nishino, T., Roberts, G.T., Zhang, X. (2008) Transient RANS and detached-eddy simulations of flow around a circular cylinder in ground effect. Journal of Fluids and Structures 24, 18-33 Petri, B.M., An, J., Moreland, V. (2011) UV system checkpoint bioassays: looking back and moving forward, applying the lessons. WEFTEC 9, 6236-6244 Powell, C. and Lawryshyn, Y. (2015) A method for determining the optimal discretization of UV lamps for emission-based fluence rate models. Water Science and Technology 71(12) 1768-1774 Shih, T. H. , W. W. Liou, A. Shabbir, Z. Yang, and J. Zhu (1995) A new k-ε eddy viscosity model for high Reynolds number turbulent flows—Model development and validation. Computers Fluids 24(3):227-238 Sozzi, D. and Taghipour F. (2006) UV reactor performance modeling by Eulerian and Lagrangian methods. Environmental Science and Technology 40(5):1609-15 United States Environmental Protection Agency (2006) Ultraviolet Disinfection Guidance Manual for the Final Long Term 2 Enhanced Surface Water Treatment Rule Wilcox, D.C. (1998). "Turbulence Modeling for CFD". Second edition. Anaheim: DCW Industries Wols, B.A., Uijttewaal, W.S.J., Hofman, J.A.M.H., Rietveld, L.C., van Dijk, J.C. (2010) The weaknesses of a k–ε model compared to a large-eddy simulation for the prediction of UV dose distributions and disinfection. Chem. Eng. J. 162(2), 528-536 Wols, B.A., Hofman, J.A.M.H., Beerendonk, E.F., Uijttewaal W.S.J., van Dijk, J.C. (2011) A systematic approach for the design of UV reactors using computational fluid dynamics. JAIChE J. 57(1), 193-207 Wright, N.G., Hargreaves, D.M. (2001) The use of CFD in the evaluation of UV treatment systems. Journal of Hydroinformatics 3 (2), 59–70 Younis, B.A. and Yang, T.-H. (2011) Prediction of the effects of vortex shedding on UV disinfection efficiency. Journal of Water Supply: AQUA 60.3 147-58

54

Chapter 3 – Fluence Rate Models

3 Introduction The published manuscript in this chapter provides a standard methodology for determining the optimal discretization

for UV lamps when trying to calculate the fluence rate, the total amount of light incident on a particle area, which

allows the calculation of the UV dose of particles. The purpose of this work was to reduce the computational time

required for fluence rate calculations in the RPP.

3.1 Key Results

The manuscript shows that there are equations that can be used to find the optimal number of point or segment

sources, nopt, that is computationally inexpensive yet still accurate when implementing fluence rate models. These

equations were implemented in the paper and found to dramatically reduce the computational cost of fluence rate

calculations as compared to previous studies. Even when higher complexity turbulence models were used, the time

to calculate the dose of each particle accurately using the MPSS FRM was reduced dramatically.

3.2 Implications

The results of this paper make it clear that for most lamp lengths and radii, the number of point or segment sources

used in past studies was much higher than needed. As a result, significant computational cost reductions can be

achieved by following the equations developed in Powell & Lawryshyn (2015). To determine the precise impact of

this model for an optimal NPS/NSS, the number of “computations” that were required to calculate the fluence rate in

the UV reactor was found. Approximately 300 “time” steps (the reactor was steady-state) were produced in the

particle track file from Fluent for each particle. For each time step for each particle, the fluence rate equation needs

to be calculated with respect to each of those point or segment sources. When this number is 1000, 300 billion

computations are required for a million particle simulation (or 300,000 computations per particle). Conversely, when

the number is 267 (the number found for this typical UV lamp used in the paper, the number of computations for a

million particle simulation is approximately 800 million (or approximately 800 computations per particle).

55

3.3 Manuscript

The paper “A method for determining the optimal discretization of a UV lamp for emission-based fluence rate

models” (Powell& Lawryshyn, 2015) is shown below as published.

3.3.1 Abstract

A method for optimizing the number of segment sources needed to discretize UV lamps for fluence rate modeling

and dose calculations when using the multiple segment source summation (MSSS) fluence rate model (FRM) is

presented. An ideal location for determining the optimal number of point or segment sources was found using the

multiple point source summation (MPSS) method with no reflection and refraction. This location was then used to

conduct a fast discretization study for the MSSS FRM. A lower than previously used number of segment sources

was required. This method reduced the time needed to perform a discretization study and thus for fluence rate and

dose distribution calculations in UV reactors.

3.3.2 Introduction

The numerical modeling of using ultraviolet (UV) radiation to disinfect drinking water and wastewater has been

studied widely (Qualls & Johnson, 1983; Linden & Darby, 1997; Blatchley et al., 2008; Bohrerova et al., 2005).

This modeling uses computational fluid dynamics (CFD) to simulate the flow field and fluence rate models (FRMs)

to determine the fluence rate at a specific location to calculate UV dose, defined as the fluence rate integrated over

time. Various FRMs have been used, including the multiple point source summation (MPSS) and multiple segment

source summation (MSSS) FRMs, which are the most popular and accurate (Liu et al., 2004; Li et al., 2011; Li et al.

2012; Rahn et al., 2006). These FRMs require the UV lamp to be modeled as a series of point or segment sources.

However, the number of point or segment sources (NPS/NSS) required to accurately model the fluence rate has not

been formalized. Table 3-1 summarizes a sampling of the NPS/NSS used previously, as well as relevant lamp

characteristics, and also includes the optimal number of point and segment sources, nopt, as calculated by the

methodology outlined later in this paper. Note that the nopt calculated using our methodology is significantly lower

than used in the papers.

56

Table 3-1 – Use of MPSS and MSSS FRMs in literature.

FRM used Reference Lamp length (m)

Sleeve diameter (mm)

NPS/ NSS1

nopt, PS nopt, SS

MPSS Bolton (2000) 0.645 100 1001 23 36 MSSS Quan et al.

(2004) 0.762 25.5 1E6 101 177

UVCalc®3D3 and MSSS

Liu et al. (2004) 0.28 37 1000 27 276

UVCalc®3D Jin et al. (2005) 0.375 n/p2 1000 - - MPSS Rahn et al.

(2006) 0.15 64 10014 9 66

UVCalc®3D Liu et al. (2007) <0.4775 n/p 1000 - - UVCalc®3D Munoz et al.

(2007) 1.1971 68 1000 60 104

MPSS Wols et al. (2010a)

<0.15 n/p 200 - -

MSSS Wols et al. (2010b)

<0.46 50 n/p 28 36

UVCalc®3D Li et al. (2012) 0.297 23 1000 44 49 1NPS = number of point sources; NSS = number of segment sources 2n/p = not provided; the information was not provided in the paper or difficult to infer. 3UVCalc®3D (Bolton Photosciences, Inc: Edmonton, Canada) is a commercially available software version that implements the MSSS FRM with 1000 segment sources. 4This particular value was inferred based upon references in the text to Bolton (2000). 5This particular value was inferred based on diagrams provided in the text; the length could not have been longer than the value given above.

6This particular value is not valid, however, because the methodology proposed here is not valid for lamp lengths less than nine times the lamp sleeve radius.

In this paper, we obtain an ideal location for optimizing the discretization of a UV lamp; this can be then used to

find an optimal number of point or segment sources for the MPSS and MSSS FRMs, respectively. This is then used

to compare the dose distribution for a single-lamp UV reactor using the segment sources calculated using our

methodology and 1000 segment sources, considered “accurate” here.

3.3.3 Methodology

Bolton (2000) was the first to conclude that 1001 point sources were required to calculate the fluence rate using the

MPSS FRM when reflection and refraction were ignored. This number was determined by increasing the NPS until

the fluence rate was within 1% of the fluence rate calculated using the line source integration (LSI) FRM (this is

known as a discretization study). In this section, we derive an implicit function that can be used to determine the

NPS that calculates the fluence rate within a specific tolerance, α, of the LSI FRM.

57

3.3.3.1 MPSS discretization without reflection and refraction

The error equation to solve for nopt is derived from the method used by Bolton (2000) and is shown below as

𝐸8,mEnn − 𝐸8,3no = 𝛼𝐸8,3no (3-1)

When the equations for the FRMs are substituted (from Liu et al. (2004)), the equation to solve for nopt is given by

JL?qrst9=

Iu?qr5fg − E

st3.tanDg

xIy<

.+ tanDg

xID<

.=

α Est3.

tanDgxIy<

.+ tanDg

xID<

. (3-2)

Equation (3-2) is then simplified considerably to make it a function of nopt, L, r, and also p, where p is the axial

distance from the centre of the lamp divided by lamp length. A formula for hi as a function of nopt was also

determined:

ℎ5(𝑛8|}) = 𝑖 − 𝑚𝑜𝑑(𝑛8|} + 1,2 − u?qry Dg L?qr�N

�) 3u?qr

− 𝑝𝐿 (3-3)

where mod(x, y) is defined as a function that returns as 0 if the quotient of x and y is an integer, and 1 otherwise.

The final implicit function for nopt is given below and can be solved using numerical methods:

1𝑛8|}

1

𝑟� + ( 𝑖 − 𝑚𝑜𝑑(𝑛8|} + 1,2 −𝑛8|} + −1 u?qryg

2 ) 𝐿𝑛8|}

− 𝑝𝐿)�

u

5fg

−

1𝐿𝑟

tanDg𝐿2 + 𝑝𝐿𝑟

+ tanDg𝐿2 − 𝑝𝐿𝑟

−

�3.

tanDgxIy|3

.+ tanDg

xID|3

.= 0 (3-4)

58

Numerical experiments showed a higher NPS was required for: 1) locations closer to the lamp sleeve; 2) lamps with

larger lamp sleeve radii; and 3) longer lamps. Most importantly, we also found that for a given lamp sleeve radius,

there is a location where nopt is highest: this ensures an accurate fluence rate everywhere around the lamp. This

location is given by a radial and axial location with respect to the centre of the lamp. The radial location of nopt is

always found just outside the lamp sleeve radius, at

r = rsleeve + ε (3-5)

where ε is a very small number. The axial location (along the lamp axis) of nopt is just outside the lamp arc. It is a

linear function of the lamp sleeve radius and was found through a least squares fit with an R2 = 0.987 for all lamp

length and radii where 1% tolerance is desired:

𝑝 = 0.5 + +.,-./011213

(3-6)

Figure 3-1 shows the percent difference between the fluence rate at NPS = 5000 and at various NPS for a 0.0250 m

lamp radius and 1.0 m lamp length and at UVT = 100%. The nopt calculated by equation (3-4) was NPS = 68 for this

scenario. The location of highest error matches with equations (3-5) and (3-6) and is just outside the lamp arc. The

white space in the middle is the lamp.

Figure 3-1 – Percent error in fluence rate (from NPS = 5000) for various NPS using the MPSS without reflection and refraction. (Lamp length= 1.0m; lamp sleeve radius = 0.0250 m; UVT = 100%)

Additionally, it was found nopt varies linearly with lamp length. This means that calculating the nopt for a given lamp

sleeve radius using a 1.0 m lamp produces a point source per lamp length (PSLL), which can be multiplied by the

length of the lamp to get a new nopt for a smaller or larger lamp without having to solve equation (3-4) again.

59

To summarize, the location of nopt can be found using equations (3-5) and (3-6). These equations are based on the

fluence rate without reflection and refraction, but as is shown in the next section, using this location to obtain the nopt

when reflection and refraction are considered will also produce an accurate fluence rate and dose distribution.

3.3.3.2 MSSS discretization with reflection and refraction

The MSSS FRM with reflection and refraction is the most accurate fluence rate model with respect to experimental

methods (Liu et al., 2004; Li et al., 2011; Li et al. 2012; Rahn et al., 2006). This fluence rate, using more than 1000

segment sources, is the “accurate” fluence rate to which the fluence rate calculated using nopt, with our proposed

methodology, is compared. The MSSS FRM is given by Liu et al. (2004):

𝐸8,mEnn = (gD�N,=)(gD�I,=)

JL

tI(9N,=y9I,=y9�,=)I𝑇�@�,=�.�N𝑇�

@I,=�.�N𝑐𝑜𝑠𝜃g,5.u

5fg (3-7)

Due to the complexity of the MSSS FRM, an implicit equation to determine nopt cannot be formulated. Therefore, a

discretization study must be performed. To reduce computational time, we can find the nopt for the MSSS FRM by

using the location of the optimal discretization using equations (3-5) and (3-6).

