by Patrick C. Young - University of Toronto T-Space...Patrick C. Young Master of Applied Science...

Investigating the Validity of UV Reactor Additivity

by

Patrick C. Young

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Chemical Engineering and Applied ChemistryUniversity of Toronto

© Copyright 2013 by Patrick C. Young

Abstract

Investigating the Validity of UV Reactor Additivity

Patrick C. Young

Master of Applied Science

Graduate Department of Chemical Engineering and Applied Chemistry

University of Toronto

2013

Ultraviolet (UV) light reactors or banks are often arranged in series in order to meet microbial inac-

tivation credit requirements. It has been assumed that UV doses given by each reactor in series are

mathematically additive, though work done to substantiate the hypothesis has been inconsistent. Based

on previously developed theory of reactor additivity and the reactor additivity factor (RAF ), three types

of UV reactors are modelled using computational fluid dynamics and their RAF s are computed. It is

noted that the assumption of perfect mixing may not be valid depending on the distance between reac-

tors in series. It is discussed that the original formulation of the RAF is inadequate when dealing with

wastewater. It is shown unexpectedly that even with perfect mixing performance, worse than additivity

would be achieved. A new performance factor (PF ) is introduced and the implications of this are further

discussed in the context of UV reactor validation.

iii

The author is very grateful for the generous support of Ontario Centres of Excellence, Trojan Tech-

nologies, the Centre for Management of Technology and Entrepreneurship, the University of Toronto,

and for Yuri Lawryshyn, who has mentored him for three years.

iv

Contents

1 Introduction 1

1.1 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Important Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 A CFD Analysis of Placing UV Reactors in Series 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.2 Numerical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 Correlated Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Wastewater Reactor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.3 Drinking Water Reactor System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 A More Rigorous Look at Reactor Additivity 27

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.2 Reactor Additivity Factor (RAF ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

v

3.3.3 Performance Factor (PF ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.1 Numerical Analysis of the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4.2 CFD Analysis of the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Conclusion 49

4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Appendices 51

A Reactor Additivity Factor (RAF ) Proofs 52

A.1 Proof of RAF 6= 1 for ρ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

A.2 Proof of RAF < 1 for ρ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.3 Special cases where RAF = 1 for ρ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A.3.1 Special Case: β = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.3.2 Special Case: β = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.3.3 Special Case: kf = kp = k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.4 Proof that RAF monotonically decreases as ρ increases. . . . . . . . . . . . . . . . . . . . 59

A.5 Proof that RAF tends to 1 as α tends to 0 for ρ = 0 . . . . . . . . . . . . . . . . . . . . . 60

vi

List of Tables

2.1 Trends in ρ and k on RAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Software used in workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Positively and negatively correlated reactor geometry . . . . . . . . . . . . . . . . . . . . . 14

2.4 Calculated RED and RAF values of the correlated systems . . . . . . . . . . . . . . . . . 19

2.5 Calculated RED and RAF values of the wastewater reactors . . . . . . . . . . . . . . . . 20

2.6 Calculated RED and RAF values of the drinking water reactors . . . . . . . . . . . . . . 20

3.1 Summary of theoretical results on the RAF . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Generated DRC parameters and ktest,c results . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Coliform bacteria experimental parameters for the double-exponential model, PF and ktest,c 41

vii

List of Figures

2.1 Positively correlated reactors in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Negatively correlated reactors in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Wastewater reactor cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Wastewater reactor mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Drinking water reactor cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.6 Drinking water reactor mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 Correlated reactor normalized dose histograms . . . . . . . . . . . . . . . . . . . . . . . . 18

2.8 Wastewater reactor normalized dose histograms . . . . . . . . . . . . . . . . . . . . . . . . 20

2.9 Particle tracks in the wastewater reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.10 Drinking water reactor normalized dose histograms . . . . . . . . . . . . . . . . . . . . . . 21

2.11 Particle tracks in the drinking water reactor . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1 Collimated beam dose response curves for the first-order and double-exponential models . 29

3.2 PF as a function of k/ktest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 RAF as a function of kf , kp, and ρ (units for kf and kp are in cm2/mJ). Dmin = 5

mJ/cm2, µ = 2.97 mJ/cm2, σ = 0.703 mJ/cm2 . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 RAF as a function of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 RAF as a function of β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Generated DRCs and extreme cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.7 ktest,c surfaces as a function of kf , kp, for the good reactor (Dmin = 5 mJ/cm2, µ = 1.5

mJ/cm2, σ = 0.3 mJ/cm2) and the worst case reactor (Dmin = 5 mJ/cm2, σ →∞) . . . 44

3.8 Wastewater reactor configuration with particle tracks . . . . . . . . . . . . . . . . . . . . . 45

3.9 RAF surfaces for the CFD wastewater reactors (units for kf and kp are in cm2/mJ).

Dmin = 6.21 mJ/cm2, µ = 3.00 mJ/cm2, σ = 0.534 mJ/cm2 . . . . . . . . . . . . . . . . 45

viii

Nomenclature

α Non-dimensionalizing coefficient (light model) m-1

α Reactor dose scaling parameter (dose model)

β Fraction of particle-associated microbes

ηL Lamp efficiency

µ Dose distribution parameter mJ/cm2

D1 Average dose of the first reactor mJ/cm2

D2 Average dose of the second reactor mJ/cm2

φ(z) Standard normal density function

ρ Correlation

σ Dose distribution parameter mJ/cm2

τs Lamp sleeve transmittance

AP Actual performance

D UV dose delivered mJ/cm2

D1 Dose distribution for a single reactor mJ/cm2

D2 Dose distribution for two reactors in series mJ/cm2

DH Hydraulic diameter m

Di Dose of the ith particle mJ/cm2

D1,i Ranked doses of the first reactor in series mJ/cm2

D10 Dose required for a 1-log reduction of a microbe mJ/cm2

D2,i Ranked doses of the second reactor in series mJ/cm2

Dmin Minimum dose mJ/cm2

EP Expected performance

I UV light intensity W/m2

IL Power intensity per unit length W/m

k Inactivation constant cm2/mJ

ix

kf Inactivation constant for free microbes cm2/mJ

kp Inactivation constant for particle-associated microbes cm2/mJ

klin Equivalent linear inactivation constant cm2/mJ

ktest,c Critical test inactivation constant cm2/mJ

ktest Inactivation constant for a test organism cm2/mJ

N Number of microbes

N/N0 Fraction of surviving microbes

PF Performance factor

R Radius of lamp sleeve m

r Radial distance from lamp center m

RAF Reactor additivity factor

Re Reynolds number

RED Reactor equivalent dose mJ/cm2

RED1 Reactor equivalent dose for the first reactor in series mJ/cm2

RED2 Reactor equivalent dose for the second reactor in series mJ/cm2

RED12 Reactor equivalent dose for the system mJ/cm2

REDtest Reactor equivalent dose for a validation test organism mJ/cm2

T UV transmittance

z Standard normal random variable

z1 Standard normal random variable

z2 Standard normal random variable

CFD Computational fluid dynamics

DRC Dose response curve

LPHO Low pressure high output

NWRI National Water Research Institute

UV Ultraviolet

UVDGM Ultraviolet Disinfection Guidance Manual

x

Chapter 1

Introduction

Ultraviolet (UV) light disinfection is commonly used in both wastewater and drinking water treatment

processes. UV disinfection is capable of easily inactivating many pathogenic bacteria and viruses. Often

this is accomplished by placing multiple reactors / banks in series in order to meet inactivation require-

ments. It is industry practice to calculate the overall system performance as the sum of the validated

performance contribution of each individual reactor. However, little work has been done to substantiate

the accuracy of this performance summation, or additivity, practice.

The application of UV reactors to drinking water treatment is relatively easy compared to other

waters due to its inherit clarity and lack of particulate matter. However, because drinking water must

be made fit for human consumption, the regulations governing pathogen levels are incredibly stringent.

The pathogen adenovirus is extremely resistant to UV radiation and requires several times the UV dose

that other pathogens require to achieve the same inactivation credit (Gerba et al., 2002). In order to

meet the requirement of 4-log adenovirus inactivation credit, the treated water must receive a reactor

equivalent dose (RED) of 186 mJ/cm2 (United States Environmental Protection Agency, 2006). The

National Water Research Institute (NWRI) guidelines allow for multiple validated reactors / banks to be

installed in series, and each reactor / banks delivers a portion of the required UV dose (National Water

Research Institute, 2012). If it can be shown that the reactors are hydraulically independent, then the

total dose is calculated as the sum of the doses of the individual reactors.

Despite the regulations allowing for reactors to be placed in series, there has been little scientific

literature to substantiate the hypothesis that UV doses between reactors are truly additive. To address

this, in the context of drinking water Lawryshyn and Hofmann (2013) developed new theory and intro-

duced a reactor additivity factor (RAF ) to quantify the degree of reactor additivity. In addition, they

1

2 Chapter 1. Introduction

showed that under perfect or complete mixing conditions, the RAF would equal one, meaning that the

REDs of the reactors in series are perfectly additive, as the NWRI guidelines assumed. However, they

showed that when the mixing between reactors was incomplete, the RAF would necessarily be less than

one and the RED of the system would be less than the sum of the REDs of the individual reactors.

The authors provided a way to quantify the effect of placing reactors in series by calculating the RAF

provided that certain assumptions were true. Most significantly, the authors assumed that the reactors

/ banks in series would share identical dose distributions.

The theory presented by Lawryshyn and Hofmann (2013) was specifically formulated for the discus-

sion of drinking water reactors, though placing reactors / banks in series is not exclusive to drinking

water. The assumption that made their theory drinking water specific was that they used a first-order

microbial inactivation model to develop the RAF . It is known that the dose response curve (DRC)

for target pathogens of wastewater differs from the first-order disinfection kinetics typical of drinking

water. Wastewater typically has a portion of pathogens embedded in and shielded by particulate matter

which results in a subset of pathogens that are more difficult to inactivate (Qualls et al., 1985). This

results in a DRC for wastewater that has an initial region that is relatively easy to inactivate, and a

tailing region where pathogens require a significantly higher UV dose to inactivate. Therefore, in order

for the discussion of the RAF to be accurate for wastewater, the RAF should first be extended to use

a microbial inactivation model that incorporates the tailing phenomenon.

This thesis is written as the culmination of two papers. The first paper examined the theory of

Lawryshyn and Hofmann (2013) using computational fluid dynamics (CFD) to compute the RAF of

different UV reactors. It also served to substantiate the assumption that reactors / banks in series share

identical dose distributions and acts as a companion paper for those who have difficulty following the

mathematics of the theory. The second paper extended the theory by accounting for double-exponential

microbial inactivation kinetics, typical in wastewater treatment.