In Figure 3-2, the percent difference from the “accurate” fluence rate (NSS = 5000) for different NSS is shown. The

nopt for this particular lamp length and radii (L = 1.0 m, rsleeve = 0.0250 m, UVT = 70%) is 267, calculated using the

methodology outlined above. The same trend with the MPSS without reflection and refraction is seen: the highest

error occurs just outside the lamp arc and follows equations (3-5) and (3-6). However, for the nopt there is slightly

higher error in the middle of the lamp; this is not seen for NSS = 1000. This slightly higher error (still below 1%) in

the middle of the lamp should not affect dose distribution statistics or disinfection because particles in this area

should receive a higher dose. These higher dose particles do not have a large effect on disinfection for test

organisms like MS2, as demonstrated previously in this paper.

60

Figure 3-2 - Error in fluence rate (from NSS = 5000) for various NSS using the MSSS with reflection and refraction. (Lamp length= 1.0m; lamp sleeve radius = 0.0250 m

The nopt still varied linearly with lamp length for the MSSS FRM, but only for lamps with a length more than

approximately nine times the radius of the lamp, as shown in Figure 3-3. More concisely, for very small lamps, nopt

does not vary linearly with lamp length – but for these very small lamps, a discretization study is not onerous so

could be performed as normal.

61

Figure 3-3 - The number of segment sources as a function of very small lamp lengths for the MSSS FRM.

Two other sensitivity analyses concerning the effect of UVT on discretization and the effect of multiple lamps on

discretization were conducted. First, we examined the effect of UVT on nopt for very low-UVT applications. Figure

3-4 shows the nopt for the same 1.0 m lamp discussed above at different UVT; the UVT is shown on a logarithmic

scale. The fluence rate was calculated at the location given by equations (3-5) and (3-6).

62

Figure 3-4 - The optimal number of segment sources as a function of UVT.

As is shown, the nopt increases as the UVT decreases. At 100% UVT, nopt = 100; at 0.0001% UVT, the nopt increases

to 140. For the most practical applications when UVT is greater than 50%, there is little change in the discretization

required. However, the discretization requirements increase when very low UVT applications are considered.

Second, a two-lamp configuration was examined in a reactor and there was no difference in the fluence rate and

dose distribution error. This is because the algorithm used to calculate fluence rate and dose when more than one

lamp is present only considers the contribution from one lamp at a time, when shadowing is not considered.

Additionally, as noted before, because the highest error (but still less than 1%) is seen near the lamp, a particle that

is close to one lamp would not be close to another lamp – this mitigates any potential error compounding caused by

multiple lamps.

63

3.3.4 Effect on dose calculations

To demonstrate the effectiveness of our proposed methodology, the dose distribution was calculated for a single-

lamp parallel flow rectangular UV reactor with 5000 particles using 267 SS (which is the nopt found using the

methodology described here) and 1000 SS for the MSSS FRM. The lamp characteristics are provided in Table 3-2.

The dose distributions using the different NSS are shown in Figure 3-5; there is no discernible difference.

Figure 3-5 – Dose distribution of a reactor calculated using NSS = 267 (-o-) and NSS = 1000 (-*-).

The root mean square error (RMSE) of the dose of each particle between NSS = 267 and NSS = 1000 is 0.05% and

when two different microorganisms (MS2 and T1) are considered, the log reduction for each is within 0.3% when

comparing the two NSS values. The dose calculations in Matlab using the ode45 function (see Matlab (2011)) took

93 minutes or 1.1 seconds per particle for 267 SS and 9.25 hours or 6.7 seconds per particle for 1000 SS. Thus, the

methodology outlined here maintains accuracy and reduces computational cost for even the most basic dose

calculations.

64

Table 3-2 – Technical specifications of the lamp and reactor used.

Specifications Reactor length (m) 5.50 Reactor width (m) 1.5 Reactor height (m) 0.15 Flow rate (m3/h) 4.75

Lamp sleeve radius (m) 0.0250 Total lamp power (W/m) 225

UV Lamp efficiency 67% Sleeve transmittance (%) 96

UV Transmittance (%/cm) 70% Turbulence model k-epsilon Realizable

Lamp arc length (m) 1.50 m

The two steps in the dose calculation that require the greatest computational time are: 1) solving the flow field to

obtain the particle tracks (here, we used Fluent (2006)) and 2) calculating and integrating the fluence rate at each

location to obtain the dose. The simulation above was solved using a steady-state k-ε model and required

approximately 30 minutes to converge for a 400000 element mesh, which is much less than the time to calculate

dose using both of the discretization values above. We also used a more complicated geometry from a separate

project: a 1.2M element transient Large Eddy Simulation (LES) simulation of a single-lamp cross flow reactor

required approximately 6 days to solve for five hydraulic residence times (HRTs). Over 5 M particles were injected

during the simulation; using 50 SS to discretize the lamp (nopt found using the methodology here), the dose

distribution was calculated in a total computer run time of 19 days using the MSSS FRM (the calculations were run

simultaneously on different machines using eight cores for parallel processing). Using the same time scaling as

above, the run time would be over a year when using 1000 SS. The advantage of our approach outlined here is even

more evident for simulations with a very large number of particles.

3.3.5 Conclusions

In this paper, a method for finding the nopt for the discretization of UV lamps for fluence rate calculations was

presented. Using the best location for this optimization, found by using the MPSS fluence rate without reflection and

refraction, a discretization study for the nopt is faster and more systematic. This method requires the user to calculate

the fluence rate at only one location with one lamp length using successively higher numbers of segment sources

until a “converged” fluence rate is found. This method produces an accurate fluence rate with fewer segment

sources, which reduces computational time dramatically. Two sample geometries demonstrated significant

65

computational time-savings for the approaches discussed here. When multiple lamps or millions of particles are

used (such as in LES or other transient simulation), the time to calculate the dose outweighs the time required to

solve the flow field; this is when fewer segment sources may save computational time.

66

3.4 References

ANSYS Fluent (2006). ANSYS Fluent User Manual v. 12.1. Blatchley, E. R., III, Shen, C., Scheible, O. K., Robinson, J. P., Ragheb, K., Bergstrom, D. E., Rokjer, D. (2008) Validation of large-scale, monochromatic UV disinfection systems for drinking water using dyed microspheres. Water Res. 42:677–688. Bohrerova, Z., Bohrer, G., Mohanraj, S. M., Ducoste, J., Linden, K. (2005) Experimental measurements of fluence distribution in a UV reactor using fluorescent microspheres. Environ. Sci. Technol. 39:8925– 8930. Bolton, J. R. (2000) Calculation of ultraviolet fluence rate distribution in an annular reactor: significance of refraction and reflection. Water Research 34(13) 3315-3324. Jin, S., Linden, K.G., Ducoste, J., and Liu, D. (2005) Impact of lamp shadowing and reflection on the fluence rate distribution in a multiple low-pressure UV lamp array. Water Research 39 (12): 2711–21. Li, M., Qiang, Z., Li, T., Bolton, J.R. and Liu, C. (2011) In situ measurement of UV fluence rate distribution by use of a micro-fluorescent silica detector. Environ. Sci. & Technol. 45:3034-3039. Li, M., Qiang, Z., Bolton, J., and Weiwei, B. (2012) Impact of reflection on the fluence rate distribution in a UV reactor with various inner walls as measured using a micro-fluorescent silica detector. Water Research 46 (11): 3595–602. Linden, K. G. & Darby, J.L. (1997) Estimating effective germicidal dose from medium-pressure UV lamps. J. Environ. Eng. 123:1142– 1149. Liu, D., Ducoste, J., Jin, S., Linden, K. (2004) Evaluation of alternative fluence rate distribution models. J. Water Supply Res. Technol. 53(6) 319-408. Liu D., Wu, C., Linden, K., and Ducoste, J. (2007) Numerical simulation of UV disinfection reactors: evaluation of alternative turbulence models. Applied Mathematical Modelling 31(9): 1753–1769. Matlab 2012a (2011). Matlab User Manual. Munoz, A., Craik, S., and Kresta, S. (2007) Computational fluid dynamics for predicting performance of ultraviolet disinfection – sensitivity to particle tracking inputs.” Journal of Environmental Engineering and Science 6(3): 285–301. Qualls, R. G. & Johnson, J. D. (1983) Bioassay and dose measurement in UV disinfection. Appl. Environ. Microbiol. 45: 872– 877. Quan, Y., Pehkonen, S., and Ray, M. (2004) Evaluation of three different lamp emission models using novel application of potassium ferrioxalate techniques. Industrial and Engineering Chemistry Research 43(4): 948-955. Rahn, R., Bolton, J., and Stefan, M. (2006) The iodide/iodate actinometer in UV disinfection: determination of the fluence rate distribution in UV reactors. Photochemistry and photobiology 82(2): 611-15. Wols, B.A., Shao, L., Uijttewaal, W.S.J., Hofman, J.A.M.H., Rietveld, L.C., van Dijk, J.C. (2010a) Evaluation of experimental techniques to validate numerical computations of the hydraulics inside a UV bench-scale reactor. Chemical Engineering Science 65 4491-4502. Wols, B.A., Uijttewaal, W.S.J., Hofman, J.A.M.H., Rietveld, L.C., van Dijk, J.C. (2010b) The weaknesses of a k–ε model compared to a large-eddy simulation for the prediction of UV dose distributions and disinfection. Chem. Eng. J. 162(2) 528-536.

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Chapter 4 – Reactor Performance Models

4 Introduction

The manuscript in this chapter to aims to develop a viable alternative to the RED to calculate the reactor

performance by developing a new mixing metric to describe UV reactor mixing. Further, it successfully quantifies

mixing in UV reactors with a strong positive correlation between mixing and UV reactor performance. This

mixedness metric thus can be used as a proxy for RED, dramatically reducing the computational cost associated

with the reactor performance calculation.

4.1 Key Results

The manuscript shows that:

1) there is a very strong positive correlation between mixedness and the RED of UV reactors;

2) the mixedness developed in Powell & Lawryshyn (2017) produces a stronger positive correlation with

RED than previous mixedness metrics;

3) a reactor simulated to be perfectly mixed according to the modified mixedness produced a 50% higher

RED than a reactor simulated to be perfectly mixed using the original mixedness.

4.2 Implications

The results of this paper, in particular the very strong linear correlation between the mixing metrics and the reactor

performance metric, RED, show that mixing can be used as a proxy for reactor performance in certain scenarios.

This means that the performance of two reactors with the same geometry but slightly different lamp configurations

can now be compared using the mixing metric, negating the need for a large number of fluence rate calculations –

the largest computational cost for a reactor performance calculation. It is important to recognize here though that

there are still other factors that have to be examined when looking at reactor performance. Fluence rate has a

68

significant impact on performance, so at least one RED calculation is required to establish that baseline correlation.

And indeed, it would almost be possible to have perfect positive correlation between mixedness and RED: mixing is

not the only factor affecting UV reactor performance.

Once the particle tracks are obtained from the CFD models, the calculation to determine the mixedness depends on

the number of divisions by which the geometry cross-section is divided, and not the number of particles in the

simulation (however, a minimum number of particles are required to ensure convergence). For each region, there are

approximately 30 computations required – thus, for the 400 region system discussed in the paper, there are

approximately 12,000 computations required. This number is drastically lower than the number of computations

required to calculate the fluence rate (and thus RED) for a one-million particle system (300 billion previously or 800

million with our method).

4.3 Manuscript

The paper “A modification of the Shannon entropy mixedness to quantify mixing in UV reactors” (Powell &

Lawryshyn, 2017) is shown below as accepted, but not yet published.