1.1 Thesis Objectives

A new framework for the analysis of placing UV reactors in series had been introduced by Lawryshyn

and Hofmann (2013), but no numerical experiments had been done to examine how real-world reactors

in series performed. The objectives of this thesis are to:

1. Test through numerical experiments the interaction of UV reactors in series by utilizing computa-

tional fluid dynamics (CFD) models;

1.2. Important Recommendations 3

2. Investigate how the extension of the theory of Lawryshyn and Hofmann (2013), through the intro-

duction of double-exponential inactivation kinetics, impacts UV system performance in a wastew-

ater context.

It is thought that the assumption of reactor additivity is valid and that UV operators are not at risk

of failing to meet pathogen inactivation requirements because of this assumption. The implications of

the findings will be discussed in the context of reactor validation and several recommendations will be

made.

1.2 Important Recommendations

As a result of the work done in the thesis, a few key recommendations relevant to the industry were

identified. The recommendations are summarized below.

Drinking water related recommendations

• The theory of Lawryshyn and Hofmann (2013) holds

– The dose distributions for reactors / banks in series tend to not be significantly different

– Even in the worst case of no mixing (ρ = 1) additivity is preserved for reactors / banks in

series if the reactor is sized with an organism whose sensitivity is less than half that of the

challenge organism. Therefore, validation for MS2 is acceptable for adenovirus inactivation

credit

Wastewater related recommendations

• With the assumption of good mixing between reactors / banks, it is recommended that reactors

be sized based on bioassay validation where the validation test organism’s inactivation constant is

greater than the critical test inactivation constant.1

1.3 Thesis Outline

The remainder of the thesis is organized as follows. Chapter 2 contains the paper A CFD Analysis

of Placing UV Reactors in Series authored by Patrick Young and Yuri Lawryshyn. This article was

submitted to the Water Quality Research Journal of Canada. Presented is an analysis of the RAF

that uses CFD to experimentally determine the performance of reactors in series under different mixing

1The critical inactivation constant is defined in Chapter 3.3.3 and is the negative of the slope of the line drawn fromthe origin to the point where the DRC plotted on a semi-log scale and the target inactivation level intersect.


conditions. Three different sets of reactors are modelled. The RAF s for these sets of reactors are

computed and the results are discussed relative to what one would expect based on the theory of

Lawryshyn and Hofmann (2013). Conforming to the second research objective, the paper A More

Rigorous Look at Reactor Additivity authored by Patrick Young and Yuri Lawryshyn is presented in

Chapter 3. This paper was submitted to Water Environment Research. In this paper, the RAF is

reformulated for wastewater using a microbial inactivation model that allows for the dose response

phenomenon of tailing. The results of the reformulation were unexpected and motivated the introduction

of a new performance factor (PF ). Recommendations for validation are made, and PF s are calculated

based on the CFD wastewater reactors of Chapter 2 and dose response curves taken from literature. The

thesis concludes with Chapter 4, which presents final remarks and provides recommendations for future

work.

REFERENCES 5

References

Charles P Gerba, Dawn M Gramos, and Nena Nwachuku. Comparative Inactivation of Enteroviruses

and Adenovirus 2 by UV Light Comparative Inactivation of Enteroviruses and Adenovirus 2 by UV

Light. 68(10), 2002. doi: 10.1128/AEM.68.10.5167.

Yuri A. Lawryshyn and Ron Hofmann. A Theoretical Look at Adding UV Reactors in Series. submitted,

pages 1–32, 2013.

National Water Research Institute. Ultraviolet Disinfection Guidelines for Drinking Water and Water

Reuse. Third edit edition, 2012.

RG Qualls, SF Ossoff, and JCH Chang. Factors controlling sensitivity in ultraviolet disinfection of

secondary effluents. Water Pollution Control Federation, 57(10):1006–1011, 1985.

United States Environmental Protection Agency. Ultraviolet Disinfection Guidance Manual for the Final

Long Term 2 Enhanced Surface Water Treatment Rule. 2006.

Chapter 2

A CFD Analysis of Placing UV

Reactors in Series2

2.1 Introduction

Ultraviolet (UV) light disinfection is a proven technology for wastewater and drinking water disinfection.

In order to meet UV dosing requirements, several UV reactors, or banks, are often placed in series.

However, few studies have addressed how putting UV reactors in series impacts their validation protocols.

Health Canada and the United States Environmental Protection Agency recommend an inactivation /

removal of at least 4-log for enteric viruses, i.e. adenovirus, for groundwater and surface water sources

(United States Environmental Protection Agency, 1989; Health Canada, 2012). There is some regulatory

concern that UV disinfection alone is not enough to effectively disinfect adenovirus in drinking water in

the absence of chlorine and filtration processes.

Although adenovirus is extremely resistant to UV disinfection, it is not immune. The Ultraviolet

Disinfection Guidance Manual (UVDGM) (United States Environmental Protection Agency, 2006) spec-

ifies a reactor equivalent dose (RED) of 186 mJ/cm2 to achieve 4-log “virus” inactivation credit. Since

few validated UV disinfection reactors exist that are able to deliver such a high UV dose, the National

Water Research Institute (NWRI) guidelines allow for UV drinking water and wastewater reactors to

be installed in series and the UV dose delivered is calculated as the cumulative dose of the individual

reactors (National Water Research Institute, 2012). The reactors in series must be shown to be hydrauli-

cally independent or the reactors in series must be validated in such a way that the installed system is

2Submitted to the Water Quality Research Journal of Canada

7

8 Chapter 2. A CFD Analysis of Placing UV Reactors in Series

identical to the validated reactor. Therefore, the guidelines allow for a drinking water system to meet

the requirement of 4-log adenovirus inactivation by installing multiple UV disinfection units in series.

The UVDGM specifies that “good mixing should be confirmed” when placing UV reactors in series.

However, the underlying assumption that UV doses are additive is inconsistent in literature and has not

been thoroughly investigated.

2.2 Literature Review

To date, there exist a few papers that consider the impact of placing UV reactors in series on overall

UV system performance. Tang et al. (2006) investigated the interaction between multiple UV banks

in series for an open channel configuration using MS2 as a test organism. The delivered UV doses

from multiple reactors were shown to not be exactly additive, with the overall dose being greater than

additive. However, no explanation for the result was given. Ferran and Scheible (2007) found that for

two low pressure high output (LPHO) UV reactors in series in an open channel, the RED was twice

the RED of a single LPHO reactor. Ducoste and Alpert (2011) numerically evaluated the RED of UV

reactors in series for both open channels and closed conduits. It was shown that additivity may only be

assumed provided sufficient mixing between reactor banks. They also commented that the UV response

kinetics of the target microorganism will impact the degree of additivity. However, their results only

looked at what happens for the two cases of perfect mixing and no mixing between reactors.

Recently, Lawryshyn and Hofmann (2013) looked at UV reactor additivity from a completely theoret-

ical perspective. A reactor additivity factor (RAF ) was introduced to quantify the degree of additivity.

RAF was defined as the RED of two reactors in series divided by twice the RED of a single reactor.

Thus, a RAF of one means exact additivity, whereas a RAF greater than means better than exact

additivity and RAF less than one means less than exact additivity – i.e. a RAF greater than one implies

that the RED of the system is greater than the sum of the RED of the original reactors, and vice-versa

for a RAF less than one. For two reactors in series with perfect mixing between the reactors, it was

shown that the RAF will necessarily be one. Furthermore, it was shown that for systems with a negative

correlation among the dose paths between the two reactors, the RAF will be greater than or equal to

one, whereas for a positive correlation, the RAF will be less than or equal to one. Additionally, it

was shown that in the extreme case of perfect positive correlation, which is considered to be a worst

case scenario, if the test organism is two or more times more sensitive than the target organism, then

the RAF will necessarily be greater than one. The implication of this result is that if two identical

reactors, that each can deliver a RED of 93 mJ/cm2 with MS2, are put in series, the resulting system

2.3. Methodology 9

will necessarily achieve a RED of at least 186 mJ/cm2 based on adenovirus.

It has been well established that computational fluid dynamics (CFD) analysis coupled with fluence

rate modelling is a reliable method for evaluating UV reactor performance. Furthermore, it is infeasible

to perform experiments to gain the insights of reactor additivity, being sought as part of this work.

Therefore, in this paper, the topic of reactor additivity will be explored from a numerical and CFD

perspective. Two simple reactor configurations will be considered to investigate both positive and

negative dose correlation (and will henceforth be referred to as the “correlated systems”). Additionally,

two real world reactor configurations will be investigated to demonstrate the phenomenon of reactor

additivity in practice.

As mentioned previously, the use of CFD for evaluating UV reactor performance is an established

practice. Unluturk et al. (2004) coupled CFD velocity fields with fluence rate models to compute the

UV dose delivered to apple cider, and found reasonable agreement between simulated and experimental

values. Lawryshyn and Cairns (2003) showed that CFD UV reactor models can be carefully used in place

of the bioassay testing of prototype reactors in order to speed up development and reduce associated

prototyping costs. Sozzi and Taghipour (2006) further demonstrated that CFD flow fields were in

agreement with particle image velocimitry measurements and that the resulting microbial inactivation

was consistent with experimentally obtained biodosimetry results. Other studies also support the use of

CFD to predict reactor performance (Chiu et al., 1999; Lyn et al., 1999; Pareek et al., 2003).

2.3 Methodology

In this section, theory is introduced that will allow us to discuss the performance of UV reactors placed

in series. Additionally, the reactor models used will be introduced.

2.3.1 Theory

It has been well established that UV reactors deliver a distribution of doses to the microbes traversing

the reactor (Wright and Lawryshyn, 2000). The path that a microbe takes as it flows through the reactor

in relation to the lamps results in its obtained UV dose. Consider the two correlated systems depicted

in Figures 2.1-2.2. Suppose that there exist two identical UV reactors, R1 and R2, in series and that

the dose distribution through each is identical. A microbe that receives a certain dose from R1 will

not necessarily receive the same dose from R2 due to mixing between reactors. The amount of mixing

between R1 and R2 may be characterized by a correlation (represented mathematically by ρ) of the doses

delivered by R1 and subsequently R2 to a given microbial path. For the theoretical case of absolutely


Figure 2.1: Positively correlated reactors in series

no mixing between reactors, one would expect perfect positive correlation in dose paths between R1

and R2 - i.e. a given microbe would receive exactly the same dose from each of the two reactors. For

positively correlated reactors, a microbe that receives a high dose from R1 would be expected to receive

a high dose from R2. Similarly, particles that receive a low dose from R1 would be expected to receive

a low dose from R2. Perfect mixing between reactors implies zero correlation between dose paths. With

zero correlation, it is not possible to estimate the dose that a microbe receives from R2 given the dose

that it receives from R1. There also exists an opportunity for the dose paths through R1 and R2 to be

negatively correlated. In this case, a microbe that receives a high dose in R1 is expected to receive a low

dose in R2 and vice versa. While this may be difficult to achieve in practice, it is possible to construct

such a reactor for demonstration purposes. Figure 2.1 and Figure 2.2 depict two very simple cross flow

reactor configurations where one would expect positive and negative dose path correlations, respectively.

In the positively correlated case, the lamps in both R1 and R2 are positioned to be near the lower wall so

that particles receiving a high dose in R1 will also receive a high dose in R2. In the negatively correlated

case, the lamp in R1 is positioned near the bottom wall and the lamp in R2 is positioned near the top

wall so that particles that receive a high dose in R1 will receive a low dose in R2.

The performance of a UV reactor is characterized by a distribution of doses microbes are expected

to receive as they traverse through the reactor. Therefore, it is generally not useful to characterize a

reactor’s inactivation potential by an average dose. UV reactor performance is often characterized by

the RED or “dose equivalent”. For first order microbial inactivation kinetics, RED is given by,

RED = −1

kln

(∫ ∞0

f(D)e−kD dD

)(2.1)

2.3. Methodology 11

Figure 2.2: Negatively correlated reactors in series

where f(D) is the probability density function, i.e. the dose distribution for a given reactor, and k is the

microbe specific inactivation constant (see Wright and Lawryshyn (2000) or Lawryshyn and Hofmann

(2013) for more details).

In the negatively correlated reactor case, although the reactors are effectively reversed in their align-

ment, the water layers on each side of the lamps of R1 and R2 are the same. On an individual basis, it is

expected that the RED for R1 and R2 will be very similar. However, across the entire system, the RED

for the two reactors in series will be different. Thus, the two reactor configurations, i.e. the positively

and negatively correlated configurations, will have different additivities. Let us define a dimensionless

additivity factor (RAF ) given by,

RAF ≡ RED12

2RED1(2.2a)

RAF ≡ RED12

RED1 +RED2(2.2b)

where RED12 is the RED calculated across the entire system, RED1 is the RED of the first reactor,

and RED2 is the RED of the second reactor. The first form of the equation is an idealization that

assumes that the dose distributions for R1 and R2 are identical. In practice, R1 will influence R2 and

the dose distributions are not the same. Thus, (2.2b) should generally be used. It should be noted that

in the case of different test versus target organisms, it makes sense that the RED of the system, i.e.

RED12, be calculated based on the target organism, whereas RED1 and RED2 be based on the test

organism. As will be shown in the results, we will also present RAF calculated with RED12 based on

adenovirus and RED1 and RED2 based on MS2.

Although the proof has been omitted (see Lawryshyn and Hofmann (2013)) it is intuitive that the


negatively correlated reactor configuration will have a higher RED and superior inactivation perfor-

mance to that of the positively correlated reactor configuration. In the positively correlated reactor

configuration, there is extreme short circuiting along the top of the reactor. Microbes flowing along the

bottom of the reactor will receive high UV doses from both R1 and R2, whereas particles flowing along

the top will receive very low doses through each of the two reactors. In the negatively correlated reactor

configuration, the microbes flowing along the bottom of the reactor will first receive high doses from R1

and then low doses from R2 with the reverse happening along the bottom of the reactor. As corollary,

it is clear that the reactor additivity factor will be greater for the case of negative correlation than that

of positive correlation. In fact, for any given value of k, the reactor additivity factor is greater than one

for negative correlation, less than one for positive correlation, and equal to one for zero correlation.

The interactions with the inactivation constant are not as simple, however. In the case of a UV

resistant organism (k → 0) the reactor additivity factor will approach one. For a sensitive organism

(k → ∞) the reactor additivity factor will also approach one except for the case of ρ = −1, in which

case the reactor additivity factor will necessarily be greater than one. The effects of correlation and

the inactivation constant on the reactor additivity factor are summarized in Table 2.1. To compute

correlation between dose paths for reactors in series, Spearman’s rank correlation3 was used as the dose

distributions are non-normal.

Table 2.1: Trends in ρ and k on RAF

ρ = −1 ρ < 0 ρ = 0 ρ > 0

k → 0 RAF → 1 RAF → 1 RAF = 1 RAF → 1

k →∞ RAF > 1 RAF → 1 RAF = 1 RAF → 1

2.3.2 Numerical Model

CFD work was done using Workbench 14.5 (ANSYS, 2011). All CFD models used the velocity-inlet

boundary condition on the inlet, pressure-outlet on the outlet, and the wall boundary condition (no-

slip) was used on all wall and lamp surfaces. A pressure-based solver was used in Fluent, and turbulence

was modelled with the realizable k−ε model. Particle tracking was done by Fluent and dose calculation

code was developed and executed using MATLAB R2008a. The process is summarized in Table 2.2

below.

The radial model (Blatchley, 1997) was used for all fluence rate calculations for its simplicity and

3Spearman’s rank correlation is calculated as ρ =∑i(D1,i−D1)(D2,i−D2)√∑

i(D1,i−D1)2∑i(D2,i−D2)2

, where D1,i and D2,i are the ranked

doses and D1 and D2 are the means of the first and second reactors respectively.

2.3. Methodology 13

Table 2.2: Software used in workflow

Operation Vendor Software

Geometry ANSYS DesignModeler

Meshing ANSYS Meshing

Physics & Particle Tracking ANSYS Fluent

Fluence Rate Modeling & Dose Calculations Mathworks MATLAB

low computational cost. Despite the fact that it is not as accurate as other fluence rate models, the

objective of this study was to show the relative trends in dose, and this is easily achievable with the

radial model. It is defined by,

I(r) =τsηLILT

α(r−R)

2πr, (2.3)

where I is the UV light intensity, r is the radial distance from the centre of the lamp, R is the radius of the

lamp sleeve, τs is the lamp sleeve transmittance, ηL is the lamp efficiency, IL is the power intensity per

unit length of the lamp, T is the UV transmittance (UVT) of the water, and α is a non-dimensionalizing

coefficient dependent on the UVT unit of measurement. For a discrete dose distribution, RED is

calculated as

RED = −1

kln

(1

N

N∑i=1

e−kDi

), (2.4)

where N is the total number of particles, and Di is the dose of the ith particle (Wright and Lawryshyn,

2000). In this chapter, D10 (dose required for a 1-log reduction of microbes) will primarily be used in

place of k. D10 is defined as follows,

D10 =ln(10)

k. (2.5)

Correlated Systems

The two correlated systems were modelled as described in Table 2.3. The reactors were modelled in

three dimensions, with the lamps and lamp sleeves extending from wall to wall. A length of 10 hydraulic

diameters (DH) was given for flow to develop before the lamps, and 5 DH was given after the lamps

and before the outlet. A mesh was created of 7.0× 105 hex elements. The minimum orthogonal quality

was greater than 0.50 and grid convergence was achieved. Turbulent flow was achieved with a Reynolds

number (Re) greater than 105 based on DH . A sufficient number of massless particles were injected

uniformly across the inlet surface to achieve convergence.


Table 2.3: Positively and negatively correlated reactor geometry

FeatureCorrelation

Positive Negative

Reactor width 1.5 m 1.5 m

Lamp diameter 0.10 m 0.10 m

Inlet length 10 DH 10 DH

Inter lamp length 10 DH 10 DH

Lamp 1 top water layer 0.15 m 0.15 m

Lamp 1 bottom water layer 0.05 m 0.05 m

Lamp 2 top water layer 0.15 m 0.05 m

Lamp 2 bottom water layer 0.05 m 0.15 m

Wastewater Reactor System

The wastewater reactor was modelled to mimic a UV system used in practice. Lamps were placed in a 4

by 4 bank arrangement, and include lamp supports. The flow direction was parallel to the lamps. The

lamps were 1.5 m in length, had a diameter of 0.10 m, and formed a square grid with a spacing of 0.20 m

between lamp centres. Lamps were spaced such that there is a 0.05 m water layer between the sides, top

and bottom, as can be seen in Figure 2.3 (dashed lines represent symmetry planes). The spacing between

banks is left as a parameter between 0.5 m and 6 m, the effect of which will be discussed later. The

meshing algorithm used adaptive refinement in order to give detail to the more complicated geometry

features such as lamp supports (see Figure 2.4). Grid convergence was achieved with 3.9×106 tetra cells,

and the minimum orthogonal quality was 0.18. Symmetry planes were used to model the open channel’s

free surface and to reduce computational time by creating a virtual 4 by 4 lamp arrangement from a

2 by 4 lamp arrangement. The flow was turbulent such that the Reynolds number based on hydraulic

diameter was greater than 105. Convergence was achieved by injecting a sufficient number of massless

particles.

Drinking Water Reactor

A drinking water reactor was modelled after a generic residential system with two reactors placed in

series. Each reactor had an annular configuration, with the lamp placed in the centre and parallel to

flow. The lamp length was 0.4 m, the sleeve radius was 0.01 m and the wall radius was 0.04 m, as can

be seen in Figure 2.5. Meshing was done using adaptive refinement on proximity and curvature, and

inflation was used to give more detail near the walls and lamps; as shown in Figure 2.6. Grid convergence

was achieved with 5.3 × 105 mixed tetra and hex cells, and the minimum orthogonal quality was 0.14.

2.3. Methodology 15

Figure 2.3: Wastewater reactor cross-section


Figure 2.4: Wastewater reactor mesh

The flow was turbulent with a Reynolds number greater than 1.7× 104 based on the reactor’s diameter.

A sufficient number of particles were injected for disinfection analysis.

2.4 Results and Discussion

CFD models were created for the correlated systems, the wastewater reactor, and the drinking water

reactor. In all cases, the reactor additivity factor was calculated by computing the individual RED

for one reactor (or bank), followed by the individual RED of the subsequent reactor, and then the

RED of the total system and using equation (2.2b). Additionally, the correlation between the two

dose distributions was calculated. In this section, the results obtained are compared to the theoretical

results presented by Lawryshyn and Hofmann (2013) with the intent of verifying them through numerical

experiments.