4.3.1 Abstract

Mixing or mixedness in ultraviolet (UV) wastewater reactors has been identified by many as a key factor in reactor

performance. However, the current methods for determining mixing are either qualitative, and cannot be correlated

directly with reactor performance, or use ideal mixing states that are not practical. In this paper, a modified

Shannon entropy mixedness metric was used to quantify mixing in UV reactors. Contrary to the current literature

which showed that the mixedness only decreased when the number of regions by which the system is divided was

increased, it was shown here numerically that mixedness can increase or decrease when the number of regions are

increased. As well, it was proven numerically and theoretically that the mixedness will converge to a finite value

with a sufficiently high number of regions. Furthermore, a highly simulated system that achieved perfect mixing

under the modified mixedness equation produced a reduction equivalent dose (RED) almost 50% higher than the

perfect mixing state under the original mixedness. The modification made to the mixedness equation suits the need

69

of a UV reactor, but can also be applied to other process flow systems where mixing and reactor performance are

linked. Finally, a very strong correlation between mixedness and UV reactor performance was found for two

idealized single-lamp UV reactors. This means that the mixedness calculated here can be used as a proxy for reactor

performance, allowing modelers to significantly reduce the computational cost required when comparing the

performance of different reactors.

4.3.2 Introduction

Using ultraviolet (UV) light to disinfect wastewater is now commonplace in many North American municipalities.

The design and modeling of UV wastewater systems requires careful consideration of UV lamp and channel

configurations that create atypical hydraulics, ultimately affecting mixing in a UV reactor, which has been identified

by many to be a critical factor in UV reactor performance (Courtelyou et al., 1954; Haas & Sakellaropoulos, 1979;

Severin et al., 1984).

To date, the discussion of mixing and UV reactor performance has been qualitative. Cortelyou et al. (1954) noted

that an increase in turbulence in the reactor increased the inactivation of microorganisms and thought that ideally,

reactor design should promote the movement of organisms towards the UV lamp(s). Qualls & Johnson (1985) raised

the importance of radial mixing, that is, movement of organisms perpendicular to the direction of flow, in increasing

reactor performance. Similarly, Haas & Sakellaropoulos (1979), Thampi & Sorber (1987) and White et al. (1986) all

discussed that maximizing plug flow (maximizing radial mixing and minimizing axial mixing) in a UV wastewater

reactor resulted in a better performing reactor. It is clear in the literature and in practice that increased radial mixing

is desired in order to improve reactor performance, but there have been few studies examining how to quantify this

mixing and its effect on UV reactor performance.

Severin et al. (1984) were the first to attempt to quantify mixing in UV reactors with kinetic models and the first to

relate the quantification of mixing to UV reactor performance. Their models related an ideal mixing condition, like

complete radial mixing or both complete radial and axial mixing to determine a survival ratio of organisms. The

models fit well with experimental bioassay data, but the determination of mixing was dependent on the UV

disinfection process itself. As well, the models only used idealized mixing conditions, not mixing conditions from

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actual UV reactors, to determine the relationship between mixing and reactor performance. Ultimately, a quantity of

mixing was never determined independently and then related to the reactor performance, so the methodology could

not necessarily be used to determine “mixedness” levels in a UV reactor. Gandhi et al. (2012) also quantified mixing

using the covariance of the concentration of a chemical tracer, but that methodology used experimental results from

3D particle image velocimetry, which is not a trivial undertaking. As well, the mixing performance is not directly

comparable to reactor performance because the tracer is chemically reactive whereas the UV light is physically

reactive. The different reaction pathways and localized nature of tracer studies mean that comparing the mixedness

with overall reactor performance is not logical. Because mixing is such an important factor in UV reactor

performance, it would be useful to quantify its impact, yet few have attempted this quantification beyond an implicit

understanding that more radial mixing provides greater reactor performance (Haas & Sakellaropoulos, 1979;

Thampi & Sorber, 1987; White et al., 1986). Thus, the objective of this paper to quantify mixing in a UV reactor and

to quantify the relationship between mixing and UV reactor performance.

Here, Shannon entropy, from information theory is used as a metric that defines the randomness, uncertainty, or

mixedness in systems. The general form of Shannon entropy is given by (Ogawa & Ito, 1973):

𝐻5 = − 𝑉l∗𝑝5l𝑙𝑜𝑔�𝑝5l

u

lfg

(4-1)

where Hi is Shannon entropy of the particles that entered the UV reactor in region i on the entrance cross-section; pij

is the discrete probability density that particles left region i on the entrance cross-section and entered region j on the

exit cross-section and is equal to the conditional probability when equal-area regions are used; n is the total number

of regions into which the exit cross-section is divided; and V*j is the normalized area of region j on the exit cross-

section given by:

𝑉l∗ =𝑉l𝑉8

(4-2)

where Vj is the area of region j on the exit cross-section and Vo is the area of the region on the entrance cross-section

from which the particles were initially injected. It should be noted that Vo = Vj for equal sized regions; thus, equation

(4-2) is equal to one and, in this case, Shannon entropy is given by:

71

𝐻5 = − 𝑝5l𝑙𝑜𝑔�𝑝5l

u

lfg

(4-3)

A further description of the dynamics of Shannon entropy equation is presented later. It should be noted here that the

base of the logarithm in equation (4-3) is arbitrary as long as it is consistent; the reasons for this are also described

later.

It has been used numerous times in the past. For example, Camesasca et al. (2006) used Shannon entropy to quantify

fluid mixing in flow channels under the Lagrangian framework. Khakar et al. (1997), Masiuk & Rakoczy (2006)

and Masiuk et al. (2008) used Shannon entropy to quantify mixing in powder and paint applications. Shannon

entropy, fully detailed later, is ideally suited to mixing quantification in UV reactors for three main reasons: 1) the

calculation is independent of the UV disinfection process; 2) it can provide both a local and global quantification of

mixing; and 3) it is directly relatable to UV reactor performance because it is based on particle movement. In this

paper, Shannon entropy, as well as a modification developed here, are used to determine mixedness in two simple

UV reactors.

This paper is organized as follows. The next section will provide a general review of the mixing mechanism in UV

reactors and how mixing is thought to improve reactor performance. Following that, in the methodology, two

mixing metrics are discussed: 1) one previously developed based on Shannon entropy and another developed in this

paper using a modification of that concept. Then, the two mixing metrics are applied to two UV reactor

configurations, cross-flow and parallel flow, and the relationship between mixing and reactor performance is

discussed. Finally, there is a brief examination of “perfect mixing” in a simulated UV reactor using the original and

modified mixing metrics. Conclusions and applications of these metrics are provided at the end of the paper.

4.3.2.1 Background

By the most general definition, mixing is an action by which the particles of one or more substances are combined

so that the spatial distribution of the particles is more uniform than before (Mohr et al., 1957; Kattan & Adler,

1972). It is recognized as an important factor in general reactor performance (Danckwerts, 1953; Zweitering, 1959;

72

Levenspeil, 1999) and, more specifically, as discussed before, UV reactor performance (Severin et al., 1984; White

et al., 1986; Lawryshyn & Cairns, 2003; Zhao et al., 2008).

The majority of work discussing the effect of mixing on UV reactor performance has been qualitative and

experimental (Haas & Sakellaropoulos, 1979; Severin et al., 1984; White et al., 1986; Qualls & Johnson, 1985;

Thampi & Sorber, 1987; Gandhi et al., 2012); there was clear consensus that maximizing radial mixing increased

UV reactor performance and that greater axial mixing was undesirable (Qualls & Johnson, 1985). It is perceived that

greater radial mixing reduces the variation in the average distance between the organisms and the lamp (White et al.,

1986); which produces a narrow dose distribution and a more efficient reactor.

Severin et al. (1984) first related mixing and reactor performance quantitatively using equations based on the

inactivation kinetics, UV intensity, and the standard mixing cases. The equations were developed for a number of

cases, including a uniform UV intensity, perfect plug flow, and a completely mixed reactor. While the approach

allows the user to determine the inactivation in a UV reactor based on an ideal mixing case, it does not prescribe the

precise relationship between mixing and reactor performance nor does it account for any local mixing behavior.

Knowing these characteristics of a UV reactor would be useful because they can be used to design, troubleshoot,

improve, and validate UV reactors.

As noted earlier, Shannon entropy can provide a local mixing quantity and accurately describes the mixing in the

radial direction in a UV reactor, so it is well suited to describe mixing in UV reactors. Ogawa & Ito (1973) were the

first to use Shannon entropy to determine the mixedness of a reactor system. They used the example of a solute

spreading from one region to another in a reactor and defined a probability density that the solute existed in one

region in order to calculate Shannon entropy. Shannon entropy was more recently applied to Lagrangian mixedness

quantification for general fluid flow in a channel by Camesasca et al. (2006); this application is the most similar to

quantifying mixing in UV reactors. The next section will describe the methodology to apply Shannon entropy to UV

reactors.

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4.3.3 Methodology

The methodology section is organized as follows: first, the general method to calculate Shannon entropy for a UV

reactor is described. Next, the modification to the original Shannon entropy is developed. Finally, the UV reactor

configurations and conditions that will be modelled using computational fluid dynamics (CFD) to apply the original

and modified Shannon entropy equations are outlined.

The general method for calculating Shannon entropy for a UV reactor is provided below:

1) Perform the typical CFD simulation of a UV reactor: build a reactor geometry, choose three meshes,

conduct a mesh independence test, and find the working solution.

2) Choose a region of interest between which mixing is of concern. This region is bookended by two identical

cross-sections, named the entrance and exit cross-sections; an example of the locations of these cross-

sections is shown in Figure 4-1 for the single-lamp cross-flow UV reactor. For a UV reactor, the entrance

and exit cross-sections should be at points upstream and downstream of the lamp(s), respectively, and

situated at the point where the UV intensity becomes approximately zero.

Figure 4-1 - Illustration of the division of the entrance and exit cross-sections from the single-lamp cross-flow UV reactor.

3) Divide each cross-section into n equal-size regions, so that each region on the entrance cross-section

corresponds to the same region on the exit cross-section. In Figure 4-1, for example, each cross-section is

divided into nine regions. Each region is labeled with i and j for the entrance and exit cross-sections,

74

respectively, and are functions of the two coordinates for each of the entrance and exit cross-sections as

calculated from the row and column indices of each region on the entrance cross-section, i1 and i2,

respectively, and the row and column indices on each region on the exit cross-section, j1 and j2,

respectively:

𝑖 = (𝑖� − 1) 𝑛� + 𝑖g (4-4)

𝑗 = (𝑗� − 1) 𝑛� + 𝑗g (4-5)

where nc is the total number of columns in the cross-sections.

4) Inject particles into the inlet of the flow domain of the UV reactor

5) For each region i, calculate the pij using

𝑝5l = 𝑁5l𝑁"5

(4-6)

where Nij is the number of particles leaving from region i and entering region j, and NTi is the total number

of particles that left from region i. Calculate this by counting the number of particles that entered from

region i on the entrance cross-section and passed through each region j = 1:n on the exit cross-section.

6) Using the values found in step 5, calculate the local Shannon entropy, Hi, for each region i using equation

(3).

It is useful here to examine the form of Shannon entropy equation for two extreme cases: complete mixing and no

mixing. When there is complete mixing, the position of any particle on the exit cross-section is random and the

probability density is equal throughout all n cross-sections, giving:

𝑝5,g = 𝑝5,� = 𝑝5,u = 1𝑉"∗

(4-7)

where𝑉"∗is the total normalized area, which, with equal area regions, is equal to n. Using equation (4-7) in equation

(4-3) gives Shannon entropy with complete mixing, which is also the maximum of Shannon entropy:

𝐻5,7�� = 𝑙𝑜𝑔�𝑛 (4-8)

When there is zero mixing, all of the particles that began in region i travel to only one particular region j, so pij is

one in that region j and zero everywhere else on the exit cross section. The no mixing condition causes a minimum

in equation (4-3):

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𝐻5,75u = 0 (4-9)

Because Shannon entropy, Hi, varies between zero and a maximum, a local metric for mixedness, Mi, relative to a

maximum mixing state, was defined by Ogawa & Ito (1973) by dividing Shannon entropy by the maximum entropy

to give an equation that varies between zero and one:

𝑀5 = 𝐻5

𝐻5,7��=− 𝑝5l𝑙𝑜𝑔�𝑝5lu

lfg

𝑙𝑜𝑔�𝑛 (4-10)

Mi varies from zero at the entrance cross-section to one when complete mixing is reached (Ogawa & Ito, 1973;

Masiuk et al., 2008; Camesasca et al., 2006). The base of the logarithm is arbitrary for the mixedness because

Shannon entropy is normalized as well by the same-base logarithm.

Camesasca et al. (2006) also defined a global mixedness value for the whole cross section which is the average of

the local mixedness values on the entrance cross-section:

𝑀 =1𝑛

𝑀5

u

5fg

(4-11).

One problem of the application of the original Shannon entropy and mixedness equation to UV reactors is that there

is no difference in the contribution to mixing between a particle that stays in the same region from the entrance to

the exit cross-sections (i.e. when i = j) or one that leaves that region (i.e. when i ≠ j). This is not logical as a particle

that stays within the same region should be considered to be not-mixed. For this reason, the original Shannon

entropy may not be a suitable metric to describe mixing in UV reactors. Indeed, as White et al. (1986) noted, it is

important in a UV reactor to minimize the particles’ average distance from the lamp in order to improve reactor

performance. It is not immediately clear whether achieving perfect mixing using the original mixedness would result

in a minimized average particle distance from the lamp. As well, Cortelyou et al. (1954) postulated that the design of

a UV reactor should promote movement of organisms towards the lamp(s); again it is not clear whether perfect

mixing using the original mixedness, where particle positions are random at the exit cross-section, would push more

organisms toward the lamp(s).

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As a result, in this paper, a modification to the original Shannon entropy is proposed in which a weighting system

gives greater weight to particles that move to a region j on the exit cross-section that is farthest from the region of

injection on the entrance cross-section. This weighting was chosen because it is generally thought that a particle is

“more” mixed when it moves further away from its initial injection zone – it should contribute more to the

mixedness but it currently does not with the original mixedness equation. The weighting system developed here and

described fully in the next section uses “number of blocks” as the numeric system for the weighting for simplicity –

actual distance could be used but this is numerically expensive.

4.3.3.1 Modified Shannon Entropy

The original mixedness equation, shown in equation (4-1), uses the normalized area, V*j, as a “weighting” based on

the size of the regions chosen to divide the cross section. When the regions chosen for the entrance and exit-cross

sections are of equal area, V*j is one, so particle movement to any region is treated equally. In the modified

mixedness equation developed here, a different weighting system based on the radial distance from the ith region on

the entrance cross-section to the jth region on the exit cross-section is used. With this weighting system, particles that

move further from their original position have a higher weighting in the equation.

This new weighting, denoted as Wij, is calculated as:

𝑊5l =5NDlN Iy 5IDlI Iyg5NDlN Iy 5IDlI IygL

FMN= 5NDlN Iy 5IDlI Iyg

OP,= (4-12)

where i1 and i2 are the row and column number, respectively, of the ith region on the entrance cross-section, j1 and j2

are the row and column number, respectively, of the jth region on the exit cross-section and the denominator is the

sum of all the weightings, WT,i, defined as:

𝑊",5 = 𝑛 + (𝑖g − 𝑗g)� + (𝑖� − 𝑗�)�u

lfg

(4-13)

The addition of the one in the numerator of equation (4-12) is required in order to avoid dividing by zero when the

equation is normalized by the weighting from the initial injection region, just as the original Shannon entropy

weighting, the Vj, is normalized by Vo in equation (4-2). The weighting of the initial injection region, Wo, when i=j,

would be zero without the addition of the one, but is the inverse of the sum of the weightings with the addition of the

one:

77

𝑊8,5 = 0 + 1𝑊",5

= 1𝑊",5

(4-14)

This is not a problem for the original mixedness equation because the denominator in equation (4-2), Vo, can never

be zero.

To use the different weighting system described above for the modified Shannon entropy and mixedness, equation

(4-12) must be normalized by Wo,i to obtain the W*ij analogous to the V*

j in equation (4-1):

𝑊5l∗ =

O=F

O?,== (5NDlN)Iy(5IDlI)Iyg

OP,=OP,=g= (𝑖g − 𝑗g)� + (𝑖� − 𝑗�)� + 1 (4-15)

Thus, incorporating the W*ij in place of the V*

j in equation (1) gives the modified local Shannon entropy:

𝐻5,7895:5;9 = − 𝑊5l∗𝑝5l𝑙𝑜𝑔�𝑝5l

u

lfg

. (4-16)

But in this equation, the probability density is no longer equal to the conditional probability because the probability

space has now been changed. Thus, a new probability density, 𝑝5l , is defined by normalizing the conditional

probability, Pij, by the normalized weighting in the same region:

𝑝5l = 𝑃5l𝑊5l

∗ (4-17)

where 𝑝5l is the modified probability density that a particle started in region i on the entrance cross-section and

ended in region j on the exit cross-section and Pij is the conditional probability of the same event. When equation (4-

17) is substituted into equation (4-16), the local modified Shannon entropy is found:

𝐻5,7895:5;9 = − 𝑃5l𝑙𝑜𝑔�𝑃5l𝑊5l

∗

u

lfg

(4-18)

To show the difference in the two “weightings”, the V*j and the W*

ij, of the original and modified mixedness

equations respectively, a square cross-section is divided into nine equal-area regions, as shown in Figure 4-2. The

left side of Figure 4-2 shows the V*j for each region for the original Shannon entropy and the right side of Figure 4-

2 shows the W*ij calculated using equation (4-15), for each region j for the modified Shannon entropy. For this

example, the initial injection region is in the top left hand corner.

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Figure 4-2 - The weightings for the original (a) and modified (b) mixedness equations, respectively. Both are shown when the initial injection region is in the top left hand region.

To demonstrate how the W*ij is calculated in the right side of Figure 4-2, a sample calculation follows. When the

initial injection region is in the top left corner, the W*ij for the bottom right corner where the row and column

numbers are i1 = 1, i2 = 1, j1 = 3 and j2 = 3 can be found using equation (4-15):

𝑊5l∗ = 𝑖g − 𝑗g � + 𝑖� − 𝑗� � + 1 = 1 − 3 � + 1 − 3 � + 1 = 8 + 1 (4-19)

Figure 4-3 shows the different “weighting systems” for a calculation where the initial injection region is in the

middle of the cross section. Again, for the left side of Figure 3, the weighting for the original Shannon entropy does

not change, whereas the weighting varies according to distance from the initial injection position for the modified

Shannon entropy. This means that for the modified Shannon entropy, particles that move further from their original

position and closer to the proportion matching the weights, increase the mixedness more than those that do not.

79

Figure 4-3 - The weightings for the original (a) and modified (b) mixedness equations, respectively. Both are shown when the initial injection region is in the middle region.

Similar to the original Shannon entropy, it can be shown that the modified Shannon entropy has a maximum. This

equation has a maximum when the probability that a particle is in a particular region is equal to the weighting, Wij,

in that region:

𝑃5l = 𝑊5l = (𝑖g − 𝑗g)� + (𝑖� − 𝑗�)� + 1

𝑊",5 (4-20)

Substituting equation (20) into equation (18) gives the maximum of the modified Shannon entropy:

𝐻5,7895:5;9,7��57�7 = 𝑙𝑜𝑔�𝑊",5 (4-21)

This maximum is analogous to the maximum in equation (4-8), but the n is now replaced with the sum of the

weights, WT,i. Thus, for the modified Shannon entropy, the value for a complete mixing state, depends on the

weighting of the particular regions. The minimum modified mixedness is zero.

Analogous to equation (4-10), there is a local modified mixedness. Mi,modified, that varies between zero and one; it is

given as:

80

𝑀5,7895:5;9 = 𝐻5,7895:5;9

𝐻5,7895:5;9,7��= − 𝑃5l𝑙𝑜𝑔�

𝑃5l𝑊5l

∗ulfg

𝑙𝑜𝑔�𝑊",5 (4-22)

and an analogue of equation (4-11) can be used to find the global modified mixedness, Mmodified:

𝑀7895:5;9 =1𝑛

𝑀5,7895:5;9

u

5fg

(4-23)

4.3.3.2 Examples

Below, a series of fictitious particle movement examples are used to compare the dynamics of the local original

mixedness to the local modified mixedness. For these examples, two cross-sections bookending a region of interest

of a rectangular UV reactor channel were chosen; each cross-section is divided into nine equal areas just as in Figure

4-1 and 1000 particles were “injected” from the entrance cross-section from the top left hand corner. The particle

positions at the exit cross-section that are shown in each example are simply used for demonstration purposes and

were not modeled using CFD.

As the particles travel through the “reactor”, the local mixedness in the top left hand corner of the entrance cross-

section changes in different ways for the original and modified local mixedness equations. The dynamics of a

number of different scenarios are shown in Figures 4-4, 4-5, 4-6, 4-7, and 4-8 below, along with the local original

and modified mixedness values for each.

The first simulation involves only one particle leaving the initial injection region and travelling to one of nine other

regions in the exit cross-section, while the remaining 999 remain within the initial region. Figure 4-4 shows the

number of particles passing through each region on the exit cross-section for three different cases.

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Figure 4-4 - Number of particles in each region after one particle moves to different regions. The local original and modified mixedness values are shown for each scenario.

The local original mixedness is the same for all scenarios because the equation only measures a departure from the

initial position, not the level of departure. The local modified mixedness increases slightly as the particle travels

further from the initial injection region, so the local modified mixedness equation is capturing how far the particles

move, not just that the particles move. The local modified mixedness value is lower for all of these scenarios

compared to the local original mixedness because the modified weighting system puts very little weight on those

999 particles that stay in the initial injection region, whereas for the local original mixedness, those 999 particles are

given equal weight to the one particle that did move.

In the next scenario, shown in Figure 4-5, 250 particles travel into the same regions as above. Again, the local

original mixedness values are identical to each other for this given case while the local modified mixedness

increases as the particles exit further from the initial injection region.

Figure 4-5 Number of particles in each region after 250 particles moves to different regions. The local original and modified mixedness values are shown for each scenario.

This trend continues when more particles travel into these zones, as shown in Figure 4-6. With 500 particles now

entering these same regions, the local mixedness increases for both equations. As well, the local original mixedness

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values are identical and the local modified mixedness values increase as the particles are further from the initial

injection region.

Figure 4-6 - Number of particles in each region after 500 particles moves to different regions. The local original and modified mixedness values are shown for each scenario.

In Figure 4-7, the particles are dispersed slightly more evenly, with 200 particles moving into four other regions in

the cross section. The local original mixedness values stay the same even when the 200 particles change position,

but the local modified mixedness value increases when the 200 particles that move further from the initial injection

region, so the local modified mixedness is better capturing the movement of particles farther from their original

position.

Figure 4-7 - Number of particles in each region after the particles moves to different regions. The local original and modified mixedness values are shown for each scenario.

The dynamics of the two different equations are evident from the above simple scenarios. But it is also necessary to

examine the perfect mixing states for these equations because they occur with different particle orientations. Figure

4-8 shows the perfect mixing states in the exit cross-section for the original mixedness on the left side and for the

modified mixedness on the right side.

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Figure 4-8 - Perfect mixing states for each of the two local mixedness values. The left side shows the perfect mixing state for the local original mixedness whereas the right side shows the perfect mixing state for the local modified mixedness.

Perfect mixing for the local original mixedness equation occurs when the number of particles, and thus the

probability that a particle is in a particular region, is equal in all regions. The local modified mixedness for this

scenario is lower than the original because of the lower weight given to the particles remaining in the initial injection

region. Perfect mixing for the local modified mixedness occurs when the probability that particles are in each region

is equal to the weighting system. For this scenario, the local original mixedness value is lower because only one

particle remains in the initial injection zone and is contributing very little to the mixedness value.

4.3.3.3 Equation dynamics

In the previous sections, it was shown that both equations, the original and modified mixedness, had a minimum at

zero when there was no mixing, and a maximum at one when there was complete mixing. A number of examples

were also used to compare the dynamics of the two local mixedness equations with respect to specific fictitious

particle movements. These examples provide a comparison of the two mixedness values, yet it is also necessary to

show the behavior of these equations in real systems. There are three important other factors that need to be

discussed when describing the behavior of both equations: 1) the required number of particles injected into the

system; 2) the required number of regions by which to divide the cross-sections; and 3) convergence.