2.4.1 Correlated Systems

As discussed, two systems were modelled with configurations such that it was expected that one model

would exhibit a positive correlation of dose paths between reactors, and the other model, negative. The

2.4. Results and Discussion 17

Figure 2.5: Drinking water reactor cross-section

Figure 2.6: Drinking water reactor mesh


Dose1gmJ/cm2)

Nor

mal

ized

1Fre

quen

cy

Positively1Correlated1Reactors

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Reactor11

Reactor12

Dose1gmJ/cm2)

Nor

mal

ized

1Fre

quen

cy

Negatively1Correlated1Reactors

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Reactor11

Reactor12

Figure 2.7: Correlated reactor normalized dose histograms

predicted trends in correlation were observed, and the resultant REDs and RAF s are summarized in

Table 4. RED1 and RED2 refer to the reactor equivalent doses of the individual reactors 1 and 2

respectively, and RED12 refers to the total RED of the system. Lamp power was adjusted in each

system such that the first reactor in series would have a RED of 20 mJ/cm2 for a D10 of 10 mJ/cm2.

It is important to note that in each case the dose histograms for the individual reactors are practically

identical and it is difficult to distinguish between the superimposed histograms, as can be seen in Figure

2.7. Thus, one would expect the numerical results to match the theory of Lawryshyn and Hofmann

(2013). With the exception of where the D10 is 1 mJ/cm2 (due to numerical instabilities), the RAF

tends towards 1 as the D10 increases, as is expected. For the positively and negatively correlated UV

systems tested with MS2 (D10 = 20 mJ/cm2), the reactor additivity factor when targeting adenovirus

(D10 = 46.5 mJ/cm2) was calculated to be 1.16 and 1.57 respectively. This result is consistent with the

theory presented by Lawryshyn and Hofmann (2013).

2.4.2 Wastewater Reactor System

An open channel reactor was modelled with a virtual 4 by 4 parallel flow lamp configuration. Two banks

of lamps were placed in series as was shown in Figure 2.3, and the distance between the two banks was

adjusted to be between 0.5 m and 6 m. Lamp power was adjusted such that the first bank in series

would have a RED of 20 mJ/cm2 for a D10 of 10 mJ/cm2. It was observed that the correlation between

dose paths was positive and decreased almost to zero as the inter-bank length increased. The results of

the RED and reactor additivity factor analysis are summarized in Table 2.5. The dose distributions for

reactor banks are virtually identical, as can be seen in Figure 2.8. Particle tracks coloured by velocity


Table 2.4: Calculated RED and RAF values of the correlated systems

Reactor configurationD10 RED1 RED2 RED12 RAF

ρ(mJ/cm2) (mJ/cm2) (mJ/cm2) (mJ/cm2) (mJ/cm2)

Positively correlated

1 12.15 11.85 22.87 0.94

0.54

10 20.00 19.31 34.22 0.86

20 25.45 24.51 42.50 0.84

30 29.63 28.52 49.11 0.84

40 33.02 31.76 54.71 0.85

Negatively correlated

1 11.94 11.54 26.57 1.13

-0.33

10 20.00 19.46 47.46 1.21

20 25.41 24.95 60.81 1.20

30 29.46 29.10 69.74 1.19

40 32.69 32.39 76.13 1.17

magnitude can be seen in Figure 2.9. The RAF tends towards 1 as the D10 increases. This convergence

is more rapid as ρ approaches 0. When the reactor is validated for 1-log MS2, the reactor additivity

factors on the 0.5 m, 3 m, and 6 m reactors are 1.08, 1.11, and 1.12 respectively, when adenovirus is

assumed to be the target.

2.4.3 Drinking Water Reactor System

Two single-lamp drinking water reactors were placed in series, and the effect was analysed. Lamp power

was adjusted such that the first reactor in series would have an RED of 20 mJ/cm2 for a D10 of 10

mJ/cm2. The results of the RED and RAF computation can be seen below in Table 2.6. The resultant

dose distributions for the two reactors are quite similar and can be seen in Figure 2.10. Particle tracks

of the microbes traversing the reactor are shown in Figure 2.11. As would be expected for a near zero

correlation, the RAF very rapidly approaches 1 as the D10 increases. For the drinking water UV system

tested with MS2, the reactor additivity factor, when targeting adenovirus was calculated to be 1.07.


Dose=3mJ/cm27

Nor

mal

ized

=Fre

quen

cy

Inter-lamp=Length===0.5m

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Reactor=1

Reactor=2

Dose=3mJ/cm27

Nor

mal

ized

=Fre

quen

cy

Inter-lamp=Length===6m

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Reactor=1

Reactor=2

Figure 2.8: Wastewater reactor normalized dose histograms

Table 2.5: Calculated RED and RAF values of the wastewater reactors

Inter-lamp length D10 RED1 RED2 RED12 RAFρ

(m) (mJ/cm2) (mJ/cm2) (mJ/cm2) (mJ/cm2) (mJ/cm2)

0.5

1 9.54 9.19 17.58 0.92

0.39

10 20.00 19.69 34.79 0.87

20 23.09 22.83 42.25 0.91

30 24.60 24.38 46.11 0.94

40 25.51 25.32 48.50 0.95

3

1 9.77 9.52 17.59 0.91

0.17

10 20.00 19.56 35.94 0.91

20 23.48 22.99 43.87 0.94

30 25.20 24.69 48.03 0.96

40 26.26 25.72 50.57 0.97

6

1 9.80 9.64 18.11 0.92

0.07

10 20.00 19.84 37.75 0.94

20 23.11 22.88 44.80 0.97

30 24.62 24.38 48.26 0.98

40 25.54 25.29 50.33 0.99

Table 2.6: Calculated RED and RAF values of the drinking water reactors

D10 RED1 RED2 RED12 RAFρ

(mJ/cm2) (mJ/cm2) (mJ/cm2) (mJ/cm2) (mJ/cm2)

1 14.57 10.86 25.53 1.00

0.04

10 20.00 17.71 37.28 0.99

20 21.34 19.98 41.07 0.99

30 21.95 21.00 42.78 1.00

40 22.29 21.57 43.75 1.00


Figure 2.9: Particle tracks in the wastewater reactor

Dose6(mJ/cm2)

Nor

mal

ized

6Fre

quen

cy

Drinking6Water6Reactors

0 20 40 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Reactor61

Reactor62

Figure 2.10: Drinking water reactor normalized dose histograms


Figure 2.11: Particle tracks in the drinking water reactor

2.5. Conclusions 23

2.5 Conclusions

Three fundamentally different UV reactor configurations were analysed: the correlated systems, a

wastewater reactor, and a drinking water reactor. For the correlated systems, it was shown through

the numerical experiments that a strong positive correlation leads to a RAF less than one and that

a strong negative correlation leads to a reactor greater than one. For both positively and negatively

correlated systems, the reactor additivity factor was shown to converge to unity as the D10 increases.

With the open-channel wastewater reactor it was shown that the correlation between dose paths is posi-

tive, and significant with respect to the RAF . The correlation decreased as the second bank was placed

further downstream. The implication of this is that the proximity between banks should be considered

when placing reactors in series and assuming additivity. Reactor banks should be placed as far apart

as reasonably possible. A residential scale drinking water reactor was modelled, and the correlation be-

tween dose paths was found to be near zero. The very low correlation justifies the additivity assumption

for this reactor geometry. Finally, it was shown for all systems that additivity could be achieved when

sizing was done based on MS2 validation and targeting adenovirus. More generally, this result holds

true when the reactor is sized with an organism whose sensitivity is less than half that of the challenge

organism.


References

ANSYS. FLUENT Theory Guide, volume 14. 14 edition, 2011.

E.R. Blatchley. Numerical modelling of UV intensity: application to collimated-beam reactors and

continuous-flow systems. Water Research, 31(9):2205–2218, 1997.

K Chiu, DA Lyn, P Savoye, and E.R. Blatchley. Integrated UV disinfection model based on particle

tracking. Journal of Environmental Engineeringl, (January):7–16, 1999.

J Ducoste and S Alpert. Assessing the UV Dose Delivered from Two UV Reactors in Series: Can You

Always Assume Doubling the UV Dose from Individual Reactor Validations? In Proceedings of the

2nd North American Conference on Ozone, Ultraviolet & Advanced Oxidation Technologies, 2011.

B Ferran and OK Scheible. Biodosimetry of a Full-Scale UV Disinfection System to Achieve Regulatory

Approval for Wastewater Reuse. In Proceedings of the Water Environment Federation, volume 2007,

pages 367–385, 2007.

Health Canada. Guidelines for Canadian Drinking Water Quality Summary Table Federal-Provincial-

Territorial Committee on Drinking Water of the Federal-Provincial-Territorial Committee on Health

and the Environment August 2012, 2012.

Yuri A. Lawryshyn and B Cairns. UV disinfection of water: the need for UV reactor validation. Water

Science & Technology: Water Supply, 3(4):293–300, 2003.


pages 1–32, 2013.

DA Lyn, K Chiu, and E.R. Blatchley. Numerical modeling of flow and disinfection in UV disinfection

channels. Journal of Environmental Engineering, (January):17–26, 1999.



V.K. Pareek, S.J. Cox, M.P. Brungs, B. Young, and A.a. Adesina. Computational fluid dynamic (CFD)

simulation of a pilot-scale annular bubble column photocatalytic reactor. Chemical Engineering Sci-

ence, 58(3-6):859–865, February 2003. ISSN 00092509.

D Angelo Sozzi and Fariborz Taghipour. UV Reactor Performance Modeling by Eulerian and Lagrangian

Methods. Environmental science & technology, 40(5):1609–15, March 2006. ISSN 0013-936X.

REFERENCES 25

CC Tang, Jeff Kuo, and SJ Huitric. UV Systems for Reclaimed Water Disinfection From Equipment

Validation to Operation. In Proceedings of the Water Environment Federation, pages 2930–2943, 2006.

United States Environmental Protection Agency. National primary drinking water regulations; giardia

lamblia, viruses, and legionella, maximum contaminant levels, and turbidity and heterotrophic bacteria

(Surface water treatment rule), Final rule, 43 FR 27486, 1989.



S.Kucuk Unluturk, H. Arastoopour, and T. Koutchma. Modeling of UV dose distribution in a thin-film

UV reactor for processing of apple cider. Journal of Food Engineering, 65(1):125–136, November 2004.

ISSN 02608774. doi: 10.1016/j.jfoodeng.2004.01.005.

H.B. Wright and Yuri A. Lawryshyn. An assessment of the bioassay concept for UV reactor validation.

Proceedings of the Water Environment Federation, 2000(2):378–400, 2000.

Chapter 3

A More Rigorous Look at Reactor

Additivity4

3.1 Introduction

Ultraviolet (UV) light disinfection is a well-established technology for disinfecting wastewater and drink-

ing water. Often, several UV reactors, or banks, are placed in series to meet UV dose requirements.

However, there is little information in scientific literature that addresses how placing UV reactors in

series impacts their validation. The National Water Research Institute (NWRI) guidelines allow for UV

wastewater and drinking water reactors to be placed in series, and the dose delivered to be calculated

as the cumulative dose of the individual reactors (National Water Research Institute, 2012). Hydraulic

independence must be demonstrated between the individual reactors in series, or the reactors must be

validated in such a way that the installed system is identical to the validated system. Despite these

guidelines, it has not been thoroughly investigated that UV disinfection performance is additive between

reactors / banks, and the scientific literature on the subject is inconsistent.

3.2 Literature Review

There is some published work on placing UV reactors and/or banks in series. For example, Tang et al.

(2006) used MS2 as a test organism to investigate the interaction of multiple UV banks in series in

an open configuration. The UV dose delivered was shown to be not exactly additive, and the overall

4Submitted to Water Environment Research

27

28 Chapter 3. A More Rigorous Look at Reactor Additivity

dose was greater than simple additivity. However, the result was presented without any explanation.

Ferran and Scheible (2007) tested two low pressure high output (LPHO) UV reactors in series in an

open channel, and found that the reactor equivalent dose (RED) was double the RED of a single LPHO

reactor. For open channels and closed conduits, Ducoste and Alpert (2011) calculated the RED of UV

reactors in series. They demonstrated that only under conditions of sufficient mixing can additivity

be assumed. Additionally, it was noted that the target organism’s UV response kinetics will affect the

degree of additivity. However, their results were limited to the two cases of perfect mixing and no mixing

between reactors.

Another assumption that impacts UV reactor disinfection efficiency is microbial inactivation kinetics.

In wastewater treatment, UV disinfection performance is known to be adversely impacted by suspended

solids (flocs). The flocs present in secondary treated wastewater are mainly produced during the biolog-

ical activated sludge treatment process (Urbain et al., 1993). The flocs serve as a haven for microbes,

effectively shielding the embedded microorganisms from UV disinfection by scattering and absorbing

photons (Das, 2001). Thereby, the microorganisms present in secondary treated effluent can be divided

into two groups; those that are associated with particles, and those that are not (“free” microbes).

Figure 3.1 shows a dose response curve (DRC) for a typical secondary treated effluent (Farnood, 2005).

It is commonly observed that pathogens in wastewater are initially very sensitive to UV radiation, as

can be seen from the steep slope in the first-order inactivation region. Additional UV radiation results

in less efficient inactivation as the UV dose enters the tailing region. The amount of energy required

to achieve 1-log of inactivation is significantly more in the tailing region than in the first-order region.

Not surprisingly, the DRC of solely free microorganisms resembles a straight line on a logarithmic scale,

similar to the first-order region.

The DRCs can be represented mathematically by a variety of models. The use of a first-order model is

often favoured over other microbial inactivation models for its simplicity and its ability to fit data under

certain conditions (Hijnen and Medema, 2005). While it may be accurate for low UV doses, it cannot

account for tailing and is thus an inaccurate representation of real UV dose response behaviour when flocs

are present. Cerf (1977) suggested that the tailing phenomenon of inactivation in a UV DRC was caused

by varying levels of UV resistance amongst microbes. Building on this idea of resistance, it was proposed

that the tailing phenomenon was caused by particle associated microbes (Loge et al., 1999; Qualls et al.,

1985). An elegant way to include tailing is to use a double-exponential model that allows for two

populations of microorganisms with different UV resistances (Farnood, 2005). The double-exponential

model was used with success to analyse the disinfection of surface waters containing particulate matter

and high organic content (Caron et al., 2007; Cantwell and Hofmann, 2008). Additionally, Cantwell

3.3. Methodology 29

(2007) used double-exponential inactivation kinetics to model the inactivation of secondary treated

effluent wastewater. Azimi et al. (2012) used the double-exponential model with success to show that

floc size distributions also have an impact on microbial UV resistance.

Lawryshyn and Hofmann (2013) developed theory to analyse the effect of placing UV reactors in series

to meet disinfection requirements. Using first-order kinetics, they introduced a reactor additivity factor

(RAF ) and showed under what conditions favourable interactions between reactors can be expected.

It was demonstrated that with perfect mixing between reactors, the UV doses of reactors would be

perfectly additive. This assumption of perfect additivity is assumed by the NWRI guidelines which

allow for the UV dose delivered to be calculated as the cumulative dose of individual reactors (National

Water Research Institute, 2012). However, the theory of Lawryshyn and Hofmann (2013) does not

address what happens with double-exponential microbial inactivation kinetics, typical in wastewater

treatment. In this study, the theory of Lawryshyn and Hofmann (2013) is expanded upon to account

for tailing, using the double-exponential microbial inactivation model.

log(

N/N

0)

UV Dose

100

Double-exponential model

k

First-order region

Tailing region

First-order model

Figure 3.1: Collimated beam dose response curves for the first-order and double-exponential models

3.3 Methodology

In this section, the reactor additivity factor introduced by Lawryshyn and Hofmann (2013) will be

reevaluated, and a new formulation of the RAF that accounts for tailing will be presented. The RAF


will also be found for extreme cases. Additionally, a new dimensionless number called the performance

factor (PF ) will be introduced. The performance factor allows for a way to ensure that validated reactors

in series are properly sized when accounting for the tailing phenomenon in wastewaters.

3.3.1 Theory

Most work done to date on UV reactor performance modelling has concentrated on first-order microbial

inactivation kinetics. The first-order model is given by

N

N0= e−kD, (3.1)

where k is a microbe specific inactivation constant, D is the UV dose delivered, and N/N0 is the fraction

of surviving microbes. However, as discussed above, this model cannot account for the dose response

phenomenon of tailing. The model applied to account for DRC tailing (see Farnood (2005)) is as follows,

N

N0= (1− β)e−kfD + βe−kpD, (3.2)

where β ∈ [0, 1] is the fraction of particle-associated microbes, kf ∈ (0,∞) is the inactivation constant for

free microbes, and kp ∈ [0,∞) is the inactivation constant for particle-associated microbes. Lawryshyn

and Hofmann (2013) defined the reactor additivity factor (RAF ) as

RAF ≡ RED12

2RED1, (3.3)

where RED1 is the RED of a single reactor and RED12 is the RED of two reactors in series. When the

doses delivered by two reactors in series are perfectly additive, RAF will be equal to one. An RAF value

of less than one implies that the reactor REDs are less than additive, and an RAF greater than one

implies that the reactor REDs are greater than additive. The degree of mixing between two reactors in

series determines the correlation among the dose paths delivered by the two reactors. A high correlation

implies that the high dose paths through the first reactor are likely to be the same as the high dose

paths through the second reactor. Zero correlation implies that the path a microbe takes through the

first reactor does not impact the path through the second so that the dose received in the first reactor is

independent of the second. This scenario is what is referred to as perfect or complete mixing. Although

it is difficult to achieve in practice, there is also the possibility for negative correlation between reactors.

This implies that a high dose path through the first reactor is likely to be a low dose path through the

3.3. Methodology 31

second reactor, and vice-versa. Mathematically, correlation is represented by ρ ∈ [−1, 1]. It was shown

by Lawryshyn and Hofmann (2013) that the RAF decreases monotonically as ρ increases from -1 to 1.

With perfect mixing and ρ = 0, it was also shown that RAF = 1, and it was said that the reactors

were perfectly additive. Without any mathematical rigour, this is exactly how the NWRI Ultraviolet

Disinfection Guidelines for Drinking Water and Water Reuse treat banks / reactors in series, assuming

that there will be perfect mixing between banks.

3.3.2 Reactor Additivity Factor (RAF )

Using the double-exponential inactivation kinetics, and closely following the methodology of Lawryshyn

and Hofmann (2013), new insight can be gained about placing UV reactors in series. The dose distribu-

tion for a single reactor is assumed to be log-normal and is modeled as

D1d= α

(Dmin + eµ+σz

), (3.4)

where z is a standard normal random variable, Dmin ∈ [0,∞] is the minimum dose, and µ ∈ (−∞,∞)

and σ ∈ [0,∞) are the dose distribution parameters. Note that the symbold= is used to signify that D1

is a random variable whose distribution follows the expression on the right hand side of the equation.

The parameter α ∈ [0,∞) is the reactor dose setting, a scaling parameter. Adjusting α scales the dose

distribution an increase in α can be accomplished by increasing lamp power or decreasing the flow

rate through the reactor. Mathematically, increasing α linearly increases the minimum, the mean and

the standard deviation of the dose distribution. Following Lawryshyn and Hofmann (2013), the dose

distribution for two reactors in series can be written as,

D2d= α

(2Dmin + eµ+σz1 + e

µ+σ(ρz1+√

1−ρ2z2))

, (3.5)

where z1 and z2 are two independent standard normal random variables. To find parameters RED1 and

RED12 of equation (3.3), the following procedure was followed. For a single reactor, substituting D1

from equation (3.4) into equation (3.2) gives

N

N0

d= (1− β)e−kfD1(z) + βe−kpD1(z). (3.6)


Similarly, for two reactors in series, substituting D2 from equation (3.5) into equation (3.2) gives

N

N0

d= (1− β)e−kfD2(z1,z2) + βe−kpD2(z1,z2). (3.7)

It can be shown that the overall microbial inactivation of a UV system, i.e. a measure of its perfor-

mance, can be computed by integrating the system’s dose distribution’s probability density function

multiplied by the inactivation model, as was presented by Wright and Lawryshyn (2000). Following

their methodology for the case of a single reactor, using equation (3.6) and integrating gives

N

N0=

∞∫−∞

((1− β)e−kfD1(z) + βe−kpD1(z)

)φ(z) dz, (3.8)

where φ(z) is the standard normal density function given by

φ(z) =1√2π

e−z2

2 . (3.9)

Analogous to Wright and Lawryshyn (2000), the microbial inactivation for two reactors in series can be

calculated by integrating over the distribution of equation (3.7), such that

N

N0=

∞∫−∞

∞∫−∞

((1− β)e−kfD2(z1,z2) + βe−kpD2(z1,z2)

)φ(z1)φ(z2) dz1dz2. (3.10)

Inactivation with respect to RED is given by Wright and Lawryshyn (2000) as

N

N0= e−kRED. (3.11)

Equating (3.8) to (3.11) and solving for RED gives

RED1 = − 1

klinln

∞∫−∞


)φ(z) dz

, (3.12)

where klin ∈ (0,∞) is the equivalent linear inactivation constant. Similarly, the same can be done for

two reactors in series by equating (3.10) to (3.11) and solving for RED, resulting in

RED12 = − 1

klinln

∞∫−∞

∞∫−∞

((1− β)e−kfD2(z1,z2) + βe−kpD2(z1,z2)

)φ(z1)φ(z2) dz1dz2

. (3.13)

3.3. Methodology 33

The inactivation constant klin is found with the following procedure. First, inactivation is calculated

using the double-exponential model (3.2) with D replaced with REDk

(N

N0

)k

= (1− β)e−kfREDk + βe−kpREDk . (3.14)

REDk and (N/N0)k are used to solve for klin. Equation (3.8) is then equated to (3.14), resulting in

∞∫−∞


)φ(z) dz (3.15)

= (1− β)e−kfREDk + βe−kpREDk .