The original and modified mixedness values are constant if the number of regions by which the cross-section is

divided is constant and the proportion of particles present in each region is constant. This is shown in an example in

Figure 4-9.

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Figure 4-9 – Number of particles where the proportion of particles is constant for each scenario. Both the original and modified mixedness values are shown.

On the left hand side of Figure 4-9, 200 particles are present in some of the regions. In the middle, in those same

regions, 20 particles are present and on the right hand side, 2 particles are present in those same regions. For each

scenario, left to right, a total of 1000, 100 and 10 particles were injected from the top left hand corner, respectively.

The local original and modified mixedness values are shown and are constant for each scenario. This result is also

fairly intuitive: if the probability, pij, in each region is equal, the mixedness values also have to be equal when the

number of regions is constant.

In a given UV reactor, for example, it is very rare that the proportion of particles in each region stays constant if the

number of particles injected changes. However, it is reasonable to assume that when using a constant number of

regions, a particular system should have a mixedness value that converges after a sufficient number of particles are

injected. This means that there must be a “particle” convergence test performed on the mixedness values to

determine the sufficient number of particles that produce a converged mixendess value; that is, the mixedness values

must be independent of the number of particles injected. This convergence test is performed for the two test reactors

in this paper in the results.

Similarly, for a given UV reactor, the mixedness should change when the number of regions into which the cross-

sections of the system are divided is changed because the probability of a particle’s presence in particular regions

will change as the size of the regions change. Camesasca et al. (2006) presented results for a Lagrangian particle

system that showed that the original mixedness decreased as the number of regions increased. The authors claim,

plausibly, that when a system is perfectly mixed (mixedness is equal to 1) it “is as such only up to a certain scale”;

that is, it is perfectly mixed only for a certain number of regions, n. When the number of regions is increased, the

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condition of mixing is now based on a smaller subregion of mixing and is no longer perfectly mixed. They showed

that, at a given time, the mixedness of their system was higher with fewer regions and the system proceeded to

perfect mixing or a converged mixedness value faster for systems with fewer regions. Similar results were also seen

by Brandani et al. (2013) when studying mixing in fluid membranes and Perugini et al. (2015) when studying

mixing with magma. However, results presented later in this paper will show that for the UV reactors chosen, the

mixedness actually increased as the number of regions increased and then converged at a particular value. This

increase in the mixedness as the number of regions increased is potentially because there is a greater number of

regions into which a particle can travel, meaning a greater chance that a particle will move into a different region,

increasing the mixedness. This is contrary to the current literature, but, as will be shown next with a numerical

experiment, it is indeed possible.

To determine the general behavior of the mixedness equation as the number of regions changes without modeling a

large number of fluid systems, a Monte Carlo simulation was created to test different particle probability scenarios.

Using the local original mixedness equation in equation (4-10), pij values for a four-region cross-section were

generated 100,000 times. For example, for each iteration, four random numbers were generated using the “rand”

function in Matlab (Matlab, 2012), generating numbers between 0 and 1. The probabilities for particle movement

from the top left hand corner region, p1,1, p1,2, p1, 3, p1,4, were found by normalizing each of those four numbers by

the sum of those four numbers. To increase the number of regions (and thus make smaller regions), each larger

region was divided into four new, smaller, but equal-sized regions. To generate the probabilities for those smaller

regions, for example, from p1,1 to p1,1,1, p1,1,2, p1,1,3, p1,1,4, four new random numbers were generated, each was

normalized by the sum of those four numbers, and then multiplied by p1,1, the larger region probability. Subsequent

splits of the regions were performed and the probabilities were obtained the same way until the mixedness

“converged”. The probabilities of one iteration of each of the 4-, 16- and 64-region cases are presented in Figure 4-

10 and the local original mixedness values are provided under each case.

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Figure 4-10 - Example of one of the iterations of the local mixedness probabilities as the number of regions is increased from four (a), sixteen (b), and 64 (c). The local original mixedness values for each scenario are shown below each case for when the particles were injected from the top left corner region.

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It is interesting to note that in this system, the local original mixedness first decreases and then increases when the

number of regions is increased from four to sixteen and then 64. Although not shown due to brevity, with 256

regions, Mi = 0.823 and with 1028 regions, Mi = 0.829. It is shown in Figure 4-11 that the mixedness does converge

when the number of regions increased past 1028 as well.

Figure 4-11 - The local original mixedness of 100000 iterations as the number of regions increases.

This simulation was run 100,000 times; the local original mixedness for each iteration as the number of regions

increase is shown in Figure 4-11; this analysis was performed, but not shown, for the local modified mixedness

because the same results are produced.

Both an increase and decrease in local original mixedness as the number of regions increased is found. On Figure 4-

11, the proportion of iterations where the local original mixedness increased or decreased when the number of

regions increased is shown: for a given system, the local original mixedness decreases more often than it increases.

The former is consistent with the results by Camesasca et al. (2006); Brandani et al. (2013) and Perugini et al.

(2015), but shows that an increase or a decrease is possible.

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Additionally, it is shown in Figure 4-11 that the mixdeness values converge after a sufficient number of regions.

Camesasca et al. (2006); Brandani et al. (2013) and Perugini et al. (2015) made no mention of the mixedness values

converging, but it is evident here in this numerical simulation. Convergence with a very high number of regions is

difficult to prove using numerical simulation though because of the high computational cost associated, which

increases exponentially as the number of regions increases. But convergence is expected to continue because at a

very high number of regions, the denominator of equations (4-10) and (4-22) dominates and any change in the

probability of a particle being in a particular region (the numerator) when the number of regions is increased is very

small compared to the change in the denominator.

It is, however, possible to prove theoretically that the equation must converge to a finite value. When the original

and modified mixedness, equations (4-10) and (4-22), respectively, are evaluated as n, the number of regions,

approaches infinity, the summations in the equations turn into integrals. It can be assumed that the probability mass

function may or may not become a probability density function, but no matter the function, the integral must be

finite as well. This work is explained in more detail in Appendix A. This means that with sufficient division of the

region, the mixedness equation will converge to a finite value. It should be noted however, that due to numerical

limitations, this would not be observed if the number of particle paths do not increase proportionally with the

number of regions.

Summarily, this section showed that when applying the local and global mixedness equations shown in this paper,

the mixedness values may increase or decrease when the number of regions is changed and will also converge to a

finite value when a sufficient number of regions are used, but finding this specific number of regions requires

significant computational cost.

In the next section of the methodology, the UV reactors simulated using CFD that were used to demonstrate the

original and modified mixedness values are described, along with the associated CFD, flow and disinfection models.

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4.3.3.4 Geometry

In this paper, two single-lamp UV reactors were used to demonstrate the original and modified mixedness. One

reactor has a single UV lamp oriented perpendicular to the flow (this geometry was shown previously in Figure 4-1)

and the other has a single UV lamp oriented parallel to the flow (this geometry is shown in Figure 4-12). Both

reactors were modelled with sufficient entrance and exit lengths to ensure developed flow.

Figure 4-12 - Geometry of the Parallel UV reactor

Other lamp and reactor characteristics for these two reactors are shown in Table 4-1.

The simulations performed here were also done at 50% and 90% UVT and there was no significant difference in the

outcomes so are not shown.

Table 4-1 - Lamp and flow conditions for both UV reactors

Operating Parameters Flow rate 0.00397 m3/s (Re = 20000) Quartz sleeve radius 0.0250 m Lamp length 0.1 m Transmittance, water 70% Transmittance, quartz 96% Effective lamp power 300 W/m Inactivation rate constant, k, for MS2

0.14 cm2/mJ (US EPA, 2006)

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4.3.3.5 Flow model

To limit computational time, a steady-state Reynolds-Averaged Navier-Stokes (RANS) approach coupled with the

realizable k-ε turbulence model was used within FLUENT v. 14.5 (Ansys Inc., 2013), a CFD software, to resolve the

flow fields. The k-ε model solves two equations for the turbulent kinetic energy, k, and the rate of dissipation, ε, in

addition to the RANS equations; these equations are available in Wilcox (1985). Further information on the

implementation of this model is available in the ANSYS FLUENT manual (Ansys Inc., 2012). The inlet velocity

was only in the direction of flow and was uniform across the inlet surface. On all solid boundaries, including the

lamp, no-slip conditions were imposed.

4.3.3.6 Disinfection model

The Lagrangian framework, often called the particle tracking method, was used here because it is required to

implement the entropy-based mixing numbers. FLUENT (Ansys Inc., 2013) was used to implement particle tracking

in the UV reactor. The Discrete Random Walk model in Fluent was utilized in order to introduce small-scale

turbulence. It was assumed particles were massless and smaller than the Kolmogorov scales so that they moved with

the flow, while not disturbing it.

The dose of each particle was calculated by integrating the UV irradiance received over the path through the reactor.

The UV irradiance was calculated using the radial model (Liu et al., 2004). The dose delivered in the reactor is

quantified with the reduction equivalent dose (RED), derived from Crick-Watson like first-order inactivation

kinetics (Wright and Lawryshyn, 2000). The RED is a measure of inactivation that is a far more useful than the

mean of the dose as a descriptor of the dose distribution and is calculated by:

𝑅𝐸𝐷 = −1𝑘^ln(

𝑒Dbcd�eP.fg𝑁"

) (4-24)

where kµ = inactivation rate constant in units of [cm2/mJ], Dr is the UV dose of each particle in units of [mJ/cm2],

and NT is the total number of particles injected. A new parameter, the local RED, REDij, is introduced here in order

to measure the reactor performance in the specific regions by which the cross-sections are divided. It is simply a

calculation of the RED from the particles that enter on the entrance cross-section region, i, and leave through the exit

cross-section region, j and is given by:

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𝑅𝐸𝐷5l = −1𝑘^ln(

𝑒Dbcd�e=F.fg𝑁5l

) (4-25).

As with any CFD simulation, the solved flow fields must be independent of the mesh and, for a UV reactor, the

number of particles injected in the reactor. For mesh independence, three meshes of progressive fineness were used

for each lamp configuration and the mesh was chosen to ensure the RED changed less than 1% from a finer mesh.

To test particle independence, using the mesh that provided mesh independence, an increasing number of particles

were injected to again ensure the RED did not change more than 1%. At least 5000 particles were needed to obtain

a converged RED.

Additionally, as noted before, the mixedness values must be independent of the number of regions and the number

of particles injected. Due to computational cost, a mixedness value independent of the number of regions was not

found, but in the results, the mixedness values for cross-sections with various numbers of regions are provided. A

convergence test was performed, however, to determine a mixedness that was independent of the number of

particles injected. At least 45000 particles were required to ensure a converged mixedness value (i.e. one that

changes less than 1% when more particles are injected). In this paper, however, over two million particles were

injected in both reactors in order to ensure a converged local RED in each of the regions. When 400 regions were

used, at least 5000 particles were necessary for a converged RED in each region so two million particles were

injected overall. In this case, the RED calculation was the limiting factor that justified using more particles than was

required for the mixedness calculations.

4.3.4 Results

In this section, the performance of the two UV reactors is discussed, including the dose distribution and RED; then,

the original and modified, local and global mixedness of the reactors are shown. The correlations between local

reactor performance and the original and modified local mixedness values are also presented. Following that, the

results of a simulation where particles in a UV reactor experience perfect mixing according to the original

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mixedness equation and then the modified mixedness equation are presented. These perfect mixing scenarios are

examined for differences in reactor performance and compared to the actual CFD simulation results.

4.3.4.1 Reactor performance

The velocity magnitudes around the lamp region for the cross-flow and parallel reactors are shown in Figure 4-13.

Figure 4-13 - Velocity contour of cross-flow (top) and parallel (bottom) UV reactors in the exact centre of the reactors (in x and y position) and in the area around the lamp. The units are in m/s.

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The dose distributions and REDs calculated using the MS2 pathogen for both the cross-flow and parallel UV

reactors are shown in Figure 4-14.

Figure 4-14 - Dose distribution and RED of the cross-flow and parallel UV reactors

The cross-flow reactor achieved an RED (MS2) approximately 20% higher than the parallel reactor. This is because

the parallel reactor was not designed to optimize disinfection – the rectangular channel is more suited to a cross-flow

reactor, as there are fewer areas of low-UV intensity than the parallel reactor. In the cross-flow reactor, there is a

higher proportion of higher-dose particles and more particles in the parallel reactor that pass the lamp without

receiving a significant dose – this is seen in the dose distributions for each reactor. Even though the parallel UV

reactor is not optimal, it was chosen here solely to compare the mixedness of two different lamp orientations.

4.3.4.2 Global mixing

The original and modified global mixedness was calculated using an entrance cross-section 20 cm upstream of the

centre of the lamp and an exit cross-section at different positions downstream of the entrance cross-section to find

how the mixedness changed in the reactor zone. Only those exit cross-sections in the reactor zone that were identical

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to the entrance cross-section were chosen for the calculation (i.e. exit cross-sections that intersected the lamp were

not used because they were not identical to the entrance cross-sections). Also, the number of regions into which the

entrance and exit cross-sections were divided was varied.

Figure 4-15 shows four graphs: the top left graph shows the original mixedness for the cross-flow reactor using an

exit cross-section of the reactor zone that is at a position indicated by the x-axis; the bottom left figure shows the

original mixedness for the parallel reactor using an exit cross-section of the reactor zone that is at a position

indicated by the x-axis; the top and bottom right figures show the modified mixedness for each reactor, respectively,

with the same x-axis nomenclature used.

Figure 4-15 – The original and modified mixedness for the cross-flow (top two squares) and the parallel (bottom two squares), respectively, at various points throughout the reactor zone.

For each reactor, as the number of regions is increased (thus reducing the scale of mixing), both the original and

modified global mixedness increased. This is contrary to the results from other authors, but the Monte Carlo

simulation previously presented in this paper shows that this increase in mixedness is plausible. The increase in the

mixedness is potentially due to a collection of particles that are initially (near the entrance cross-section) clustered

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together in a region, leading to a low mixedness value, and then spread throughout the cross-section, leading to a

higher mixedness value than at the entrance cross-section. It is also possible due to the higher probability that a

given particle will leave its initial region because there are more regions into which a particle can travel.

Conversely, when there are fewer regions, that given particle is more likely to stay in that particular region which

means the mixedness could be lower.

For both reactors, the modified mixedness is lower than the original mixedness. This difference occurs because, for

these simple reactors, with close to laminar flow at points, a significant proportion of the particles do not change

regions between the entrance and exit cross-section (where i = j) or travel to a region on the exit cross-section that is

not that far from the initial region on the entrance cross-section. In the modified mixedness equation, these particles

are given a lower weighting, so this reduces the modified mixedness. Thus, for these simple geometries, the

modified mixedness should generally be lower than the original mixedness.

The parallel UV reactor also has a lower original and modified mixedness value than the cross-flow reactor. This

lower value occurs because of the difference in the length of the recirculation region caused by the orientation of the

lamp. In cross-flow UV reactors, there is a longer vortex street, which contributes to greater and more chaotic

particle movement downstream of the lamp; this is seen in Figure 4-13 comparing the velocity contours downstream

of the lamp in both reactors. This greater randomness in the particle trajectories contributes to a higher mixedness

value even for the modified mixedness equation. This effect is observed more clearly when examining the local

mixedness, discussed next.

4.3.4.3 Local mixing

While global mixedness is useful to examine the overall mixing of a reactor to compare reactors with the same

geometry, it is also valuable to examine the differences in mixing in each region of the reactor; this is done using

local mixedness values. As well, by examining the local mixedness we can determine why the parallel UV reactor

has a lower mixedness.

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Figure 4-16 shows four panels for the cross-flow UV reactor: the local original mixedness values are shown on the

left and the local modified mixedness values are shown on the right, all with a constant scale from 0 to 1 (0 being no

mixing and 1 being perfect mixing). The top two panels show the local mixedness values for the cross-flow reactor

using 16 regions and the bottom panels show the local mixedness values using 400 regions. The local mixedness

values were calculated using an entrance cross-section 20 cm upstream of the centre of UV lamp and an exit cross-

section 20 cm downstream of the centre of the lamp.

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Figure 4-16 – Local original (left) and modified (right) mixedness values for cross-flow UV reactor using 16 regions (top) and 400 regions (bottom).

For the cross-flow reactor, a different scale of mixing (i.e. different number of regions) produced a different local

mixedness profile. When 16 regions were used, the spatial differences are rather proportional (i.e. an increase in one

region is seen in both regions). But with 400 regions, the regional disparity in both mixedness metrics is very clear.

This difference in the mixedness profiles is partly due to the scale: as noted before, when the mixedness is calculated

with fewer regions, it is lower because particles then tend to stay in their own region. Therefore, some mixing (i.e.

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particle movement) is not captured when only larger scales are considered. As an aside, it is important to note that

the highest mixedness regions may appear to be slightly off-centre, even though the UV lamp is directly in the

centre of the reactor. This appearance is because there is an even number of regions in the vertical direction, so

there will not be a “centre” region that has the highest mixedness: there will always be a different number of regions

on either side of the cross-section. This applies for all cross-sections where an even number of regions is used.

As well, as is shown in Figure 4-16, particles that begin in the centre of the entrance cross-section have higher

mixing than those that enter closer to the walls; this happens for two reasons: 1) particles that start near the centre of

the reactor are subjected to higher velocities and thus higher turbulence which pushes particles further from their

initial position and 2) in the case of this particular reactor, particles that start in the centre are more likely to be

impacted by the presence of the lamp in the middle which pushes them further from their initial injection region.

Figure 4-17 shows four panels for the parallel UV reactor: the local original mixedness values are shown on the left

and the local modified mixedness values are shown on the right, all with a constant scale. The top two panels show

the local mixedness values for the reactor using 16 regions and the bottom panels show the local mixedness values

using 400 regions. The local mixedness values were calculated from the same positions as the cross-flow reactor.

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Figure 4-17 - Local original (left) and modified (right) mixedness values for parallel UV reactor using 16 regions (top) and 400 regions (bottom).

For the parallel UV reactor, the spatial differences in local mixedness that occurred in the cross-flow reactor were

seen again. The scale of mixing drastically impacts the original and modified mixedness values, with finer scales of

mixing leading to greater mixedness. Furthermore, more mixing is experienced by particles that begin in the centre

of the entrance cross-section for the same reasons as above. As well, by examining the local mixedness of the cross-

flow and parallel reactors, it is clear why the cross-flow reactor has higher global mixedness: there is a wider area

impacted by the presence of the cross-flow lamp and this leads to a higher global mixedness for that orientation.

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The local mixedness, whether original or modified, is a useful tool to examine where mixing is occurring within the

reactor. Regions of higher or lower mixing can be examined on different scales first in order to see where

opportunities to improve performance in reactors may exist or where areas of lower mixing are located. Whereas

previously, the residence time distribution (RTD) would have been used to show that some abnormal mixing is

occurring, the mixedness values using Shannon entropy can be used to find exactly where this abnormal mixing is

occurring.

4.3.4.4 Mixedness and reactor performance

One of the objectives of this paper is to relate mixing and reactor performance. Although the mixedness changes

with the number of regions by which the cross-section is divided, it is possible to show a range of the correlation

between mixing and reactor performance, here represented by the local RED, introduced previously. This local

RED is the RED of the particles that are injected from a particular region on the entrance cross-section. Figures 4-

18 and 4-19 show the local RED (MS2) normalized by the maximum RED in the cross-section divided into 400

regions, for both the cross-flow and parallel reactors, respectively, with the entrance and exit cross-section 20 cm

upstream and downstream of the centre of the UV lamp.

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Figure 4-18 - Local RED of the cross-flow reactor with 400 regions; the RED is normalized by the maximum RED in the 400 regions.

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Figure 4-19 - Local RED of the parallel reactor with 400 regions; the RED is normalized by the maximum RED in the 400 regions.

For the cross-flow reactor, it is clear that particles that started in the centre of the entrance cross-section and

interacted closely with the lamp received the highest RED. The same is seen for the parallel reactor, but there is a

much smaller area that is affected by the lamp, hence there are more regions with a lower RED and this produces a

lower RED overall. By comparing Figures 4-16 and 4-18 for the cross-flow reactor and then Figures 4-17 and 4-19

for the parallel reactor, the positive correlation between regions of higher mixing and higher RED is evident.

To further illustrate the correlation between the local mixedness and the local RED, the two parameters were plotted

against each other; Figures 4-20 and 4-21 show the original and modified local mixedness for each region for both

the cross-flow and parallel reactors, respectively. The R2 for each linear correlation is shown for 400 regions and

using MS2 to calculate the RED; the analysis was performed using a number of different regions and a number of

different organisms (to derive the kµ value in equation (4-25)), but the R2 values did not significantly change so they

were not shown.

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Figure 4-20 - Plot of local mixedness versus RED for the cross-flow reactor using the original and modified mixedness equation.

Figure 4-21 - Plot of local mixedness versus RED for the parallel reactor using the original and modified mixedness equation.

The cross-flow reactor had a lower R2 than the parallel reactor, but both reactors showed strong positive correlation

between the mixedness metric and the RED. The modified mixedness also had a slightly higher correlation than the

original mixedness with RED for both reactors. It is clear that there is a fairly strong positive correlation between

mixedness and RED: indeed, more radial mixing does lead to higher reactor performance. As well, it is clear that

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the modified mixedness, developed here to better suit the needs of a UV reactor, showed slightly better correlation

to RED than the original mixedness. However, this alone should not be used as a validation of the modified

mixedness.

4.3.4.5 Perfectly mixed reactor

To further demonstrate the validity of the modified mixedness metric, two groups of particle tracks in a UV reactor

were simulated without CFD so that the overall mixedness would be close to one, that is, the reactors achieved

perfect mixing for either the original and modified mixedness. It is important to differentiate these perfect mixing

cases from an idealized system which has a uniform dose distribution where the position of a particle is random and

thus “everywhere” in the reactor at once; the cases shown below produce mixedness values of close to one for real

reactor conditions, indicating practical perfect mixing for those metrics. The process to simulate these particle tracks

without CFD is described below.

The reactor simulated used the geometry shown in Figure 4-1, with a UV lamp placed perpendicular to flow and the

same dimensions and flow conditions as the reactor discussed previously. 50000 particle tracks were generated in

each simulation. Each particle track was randomly assigned a starting position at the entrance cross-section and then

assigned a position at the exit cross-section that generated a perfect mixing condition for either the original or

modified mixedness metric. For the original mixedness, the proportion of particles in each region on the exit cross-

section matched the left side of Figure 8 and the position of each particle in its specific region was randomized; for

the modified mixedness, the proportion of particles on the exit cross-section matched the right side of Figure 4-8 and

the position of each particle in its specific region was randomized. To “fill-in” the positions of the particles between

the entrance and exit cross-section, each particle was forced to travel a straight line; a particle position was

determined every 1 cm in the direction of flow in the reactor zone and at that position, a velocity magnitude that was

taken from the same position from the CFD simulation done in this paper was assigned to determine the time the

particle travelled in that 1 cm. If a particle hit the lamp, it was forced to move around the lamp, the distance and

direction of which was randomized.

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The simulated particle tracks were run 50 times each and with 25 and 400 regions to show the difference when

different scales of mixing are used. The dose distribution for each iteration and the average of all the REDs of each

dose distribution (using MS2) for each perfect mixing case using 25 and 400 regions are shown in Figure 4-22.

Figure 4-22 – Dose distributions and RED (MS2) of the simulated cross-flow particle tracks using 25 (top) and 400 (bottom) regions.

Using 25 regions, the average of the REDs for the perfect mixing case using the modified mixedness was only

slightly higher than the average of the REDs for the perfect mixing case using the original mixedness; both were

almost double the average of the REDs of the actual cross-flow UV reactor, shown in Figure 4-14. The difference

between the average of the REDs increases when the number of regions increases, however. With 400 regions, there

is an approximate 50% increase in the average of the RED for the modified mixedness perfect mixing case, while

the average of the REDs for the original mixedness perfect mixing case does not change significantly. The mean of

each dose distributions was almost identical when using 25 and 400 regions for both mixedness equations indicating

the same amount of overall energy reached the particles for all scenarios.