The parameter REDk in equation (3.14) can then be solved for numerically using equation (3.15). The

inactivation constant klin is then determined as follows,

klin = −ln((

NN0

)k

)REDk

. (3.16)

Graphically, this is equivalent to fitting the first-order inactivation kinetics model between the origin

and the double-exponential dose / inactivation point and computing the inactivation constant which is

equivalent to finding the slope of the first-order line on a semi-log plot (see Figure 3.1). Finally, the

RAF can be calculated using equation (3.3) by dividing RED12 by 2RED1.

Most importantly, it can be proven (see Appendix A.1 and A.2) that under perfect mixing conditions

(i.e. ρ = 0) the RAF will necessarily be less than one. The implications of this result are significant.

The result assures that the total RED of installed UV systems with reactors / banks in series is less

than what is currently assumed in industry practice. However, the UV industry currently tests with a

first-order test organism and assumes additivity in the context of double-exponential wastewaters, and

this issue will be investigated further, below. Additionally, it can be shown that when β is equal to 0

or 1, or kf = kp then the double-exponential model reduces to the first-order model and the assertion

of Lawryshyn and Hofmann (2013) that RAF = 1 when ρ = 0 holds (see Appendix A.3). Similar to

the first-order inactivation kinetics case of Lawryshyn and Hofmann (2013), it can be shown that RAF

monotonically decreases as ρ increases (see Appendix A.4). This result ensures that the RAF will not

have a local maximum or minimum as a function of ρ, which allows one to bound the worst case RAF

given an assumed mixing (ρ) scenario. Finally, it can be shown that as the reactor dose setting, α,

approaches 0, the RAF will approach 1 (see Appendix A.5). The results are summarized in Table 3.1.


Table 3.1: Summary of theoretical results on the RAF

Condition Result

ρ = 00 < β < 1 RAF < 1

ρ = 0β = 0 RAF = 1

ρ = 0β = 1 RAF = 1

ρ = 0kf = kp RAF = 1

Effect of ρ RAF |ρ>0 ≤ RAF |ρ=0 ≤ RAF |ρ<0

α→ 0 RAF → 1

3.3.3 Performance Factor (PF )

As was discussed above, even under the conditions of perfect mixing between banks (ρ = 0), the RAF

will necessarily be less than one for the double-exponential kinetics case which poses concerns for putting

UV banks / reactors in series, as is typically done in practice. However, in practice, a single UV bank

or reactor is typically validated using test microbes such as MS2 or T1 (see National Water Research

Institute (2012)), both of which exhibit first-order kinetics. To better understand the implications of

the, arguably, concerning result of RAF < 1 for ρ = 0, above, in this section, the impact of validating

a single UV bank / reactor with a test organism that exhibits linear kinetics and then placing identical

banks / reactors in series, is investigated.

As presented in Lawryshyn and Hofmann (2013), for the case of linear kinetics and a reactor or bank

that exhibits a log-normal dose distribution (equation (3.4)), the validated RED is given by

REDtest = − 1

ktestln

∞∫−∞

e−αktest(Dmin+eµ+σz)φ(z) dz

= αDmin −

1

ktestln

∞∫−∞

e−αktesteµ+σz

φ(z) dz

= αDmin −

1

ktestln(

E[e−αkteste

µ+σz]). (3.17)

where the subscript test is used to emphasize that the validation was performed using a test organism

with a linear inactivation constant, ktest. To ease the presentation of the formulation, the following

function is defined

g(k; , α, µ, σ) ≡ E[e−αkteste

µ+σz], (3.18)

3.3. Methodology 35

and since, for a given reactor, the parameters α, µ and σ can be constant, the notation gk = g(k;α, µ, σ)

will be used.

Applying the DRC of equation (3.2), for a given level of required inactivation, the required dose, D,

can easily be determined. The standard assumption in industry is that the validated dose of the reactor,

REDtest, from equation (3.17) can be substituted for D in equation (3.2), and if the inactivation level

is better than the target then the UV system will be in compliance. For two banks in series, it would be

standard practice to assume that two times REDtest will lead to the required performance. Therefore,

substituting 2REDtest for D of equation (3.2) leads to the expected performance of

N

N0

∣∣∣∣EP

= (1− β)e−2kfREDtest + βe−2kpREDtest

= (1− β)e−2αkfDmin (gktest)2kfktest + βe−2αkpDmin (gktest)

2kpktest , (3.19)

where the subscript EP is used to designate expected performance.

The actual performance of the system, however, will be determined by equation (3.10). In the analysis

presented here, the simplifying assumption is made that there is complete mixing between the reactors

/ banks such that ρ = 0 in equation (3.5) and therefore the actual performance becomes

N

N0

∣∣∣∣AP

= (1− β)e−2αkfDmin(gkf)2

+ βe−2αkpDmin(gkp)2, (3.20)

where AP is used to denote actual performance. It is noted that equation (3.20) follows from the

substitution of equation (3.5) into equation (3.10) and the fact that when ρ = 0 the double integral

simplifies to two single integrals which can be written as expectations similar to the procedure above.

Furthermore, the function g(k;α, µ, σ) only changes with the different inactivation constants (i.e. α,

µ and σ are the same for both the expected and actual performance calculations) and therefore the

subscript notation for the function is utilized for the sole purpose of ease of notation.

The performance factor, PF , is defined as follows

PF ≡NN0

∣∣∣EP

NN0

∣∣∣AP

=(1− β)e−2αkfDmin (gktest)

2kfktest + βe−2αkpDmin (gktest)

2kpktest

(1− β)e−2αkfDmin(gkf)2

+ βe−2αkpDmin(gkp)2 . (3.21)

The PF will have a value less than one when the actual performance of the system is worse than expected

and greater than one when the performance is better than expected. Clearly, PF = 1 represents the


critical value for a given reactor performance (Dmin, α, µ, σ), DRC(β, kf , kp) and target inactivation. It

is of interest to determine the critical value for ktest. Setting equation (3.21) to one, numerical methods

can be used to determine the critical ktest which will be referred to as ktest,c. Below, a few extreme cases

are investigated.

First-order Inactivation Kinetics

For cases where β equals zero or one the DRC becomes first-order and it is easy to show that for PF > 1,

ktest,c must be greater than kf when β = 0 and ktest,c must be greater than kp for the case when β = 1,

i.e.,

for β = 0, ktest,c ≥ kf

for β = 1, ktest,c ≥ kp.

Substituting β = 0 or β = 1 into (3.21) gives

PF =

(gktest,c

) kktest,c

gk> 1, (3.22)

where k is kf or kp for β = 0 or β = 1 respectively. This result is shown numerically in Figure 2 where

PF is plotted as a function of k/ktest,c. It can be seen that while k/ktest,c < 1, PF > 1 which implies

that ktest,c > k. While a rigorous mathematical proof is not developed as part of this work, a large

number of numerical experiments were run and the results remained consistent.

k/ktest

PF

10-1

100

101

102

0

0.2

0.4

0.6

0.8

1

Figure 3.2: PF as a function of k/ktest


Extreme kf and kp

Consider the case where kf →∞, kp = 0 and β is set such that the target inactivation can be achieved.

Then, equation (3.10) gives N/N0|target ≤ β. In equation (3.21), gkf = 0 and gkp = 1 and since gk will

always be less than one for any positive k, the PF is given by

limkf→∞,kp→0

PF = limkp→0

(gktest,c

) 2kpktest,c(

gkp)2 = 1. (3.23)

Therefore, for this extreme case, any value for ktest,c will ensure the desired performance.

Efficient Reactor (σ = 0)

When σ is set to zero, the dose distribution has no variance and this is often considered to be the ideal

reactor. In this case gk = E[e−αke

µ]= e−αke

µ

and equation (3.21) gives

PF =(1− β)e−2αkfDmin

(e−αkteste

µ) 2kfktest + βe−2αkpDmin

(e−αkteste

µ) 2kpktest

(1− β)e−2αkfDmin (e−αkf eµ)2

+ βe−2αkpDmin (e−αkpeµ)2 = 1. (3.24)

Inefficient Reactor (σ →∞)

As Wright and Lawryshyn (2000) discussed, reactor performance deteriorates as the standard deviation

for the dose distribution increases. It is important to consider the limiting case of the ktest,c for σ →∞

to ensure the problem is bounded. Lawryshyn and Hofmann (2013) showed that limσ→∞ gk = 1/2 .

Therefore, to find ktest,c, PF of equation (3.21) is set to one and the equation reduces to

1− 4(14

) kfktest,c

1− 4(14

) kpktest,c

= − β

1− βe−2αDmin(kp−kf ), (3.25)

and numerical methods can be used to solve for ktest,c.

3.4 Results and Discussion

In this section, numerical experiments of the theory developed are undertaken for first the RAF and

then the PF . It is shown how the RAF is affected by kf , kp, α, β, and ρ. It is of special interest to see

by what degree is RAF < 1 for ρ = 0. Next, the ktest,c is found for a family of DRCs and for reactors

of varying efficiency. Additionally, surfaces of ktest,c as a function of kf and kp are shown for various

reactor efficiencies and values of β.


In addition to numerical experiments, the theory is investigated through use of CFD models. An

open channel wastewater reactor with two banks in series was modeled by Young and Lawryshyn (2013),

and the resulting dose path correlations and dose distribution were used. The RAF calculated for

three scenarios of varying ρ. Also, the PF and ktest,c for two test organisms are calculated for the CFD

generated dose distribution and six wastewater inactivation kinetics for fecal coliform bacteria previously

presented in the literature.

3.4.1 Numerical Analysis of the Theory

A “base case” UV reactor system will be used throughout the section. The minimum dose that the

base case reactor can deliver is 5 mJ/cm2. The mean of the dose distribution is 25 mJ/cm2, and the

standard deviation is 20 mJ/cm2, which corresponds to a µ of 2.97 mJ/cm2 and a σ of 0.703 mJ/cm2.

Numerical Analysis of the RAF

Since the integrals in equations (3.12) and (3.13) cannot be solved analytically, it is not possible to look

for local minima by means of partial derivatives. Therefore, the analysis is limited to plotting ranges of

variables and interpreting the impact on the RAF . The magnitude by which RAF < 1 is the primary

focus.