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The RED for the original mixedness perfect mixing case does not change with the number of regions because the

positions of the particles on the exit cross-section are random: if the number of regions changes, the particle

positions are still completely random and an equal number of particles still travel to each region on the exit cross-

section. However, because of the weighting system in the modified mixedness equation, the positions of the particles

on the exit cross-section are randomized only within the region to which they are required to travel to achieve

perfect mixing. Thus, with a smaller number of regions, the RED for the modified mixedness perfect mixing case

should be close to the RED for the original mixedness perfect mixing case. As the number of regions increases, the

randomization of the particle position is confined to a much smaller region and the RED increases because there is

more direct control over how the particles pass the lamp.

Even though the particle tracks used in the above example do not necessarily reflect actual fluid dynamics in the UV

reactor, the exercise shows that it is possible to better tune the particle positions to achieve a higher disinfection.

This modified mixedness could be used as a design aid or troubleshooting tool for UV reactors. However, what this

exercise does not show is that there may be extra cost to achieving a UV reactor where the particles travel in the way

prescribed by the perfect mixing case of the modified mixedness. That cost-benefit ratio must be examined in the

real world.

4.3.5 Conclusions

The objective of this paper was to first quantify mixing in a UV reactor using Shannon entropy and then determine

the relationship between the mixing and reactor performance. Previously, mixing in UV reactors was described only

qualitatively, so there was a need to define a relevant metric that could be applied and to determine the relationship

between those two necessary factors.

In this paper, the original Shannon entropy was modified to give greater weight to particles that travelled further

from their original position in the reactor. The motivation behind changing the weighting system in the original

Shannon entropy was that it was postulated that when particles travel farther from their original position and across

the system, they would be more closely exposed to the UV lamp and it would result in a better reactor performance.

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It was shown numerically using a Monte Carlo simulation that the mixedness value could increase or decrease when

the number of regions increases; this is a new contribution as previous authors saw only a decrease in the mixedness

value as the number of regions increased. Furthermore, it was shown both numerically and theoretically that the

mixedness does converge when the number of regions is sufficiently high; finding this value for practical systems

may be computationally expensive, but it is possible.

Two simple single-lamp UV reactors were used to demonstrate the original and modified mixedness metrics under

practical conditions. The modified mixedness was lower than the original mixedness for the cross-flow and parallel

UV reactors shown here because geometrically, there was nothing to encourage movement towards the particular

location which had better weighting for the modified mixedness; this may change with more complicated reactor

orientations.

The reactor performance, indicated by the local RED, was calculated and compared to local mixedness; a very high

correlation was seen between the mixedness and RED for both the modified and original mixedness equations. For

both reactors, there was little difference in the R2 values between the local RED and the local modified or original

mixedness, but the R2 was slightly higher for the modified mixedness.

The modified mixedness equation produced a high correlation between reactor performance and mixedness: thus, it

is possible to use the mixedness equation as a proxy for reactor performance of UV reactors. If, for example,

comparing two different lamp configurations for a UV reactor geometry, a mixedness calculation could be used as a

preliminary check on performance difference instead of a dose calculation. The mixedness calculation is far less

computationally expensive, more than an order of magnitude difference, than a dose calculation which now does not

need to be calculated to compare the performance of the two reactors because of the correlations found in this paper;

this could be especially useful for transient simulations and especially large eddy simulations (LES), where dose

calculations may be expensive due to particle tracks with more time steps.

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However, not only is the modified mixedness equation useful for UV reactor performance determinations, the

framework for the modified weighting system can be used to gauge the extent to which any Lagrangian simulation

gets to some sort of “perfect mixing” state. That perfect mixing state can be changed dependent on the needs of the

modeler and the system can be evaluated for progression towards that particular state.

It was shown that a simulated UV reactor under a perfect mixing state described by the weighting system used in the

modified mixedness equation produced a higher disinfection than one using the equal weighting of the original

mixedness equation. This implies there may be a benefit to changing the definition of “perfect mixing” to suit the

needs of a UV reactor, but practical tests are required to determine the costs.

The modified mixedness equation developed here is a powerful tool in UV reactor simulations and can be further

modified for use in general flow systems. Using this equation, a quantifiable, strong and positive correlation

between mixing and UV reactor performance was found; this correlation means that the mixedness can be used as a

proxy for reactor performance and reduce the computational expense required to compare reactors.

4.4 Appendix A

It is possible to show that the modified mixedness equation developed in this paper converges after sufficient n. To

begin, it is prudent to show that the original mixedness equation will also converge to a value after sufficient n. The

original mixedness equation is as follows:

𝑀5 = 𝐻5

𝐻5,7��=− 𝑝5l𝑙𝑜𝑔�𝑝5lu

lfg

𝑙𝑜𝑔�𝑛 (4-26)

It was shown in the paper that the equation is bounded by 0 and 1, for the zero mixing and perfect mixing cases,

respectively.

To show that the original mixedness value converges, it is necessary to understand the behaviour of the equation at

large values of n. This can be done by taking the above equation as n approaches infinity, thereby changing the

discrete summation to an integral over the area of the entrance cross-section, A, as shown below:

109

𝐻5 =− 𝑝 𝑥, 𝑦 𝑙𝑜𝑔�𝑝� 𝑥, 𝑦 𝑑𝑥𝑑𝑦

𝑙𝑜𝑔�𝑛 (4-27)

The probability density function, given by p(x,y), is no longer discrete, but now a continuous function based on the

position on the area of the entrance cross-section. Since we know that the bounds of the integral are defined by the

area of the cross-section, and that the probability density function is a continuous function, this means that the

integral, and thus the original mixedness must converge to a finite value as n approaches infinity.

The same analysis can be applied to the modified mixedness. The modified mixedness equation is as follows:

𝑀5,7895:5;9 = 𝐻5,7895:5;9

𝐻5,7895:5;9,7��= − 𝑃5l𝑙𝑜𝑔�

𝑃5l𝑊5l

∗ulfg

𝑙𝑜𝑔�𝑊" (4-28)

Again, it was shown in the paper that this is equation is bounded by 0 and 1.

It is possible to determine if the modified mixedness equation converges at a value by examining it as n approaches

infinity. This turns the discrete summation into an integral, as shown below:

𝑀5,7895:5;9 = − 𝑝(𝑥, 𝑦)𝑙𝑜𝑔�

𝑝(𝑥, 𝑦)𝑊(𝑥, 𝑦) 𝑑𝑥𝑑𝑦�

𝑙𝑜𝑔�𝑊" (4-29)

Just as with the original mixedness, the probability is now a continuous function over the area of the entrance cross-

section. Similarly, the weighting function, W(x,y) is now also continuous over the entrance cross-section. For the

same reasons as above, with all continuous functions inside the integral, this means that the integral must converge

to a definite value at very large n.

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4.4.1 Additional Weighting Systems

Further investigation showed that the correlation between the mixedness and RED could potentially be increased if

the weighting system associated with the modified mixedness better reflected movement toward the UV lamp, not

just movement far from the original position, as is the case with the modified mixedness presented in Powell &

Lawryshyn (2017). This section is structured as follows: first, the new weighting system equation, logic and

justification are introduced; then, the new mixedness values are calculated using the same cross-flow and parallel

UV reactors as in Powell & Lawryshyn (2017) and are compared to the original and modified mixedness values in

that paper; following that, a new UV reactor, with an off-centre UV lamp, is analysed to determine the correlation of

mixedness to RED when the lamp orientation is atypical.

The mixedness equation presented in Powell & Lawryshyn (2017) is shown below for a UV reactor divided with an

entry and exit cross-section, upstream and downstream of the lamp, respectively:

𝑀5,7895:5;9 = D E=FG8HI

J=FK=F∗

LFMN

G8HIOP,= (4-30)

where i represents a particular region on the entry cross-section; j represents a particular region on the exit cross-

section a UV reactor; Pij is the probability a particle moved from region i to region j; and W*ij and WT,i are the

weighting factors as defined in Powell & Lawryshyn (2017). Those weighting factors are based on the distance

between the region i and the region j, giving highest weight to regions that are farthest from each other. Thus, this

metric calculates mixing as the degree to which all of the particles have moved farthest from their original regions;

as such, a perfectly mixed system, according to the above equation, has particles where all have moved far from

their original regions and none have stayed in the same region.

However, it has been noted that this equation does not take into account the position of the UV lamp. As White et al.

(1986) claim, the aim of an efficient reactor is to minimize the distance between the UV lamp and the particle, so it

can be assumed that an efficient, well-mixed reactor has particles that travel close to the UV lamp.

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Two new weighting systems are proposed here that incorporate a weighting factor based on the distance from the

region containing the UV lamp and a region on the exit cross-section. Only those weighting factors, W*ij and WT,i,

have changed.

4.4.1.1 Single-factor: distance from the lamp

The first new weighting system proposed aims to encourage particle travel within the UV reactor close to the UV

lamp. In the Powell & Lawryshyn (2017) modified mixedness, the highest weight was given to the region on the

exit cross-section that was farthest from the entry cross-section. To be analogous, the first new weighting system

should have the highest weight in the region on the exit cross-section that contains the UV lamp. To discourage

particle travel to regions far from the lamp, these regions should have a lower weight. Thus, the weighting system

for this first new mixing metric is calculated as:

𝑊5l∗ = g

95�}�u�;:.87}�; ¡G�7|.;H58u}8�.;H58u8u}�;;�5}�.8��D�;�}58u yg (4-31)

Some notes on the dynamics of this equation:

• The region on the entry cross-section is in does not matter, so calling the weighting system W*ij is not

accurate, but the nomenclature is kept for consistency;

• W*ij = 1 when the region of interest on the exit cross-section is the lamp region and also as its maximum;

• W*ij gets smaller further from the lamp region; and

• WT,i is still calculated as the sum of the individual weights on the cross-section.

However, upon further examination, this weighting system can produce negative mixedness values when particles

have a high probability of being in a particular region (i.e. when Pij is high) but are very far from the lamp (i.e. when

W*lj is low). In this case, the ratio of the probability to the weighting factor, which is inside the logarithm in equation

(4-22), is greater than one; this produces a logarithm that is positive and typically large, as shown below:

𝑙𝑜𝑔�E=F↑

O=F↓∗ ≫ 0 (4-32)

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This large and positive term dominates the summation part of the overall mixedness equation and produces a large

positive summation – and thus the mixedness is negative – a condition that is not possible. This means that this

particular weighting system cannot be used.

To rectify that problem, the lamp-based weighting system was combined with the weighting system in Powell &

Lawryshyn (2017), as discussed in the next section.

4.4.1.2 Dual-factor: distance from the lamp

The second new weighting system proposed also aims to encourage particle travel within the UV reactor close to the

UV lamp, but keeps the weighting system that was used in Powell & Lawryshyn (2017) as well. The second new

weighting system is calculated as:

𝑊5l∗ = 95�}�u�;:.87}�;;u}.¥�.8��D�;�}58u}8}�;;�5}�.8��D�;�}58u yg

95�}�u�;:.87}�; ¡G�7|.;H58u}8�.;H58u8u}�;;�5}�.8��D�;�}58u yg (4-33)

Some notes on the dynamics of this equation:

• W*ij = 1 when the region of interest on the exit cross-section is the lamp region and it is farthest from the

entrance cross-section; and

• WT,i is still calculated as the sum of the individual weights on the cross-section.

4.4.2 Results

Figure 4-23 compares the mixedness with all three weighting systems: original, modified (Powell & Lawryshyn,

2017), and modified (this chapter, unpublished) and the RED for the parallel UV reactor.

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Figure 4-23 - The Local RED and mixedness for the Parallel UV reactor using the three different mixedness equations.

It is noted that the R-squared values between the two modified mixedness values is almost identical: the mixedness

and RED are still strongly, positively correlated.

Figure 4-24 compares the mixedness with all three weighting systems: original, modified (Powell & Lawryshyn,

2017), and modified mixedness (this chapter, unpublished) and the RED for the cross-flow UV reactor.

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Figure 4-24- The local RED and mixedness for the cross-flow UV reactor using the three different mixedness equations.

For the cross-flow reactor, the R-squared values do change slightly more between the modified mixedness in Powell

& Lawryshyn (2017) and the new method developed in this paper. It is clear that the work in Powell & Lawryshyn

(2017) shows the strongest, positive correlation between the modified mixedness and the RED, whereas the

weighting system developed here in this particular paper is much lower than that produced using the original

mixedness equation. The lower R2 overall for the cross-flow reactor is due to the orientation of the lamp and

geometry of the reactor. More particles in the cross-flow reactor will pass close to the lamp (for a similar amount of

time) and thus receive a similar dose (and RED) than in the parallel reactor because the cylinder that faces the flow

in a cross-flow reactor has greater surface area than the circle in a parallel reactor. This leads to a slightly higher

overall RED in the cross-flow reactor, but more importantly leads to more variance in mixedness values, because

there is more area by which the obstruction can affect the flow. Because there is less variation in RED for cross-flow

reactors, but more variation in mixedness values, there are more vertical groupings of dots in Figure 4-24 compared

to Figure 4-23. These vertical groupings also produce less strongly, positive correlations, thus reducing the R2 value

for cross-flow UV reactors.

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4.4.3 Conclusions

The modifications made to the weighting system in this short paper is not suitable. For the parallel UV reactor, there

is not a significant difference in the R2 between mixedness and RED using either of the weighting systems for the

modified mixedness (that developed in Powell & Lawryhsyn (2017) and that developed in this paper, but for the

cross-flow reactor, the difference is much more evident. Even though each of the mixedness values compared

against the RED produce R2 values much lower for the cross-flow reactor than for the parallel, the modified

mixedness developed in Powell & Lawryshyn (2017) and the RED produce the best R2 for cross-flow reactors. For

the time being, the weighting system developed in Powell & Lawryshyn (2017) is absolutely suitable for single-

lamp parallel UV reactors, but could be improved for cross-flow UV reactors.

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4.5 References

ANSYS Fluent (2012). Ansys Fluent User Manual v. 14.5. Brandani, G.B., Schor, M., MacPhee, C.E., Grubmüller, H., Zachariae, U., Marenduzzo, D. (2013) “Quantifying disorder though conditional entropy: an application to fluid mixing”. PLOS ONE 8(6) Camesasca, M., I. Manas-Zloczower, M. Kaufman (2005) “Entropic characterization of mixing in microchannels”. Journal of Micromechanics and Microengineering 15(11), 2038-2044. Cortelyou, J. R., McWhinnie, M.A., Riddiford, M.S., Semrad, J.E. (1954), "Effects of Ultraviolet Irradiation on Large Populations of Certain Waterborne Bacteria in Motion." Applied Microbiology 2(227). Danckwerts, P.V. (1953). “Continuous flow systems; distribution of residence times.” Chem Eng Sci 2, 1–13. Gandhi, V., Roberts, P.J.W., Stoesser, T., Wright, H., Kim, J.-H. (2011) “UV reactor flow visualization and mixing quantification using three-dimensional laser-induced fluorescence.” Water Research 45, 3855-3862. Haas, C.N., and G.P. Sakellaropoulos (1979) “Rational analysis of UV disinfection reactors” Proceedings of ASCE National Conference on Environmental Engineers, ASCE, New York, 540-547 Kattan, A. & Adler, R.J. (1972) “A conceptual framework for mixing in continuous chemical reactors.” Chem Eng. Sci. 27(5), 1013-1028. Khakhar, D.V., McCarthy, J.J., Shinbrot, T. & Ottino, J.M. (1997) “Transverse flow and mixing of granular materials in a rotating cylinder.” Phys. Fluids 9, 31-43 Lawryshyn, Y. A. & Cairns, B. (2003) “UV disinfection of water: the need for UV reactor validation.” Water Science & Technology: Water Supply 3(4), 293–300 Levenspiel O (1999). Chemical Reaction Engineering, 3rd edn. John Wiley, New York. Liu, D., Ducoste, J., Jin, S., Linden, K. (2004) “Evaluation of alternative fluence rate distribution models.” J. Water Supply Res. Technol. 53(6), 319-408. Masiuk, S. & Rakoczy, R. (2006) “The entropy criterion for the homogenisation process in a multi-ribbon blender. “Chem. Eng. Process 45, 500–506 Masiuk, S., Rakoczy, R., & Kordas, M. (2008). “Entropy criterion of random states for granular material in mixing process.” Chemical Papers 62, 247–264 Mohr, W.D., R.L. Saxton, C.H. Jepson (1957) “Mixing in Laminar-Flow Systems.” Ind. Eng. Chemistry 49(11), 1855-1956. Ogawa, K. & Ito, S. (1973) “A definition of quality of mixedness.” J. Chem. Eng. Jpn. 8, 148–151 Perugini, D., De Campos, C.P., Petrelli, M., Morgavi, D., Vetere, F.P. Dingwell, D.B. (2015) “Quantifying magma mixing with Shannon entropy: application to simulations and experiments.” Lithos 236, 299-310. Powell, C. & Y. Lawryshyn (2017) A modification of the entropy-based mixing to quantify mixing in UV reactors. Journal of Environmental Engineering (in-press) Qualls, R.G., and J.D. Johnson (1985) “Modelling and efficiency of ultraviolet disinfection systems.” Wat. Res. 19, 1039-1046.

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Severin, B.F., Suidan, M. T., Engelbrecht, R. S. (1984) “Mixing Effects in UV Disinfection.” Journal (Water Pollution Control Federation) 56(7), 881-888 Shannon, C.E. (1948) “A mathematical theory of communication.” Bell System Technical Journal. 27:379-425, 623-656. Thampi, M.V. and C.A. Sorber (1987) “A method for evaluating the mixing characteristics of UV reactors with short retention times”. Water Research 21(7),765-771 U.S. EPA (1986) Design Manual: Municipal Wastewater Disinfection. EPA/625/1-86/021, Cincinnati, Ohio. White, S. C., et al. (1986) “A study of operational ultraviolet disinfection equipment at secondary treatment plants.” J. Water Pollut. Control Fed. 58, 181. Wilcox, D.C. (1998). Turbulence Modeling for CFD. Second edition. Anaheim: DCW Industries Zhao, Z.F., Mehrvar, M., Ein-Mozaffari, F. (2008) “Mixing time in an agitated multi-lamp cylindrical photoreactor using electrical resistance tomography”. Journal of Chemical Technology and Biotechnology 83:12 1676–1688. Zweitering T.H. (1959) “The degree of mixing in continuous flow systems.” Chem. Eng. Sci. 11:1.

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5 Chapter 5 – Conclusions and Future Work

5.1 Purpose

The purpose of this thesis was to show that, through a systematic review of the three models associated with the

calculation of the UV reactor performance, a reduction in the computational cost of those of executing those models

was possible.

The following novel contributions reduced the overall computational cost of the reactor performance calculation:

1. By providing a standard methodology for a transient UV reactor simulation, the computational cost was

further reduced 80% by the most typically-used execution.

2. By developing an equation to find the optimal NPS/NSS for fluence rate models, the computational cost

was reduced by over 80% from previous literature methods.

3. Finally, by showing that a new mixedness metric can be used as a proxy for RED, the computational cost of

the reactor performance calculation was reduced dramatically compared to the overall RED calculation

process.

5.2 Limitations of the research

The research in this thesis used very specific examples of UV reactors to demonstrate computational cost savings,

often the simplest versions of UV reactors, such as single-lamp parallel flow reactors, were used. This was done to

limit complexity, but choosing this simple model could potentially have an impact on results, but care was taken to

minimize this impact.

For the transient simulations, single-lamp reactors were used initially to determine and then demonstrate the

standard methodology for transient UV reactors. However, a multi-lamp reactor was also chosen to implement the

methodology and was found to be successful. However, attempts by the researches to use more complex turbulence

models, such as LES, have been quite difficult to date – not because of the methodology developed, but because of

the complexity of the models themselves. The idea of “convergence” for these models is not quite the same, as

mentioned by Georgidias et al. (2010) and more work needs to be done, including an expansion of computing power

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and memory to implement these models, even with the immense computational cost reductions found in Powell &

Lawryshyn (2016). Furthermore, with the prescribed methodology, some starting points for mesh size and time step

are provided; these are intended as rules of thumb only and have been suggested by other authors. These starting

points will not work for all UV reactor geometries or lamp orienations. Indeed, the same applies for the application

of the steady state solution as the initial condition; there are cases where this may not reduce computational cost, but

this would be for exceptional reactors only.

For fluence rate modeling, whether the reactor was single-lamp, or cross-flow or parallel-flow is irrelevant because

a) the fluence rate from a UV lamp is additive: the total fluence rate from two lamps is the sum of each of the

fluence rates; and b) the orientation of the lamp is irrelevant to fluence rate calculations, only the position of the

lamp with respect to the particle. This is mentioned in the Powell & Lawryshyn (2015) as well where more complex

reactors and turbulence rate models are employed to demonstrate the new discretization methodology.

With respect to mixing quantification, simple single-lamp UV reactors were used as well. While there was a clear

linear correlation between mixedness and RED for parallel UV reactors, the same cannot be said for cross-flow UV

reactors. Furthermore, while the correlation was high, it would never reach one because of the other factors involved

in reactor performance, such as the fluence rate – this is why RED is currently used. Regardless, using mixedness to

compare and to optimize reactors can significantly reduce computational cost. As well, Powell & Lawryshyn (2017)

did not test off-centre UV lamps – this lead to the follow-up communication testing new weighting systems and off-

centre UV lamp orientation. The mixing of the cross-flow UV reactor remains difficult to model using the methods

chose to quantify mixing.

In general, while using the above models with more complex reactors and turbulence models is most likely valuable,

the fact that such dramatic computational cost reductions were found using the simplest UV reactors proves that this

work is valuable. As the complexity increases, the computational cost increases, so even the modest reductions

shown here in this thesis could be compounded with different applications.

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5.3 Suggestions for Future Work

As such, further research is needed. First, it would be valuable to “fine-tune” the weighting system used in the

modified mixedness using experimental results, as is done in the short communication following Powell &

Lawryshyn (2017). This fine-tuning should come in the form of a further modification and refinement of the

weighting system for different lamp orientations and reactor configurations, as the correlation between mixedness

(as demonstrated in the paper) and RED is not constant for all lamp orientations. In Powell & Lawryshyn (2017), as

well as in the follow-up paper, a weighting system based on number of blocks travelled from the original position

was used. It is possible that using actual distance could produce a modified mixedness with an even stronger

correlation with reactor performance, but the computational cost is high because this is essentially increasing the

number of regions by which the cross-sections are divided infinitely – this increase may outweigh the benefits of no

longer having to calculate RED to quantify reactor performance. Indeed, as was shown in the Short Communication,

using the actual distance does not provide a better correlation between mixedness and RED.

Second, more complex reactors must be tested – these give the opportunity to further refine and test different

weighting systems, but also to demonstrate more complex cases, which as discussed in the limitations, could be

where a lot of the reduction in computational cost is found. Finally, different mixing applications need to be tested:

the previous uses of an entropy-based mixing quantification have been quite varied. There may be opportunities to

introduce a weighting system to ensure a perfectly mixed system beyond just UV reactors that provides the best

performance of that system. There are still numerous opportunities to continue this work.

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5.4 References

Georgiadis, N. J., Rizzetta, D. P., and Fureby, C. (2010). “Large-eddy simulation: Current capabilities, recommended practices, and future research.” AIAA J., 48(8), 1772–1784. Powell, C. & Y. Lawryshyn (2015) A method for determining the optimal discretization of UV lamps for emission-based fluence rate models. Water Science and Technology 71(12) 1768-1774 Powell, C. & Y. Lawryshyn (2016) Standard methodology for transient simulations of UV disinfection reactors. Journal of Environmental Engineering doi:10.1061/(ASCE)EE.1943-7870.0001153 Powell, C. & Y. Lawryshyn (2017) A modification of the entropy-based mixing to quantify mixing in UV reactors. Journal of Environmental Engineering (in-press)

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