The RAF was plotted as a function of two variables, kf and kp, and the process was repeated for

different levels of correlation with β = 10−3. The correlation between reactor dose paths (ρ) was left

as a parameter. The base case reactor described above was used. The result is given in Figure 3.3,

with RAF = 1 plotted as a horizontal plane for a frame of reference. It can be seen that the RAF

monotonically decreases as ρ increases from -1 to 1, and the result confirms the theory of Appendix

A.4. Of most interest is the plot where ρ = 0. The theory of Lawryshyn and Hofmann (2013) assured

unequivocally that RAF = 1 when ρ = 0 for the case of first-order inactivation kinetics. In Section 3.3.2

of this paper it was mentioned that RAF < 1 for ρ = 0, and this plot shows to what extent RAF < 1.

As to be expected, for the entire range RAF < 1, and over a large area RAF < 0.8.

The impact of the reactor dose setting (α) was also investigated. The base case dose distribution was

used. It can be seen from Figure 3.4 that similarly to the results of Lawryshyn and Hofmann (2013),

the RAF initially decreases as α increases, and then after a minimum, RAF increases as α increases.

In practice, increasing α increases the operational cost of disinfection, so a cost-benefit analysis would

have to be performed to determine an optimal α.

Additionally, RAF was also plotted as a function of β, with ρ = 0, kf = 0.460 cm2/mJ and kp =


0.046 cm2/mJ . The base case dose distribution was used. The result is shown in Figure 3.5. With the

double-exponential model first presented in equation (3.2), when β is equal to 0 or 1, the model reduces

to the first-order inactivation model of equation (3.1). Thus, when β is equal to 0 or 1, it was expected

that the theory of Lawryshyn and Hofmann (2013) would apply and that RAF would tend towards 1

as β went to 0 and 1. This result is apparent, as it can clearly be seen that RAF tends to 1 when β is

equal to 0 or 1. It is noted that for a generic wastewater, β ranging between 10−1 – 10−2 is reasonable

(see Azimi et al. (2012)).

Numerical Analysis of the PF

As mentioned previously, it is desired to have a performance factor that is greater than 1 in order to

ensure that a reactor validation result, where first-order kinetics were used, is valid in the context of

double-exponential kinetics. For a test organism with an inactivation constant of ktest, it was shown

that in order for PF > 1, ktest > ktest,c. However, no assertions have been made for how difficult this

is to achieve in practice.

A family of dose response curves were generated that all achieve a microbial inactivation of 10−3 with

a dose of 20 mJ/cm2. The DRCs can be seen in Figure 3.6, and the double-exponential parameters β,

kf , and kp are given in Table 3.2. The DRCs are bounded between two extreme cases shown as dashed

lines. The diagonal dashed line is the case where kf = kp, and the inactivation kinetics are essentially

first-order. In this case, the PF and RAF are both assured to equal one when ktest = kf since the

case has been reduced to first-order inactivation kinetics and the theory of Lawryshyn and Hofmann

(2013) applies. The horizontal dashed line is the case where the free microbes are infinitely easy to kill

(kf → ∞), the particle-associated microbes are infinitely hard to kill (kp ∈ 0), and β = 10−3. The

theory presented in Section 3.3.2 assures us that PF = 1 in this case for any ktest.

For the DRCs shown in Table 3.2, the ktest,c was computed using the method described in Section

3.3.3 for a “good” reactor and a “poor” reactor. The difference between the good and poor reactors

was the breadth-controlling parameter σ. A higher σ results in a wider dose distribution and poorer

performance. The good reactor had a dose distribution with Dmin = 5 mJ/cm2, µ = 1.5 mJ/cm2, and

σ = 0.3 mJ/cm2. The poor reactor had a dose distribution with Dmin = 5 mJ/cm2, µ = 1.0 mJ/cm2,

and σ = 1.044 mJ/cm2. As well, both reactors have the same average dose of 9.68 mJ/cm2, which

ensures their energy output is identical - one being more efficient than the other, however. Additionally,

ktest,c was calculated for a worst case reactor where σ → ∞. The ktest,c calculated for the worst case

reactor serves as a maximum lower bound for ktest > ktest,c. It can be seen in Table 3.2 that for any

given DRC, as reactor performance worsens, ktest,c increases. Also, as the dose response curve flattens


out, the ktest,c increases. It is important to note that ktest,c appears to have a lower bound of kp, and an

upper bound of kf . Finally, it is apparent that as the DRC deviates from first-order inactivation kinetics

to double-exponential inactivation kinetics, ktest,c approaches kp. Surface plots showing the relationship

between kf , kp, and ktest,c are shown in Figure 3.7 for the good reactor, and the worst case reactor and

various values of β. It can be seen that the ktest,c is considerably less for the good reactor relative to

the worst case reactor.

Table 3.2: Generated DRC parameters and ktest,c results

DRC β kf kp ktest,c good ktest,c poor ktest,c worst(cm2/mJ) (cm2/mJ) (cm2/mJ) (cm2/mJ) (cm2/mJ)

1 0.0012 2.7857 0.0099 0.0099 0.0099 0.00992 0.0023 1.7143 0.0426 0.0426 0.0426 0.04263 0.0049 1.1786 0.0805 0.0805 0.0806 0.08074 0.0120 0.8571 0.1250 0.1254 0.1281 0.13005 0.0344 0.6429 0.1781 0.1853 0.2030 0.21276 0.1233 0.4898 0.2424 0.2719 0.2959 0.3067

3.4.2 CFD Analysis of the Theory

To get an understanding for how real-world reactors are affected by the theory developed in this paper,

it is of interest to see what values RAF and PF take for an actual reactor. The wastewater reactor

described by Young and Lawryshyn (2013) will be analysed in this section. A render of the geometry is

shown in Figure 3.8, with spacing between two reactors in series. The minimum dose was 6.21 mJ/cm2,

the average dose was 29.4 mJ/cm2, and the standard deviation was 13.31 mJ/cm2. This corresponds

to µ = 3.00 mJ/cm2 and σ = 0.534 mJ/cm2.

CFD Analysis of the RAF

An analysis of the RAF was done using parameters obtained with CFD by Young and Lawryshyn (2013).

A particle associated microbe fraction of β = 10−3 was used. The inter-bank lengths of 0.5 m, 3 m,

and 6 m resulted in dose path correlations of 0.39, 0.17, and 0.066 respectively. RAF was plotted as

a function of kf and kp, and the result is shown in Figure 3.9. It should be noted that the correlation

between z1 and z2 is not necessarily the correlation between dose paths (except for ρ = 0 or ρ = 1) as

discussed in Lawryshyn and Hofmann (2013). Once again, it is apparent that the RAF is well below one,

and the performance is the worst for the configuration with the highest correlation. The true impact

on RAF will ultimately depend on the site and time specific properties of the wastewater, namely the

parameters, kf , kp, and β.


CFD Analysis of the PF

An analysis of the PF was done using the dose distribution generated with CFD by Young and Lawryshyn

(2013). Secondary treated effluent wastewater samples were taken from the Ashbridges Bay Wastewater

Treatment Plant in Toronto, Ontario. Double-exponential model parameters for coliform bacteria were

estimated for the samples using a collimated beam apparatus by Cantwell (2007). The data has been

adapted and is shown in Table 3.3.

Table 3.3: Coliform bacteria experimental parameters for the double-exponential model, PF and ktest,c

Sample kf kp β PF (MS2) PF (T1) ktest,c(cm2/mJ) (cm2/mJ) (cm2/mJ)

1 0.66 0.042 3.16× 10−4 1.36 2.03 0.0422 0.55 0.029 5.01× 10−4 1.29 1.70 0.0293 0.45 0.025 7.94× 10−4 1.26 1.60 0.0254 0.47 0.026 3.16× 10−3 1.27 1.62 0.0265 0.35 0.022 1.26× 10−3 1.23 1.53 0.0236 0.48 0.050 2.51× 10−3 1.39 2.22 0.050

Using the data collected by Cantwell (2007) and the theory presented previously, for the reactors

described by Young and Lawryshyn (2013) PF and ktest,c was calculated. The performance factor was

calculated for the test organisms MS2 and T1. The dose required for 1-log of inactivation (D10) for

MS2 and T1 are 16 mJ/cm2 and 5 mJ/cm2 which correspond to test inactivation constants of 0.14

cm2/mJ and 0.46 cm2/mJ respectively (United States Environmental Protection Agency, 2006). For

all six wastewater samples, it can be seen in Table 3.3 that PF > 1 since for all samples, the inactivation

constant for MS2 or T1 was greater than ktest,c. It should be noted that ρ was not equal to zero (which

was an assumption for calculating PF ) for the two banks in series, although ρ approached zero for the

bank-to-bank separation of 6 m. Although all six samples “passed” in terms of PF for MS2 and T1, it

would be wrong to assume PF will always be greater than one for MS2 and T1. Ultimately, the ktest,c

must be calculated for every reactor and wastewater in order to ensure that the use of a first-order test

organism is valid.


(a) ρ = −1 (b) ρ = −0.75 (c) ρ = −0.5

(d) ρ = −0.25 (e) ρ = 0 (f) ρ = 0.25

(g) ρ = 0.5 (h) ρ = 0.75 (i) ρ = 1

Figure 3.3: RAF as a function of kf , kp, and ρ (units for kf and kp are in cm2/mJ). Dmin = 5 mJ/cm2,µ = 2.97 mJ/cm2, σ = 0.703 mJ/cm2


Figure 3.4: RAF as a function of α

(a) 0 ≤ β ≤ 1 (b) 0 ≤ β ≤ 0.1

Figure 3.5: RAF as a function of β


D (mJ/cm2)

N/N

0

0 5 10 15 20 2510

-4

10-3

10-2

10-1

100

21

34

56

Figure 3.6: Generated DRCs and extreme cases

kp (cm2/mJ)kf (cm2/mJ)

ktes

t,c (

cm2 /m

J)

0.1

0.2

0.3

0.4

0.51

1.52

2.5

0.2

0.4

0.6

(a) good, β = 10−2


ktes

t,c (

cm2 /m

J)

0.1

0.2

0.3

0.4

0.51

1.52

2.5

0.2

0.4

0.6

(b) good, β = 10−3


ktes

t,c (

cm2 /m

J)

0.1

0.2

0.3

0.4

0.51

1.52

2.5

0.2

0.4

0.6

0.8

(c) good, β = 10−4


ktes

t,c (

cm2 /m

J)

0.1

0.2

0.3

0.4

0.51

1.52

2.5

0.2

0.4

0.6

(d) worst, β = 10−2


ktes

t,c (

cm2 /m

J)

0.2

0.4

0.51

1.52

2.5

0.2

0.4

0.6

0.8

(e) worst, β = 10−3


ktes

t,c (

cm2 /m

J)

0.1

0.2

0.3

0.4

0.51

1.52

2.5

0.2

0.4

0.6

0.8

1

(f) worst, β = 10−4

Figure 3.7: ktest,c surfaces as a function of kf , kp, for the good reactor (Dmin = 5 mJ/cm2, µ = 1.5mJ/cm2, σ = 0.3 mJ/cm2) and the worst case reactor (Dmin = 5 mJ/cm2, σ →∞)


Figure 3.8: Wastewater reactor configuration with particle tracks

(a) ρ = 0.39 (b) ρ = 0.17 (c) ρ = 0.066

Figure 3.9: RAF surfaces for the CFD wastewater reactors (units for kf and kp are in cm2/mJ). Dmin

= 6.21 mJ/cm2, µ = 3.00 mJ/cm2, σ = 0.534 mJ/cm2


3.5 Conclusions

It was acknowledged that the theory of Lawryshyn and Hofmann (2013) is limited to the discussion of

drinking water only, since the model for the RAF assumed first-order microbial inactivation kinetics. In

this paper, the model was reformulated to allow for double-exponential inactivation kinetics and thus the

discussion of wastewater and surface water. It was found that under perfect-mixing conditions (ρ = 0),

the RAF would necessarily be less than one. The results were then confirmed with CFD for two UV

reactor banks in series, and it was found that RAF was less than one. This unexpected result is contrary

to what one would have expected based on the results of Lawryshyn and Hofmann (2013). It assures that

the RED assumed in industry practice is greater than the actual total RED of installed UV systems

with reactors / banks in series.

The UV industry currently tests with a first-order test organism and assumes additivity in the context

of double-exponential wastewaters. In order to find when this assumption is still valid, a new factor called

the performance factor was introduced, and it was shown that a critical test inactivation constant could

be found for any DRC and a worst case reactor. Reactors sized with a test organism with an inactivation

constant greater than that of the critical test inactivation constant are ensured that they will meet their

target microbial inactivation level. If perfect mixing between reactors / banks in series is assumed to

be valid, then equation (3.25) should be used to calculate a critical inactivation constant for the UV

system and site. The PF was calculated experimentally using CFD to generate a dose distribution for

a wastewater reactor with two banks in series and dose response curves from secondary treated effluent

samples. It was found that the inactivation constants for MS2 and T1 were greater than the critical

inactivation constants for the six DRCs, and thus, the PF was greater than one. It was noted that

the results are not universally valid for MS2 and T1 and that a check must be performed for every

test organism, reactor, and DRC to ensure that that the PF is greater than one and that the use of a

first-order test organism is valid.

REFERENCES 47

References

Y Azimi, D G Allen, and R R Farnood. Kinetics of UV inactivation of wastewater bioflocs. Water

research, 46(12):3827–3836, April 2012. ISSN 1879-2448. doi: 10.1016/j.watres.2012.04.019.

Raymond E Cantwell. Impact of Particles on Ultraviolet Disinfection of Bacteria in Water. PhD thesis,

2007.

Raymond E Cantwell and Ron Hofmann. Inactivation of indigenous coliform bacteria in unfiltered

surface water by ultraviolet light. Water research, 42(10-11):2729–35, May 2008. ISSN 0043-1354.

doi: 10.1016/j.watres.2008.02.002.

Eric Caron, Gabriel Chevrefils, Benoit Barbeau, Pierre Payment, and Michele Prevost. Impact of mi-

croparticles on UV disinfection of indigenous aerobic spores. Water research, 41(19):4546–56, Novem-

ber 2007. ISSN 0043-1354. doi: 10.1016/j.watres.2007.06.032.

O Cerf. Tailing of survival curves of bacterial spores. The Journal of applied bacteriology, 42(1):1–19,

February 1977. ISSN 0021-8847.

TK Das. Ultraviolet disinfection application to a wastewater treatment plant. Clean Products and

Processes, 3, 2001. doi: 10.1007/s100980100108.

J Ducoste and S Alpert. Assessing the UV Dose Delivered from Two UV Reactors in Series: Can You

Always Assume Doubling the UV Dose from Individual Reactor Validations? In Proceedings of the

2nd North American Conference on Ozone, Ultraviolet & Advanced Oxidation Technologies, 2011.

Ramin Farnood. Flocs and Ultraviolet Disinfection. In Ian G Droppo, Gary G Leppard, Steven N Liss,

and Timothy G Milligan, editors, Flocculation in Natural and Engineered Environmental Systems.

2005. ISBN 1566706157.

B Ferran and OK Scheible. Biodosimetry of a Full-Scale UV Disinfection System to Achieve Regulatory

Approval for Wastewater Reuse. In Proceedings of the Water Environment Federation, volume 2007,

pages 367–385, 2007.

W Hijnen and G Medema. Inactivation of viruses, bacteria, spores and protozoa by ultraviolet irradiation

in drinking water practice: a review. Water Supply, pages 93–99, 2005.


pages 1–32, 2013.


FJ Loge, RW Emerick, and DE Thompson. Factors Influencing Ultraviolet Disinfection Performance

Part I: Light Penetration to Wastewater Particles. Water Environment Research, 71(3):377–381, 1999.



RG Qualls, SF Ossoff, and JCH Chang. Factors controlling sensitivity in ultraviolet disinfection of

secondary effluents. Water Pollution Control Federation, 57(10):1006–1011, 1985.

CC Tang, Jeff Kuo, and SJ Huitric. UV Systems for Reclaimed Water Disinfection From Equipment

Validation to Operation. In Proceedings of the Water Environment Federation, pages 2930–2943, 2006.



V Urbain, JC Block, and J Manem. Bioflocculation in activated sludge: an analytic approach. Water

Research, 27(5):829–838, 1993.

H.B. Wright and Yuri A. Lawryshyn. An assessment of the bioassay concept for UV reactor validation.

Proceedings of the Water Environment Federation, 2000(2):378–400, 2000.

Patrick C. Young and Yuri A. Lawryshyn. A Computational Fluid Dynamics Analysis of Placing UV

Reactors in Series. submitted, 2013.

Chapter 4

Conclusion

This thesis addresses the unanswered question left by Lawryshyn and Hofmann (2013) with respect to

reactor additivity. The numerical work in Chapter 2 serves to show how accurate the assumption of

reactor additivity is by calculating the RAF for several scenarios. Additionally, ρ was calculated for

reactors / banks in series which provided a metric for how valid the assumption of perfect mixing between

reactors / banks in series is. For the set of wastewater reactors, it was shown that ρ will be greater than

zero, and that ρ will approach zero as the distance between banks in series increases. Since the RAF is

superior for a ρ closer to zero, it is advantageous in terms of reactor additivity to place banks in series as

far apart as reasonably possible. For the drinking water reactors in series, it was shown that the mixing

between reactors was sufficient to give near additive reactor performance. Additionally, because the UV

sensitivity of MS2 is less than half that of adenovirus, it was shown numerically that a reactor validated

with MS2 would be able to achieve a RAF greater than one when targeting adenovirus.

In addition to providing a numerical backing to the theory of reactor additivity, the thesis extended

the theory from drinking water to double-exponential wastewater. When analysing the reformulated

RAF for wastewater and double-exponential microbial inactivation kinetics, it was unexpectedly shown

that the RAF would be less than one even with perfect mixing between reactors / banks in series. This

motivated the definition of the new PF . Criteria were given for which the PF is greater than one, which

implies that a UV reactor system validated with a first-order test organism such as MS2 would be valid

in the context of a double-exponential wastewater. A worst-case reactor was defined and for which a

maximum lower bound on the inactivation constant of the test organism could be found using equation

(3.25). It was noted and is strongly recommended that the methodology shown for finding the critical

inactivation constant be found for new reactor configurations and wastewaters.

49

50 Chapter 4. Conclusion

4.1 Future Work

While attempting to be robust, this thesis leaves room for more work to be done on the topic of reactor

additivity. Within the scope of this work, there is still room for additional work to be done. Beyond

the scope of this thesis, there are also opportunities to do further research. Future research areas are

summarized below.

• Use CFD to model more reactor configurations (cross-flow, different baffles, different reactor scales)

to further test the validity of the main assumption of the theory, namely identical dose distributions

among the two reactors

• Find a rigorous proof for the result in equation (3.22) as it has only been shown numerically

• Find closed form results for RAF for as kf tends to infinity, kp tends to zero, and α tends to

infinity

• Extend the RAF and PF from 2 to N reactors / banks in series and numerically test the theory

using CFD

• Find a suitable way for engineers to quantify the degree of mixing for installed reactors / banks in

series

• Field test / perform bioassays on reactors in series to compute the RAF and PF

• Determine how reactors / banks in series affect the UVDGM’s RED bias factor

To summarize, there are still many unknowns and opportunities to work on the subject of UV reactor

additivity. The models developed in this thesis can be generalized further, and the impact of them in

the context of UV reactor regulations still needs to be explored.

Appendices

51

Appendix A

Reactor Additivity Factor (RAF )

Proofs

A.1 Proof of RAF 6= 1 for ρ = 0

Proof. Inactivation is defined by

N

N0= e−kRED. (A.1)

For double-exponential inactivation kinetics,

N

N0= (1− β)e−kfRED + βe−kpRED. (A.2)

Let the dose that a single reactor receives be given by

D1d= α

(Dmin + eµ+σz1

). (A.3)

Similarly, let the dose that two reactors in series receive be given by

D2d= α

(2Dmin + eµ+σz1 + e

µ+σ(ρz1+√

1−ρ2z2))

. (A.4)

When ρ = 0,

D2d= α

(2Dmin + eµ+σz1 + eµ+σz2

). (A.5)

52

A.1. Proof of RAF 6= 1 for ρ = 0 53

Substituting D2 into the double-exponential inactivation model for RED, we get

N

N0= (1− β)e−kfα(2Dmin+eµ+σz1+eµ+σz2) + βe−kpα(2Dmin+eµ+σz1+eµ+σz2). (A.6)

Wright and Lawryshyn (2000) showed that inserting the probability density function for a given dose

and integrating over its range allows for an alternate computation of inactivation such that,

N

N0=

∞∫−∞

e−kD(z)φ(z) dz. (A.7)

Analogously, the same can be done for a double-exponential inactivation model and two reactors in

series, such that

N

N0=

∞∫−∞

∞∫−∞

((1− β)e−kfD(z1,z2) + βe−kpD(z1,z2)

)φ(z1)φ(z2) dz1dz2. (A.8)

Substituting in the dose model in (A.5) for two reactors in series when ρ = 0 gives

N

N0=

∫ ∞−∞

∫ ∞−∞

((1− β)e−kfα(2Dmin+eµ+σz1+eµ+σz2)

+ βe−kpα(2Dmin+eµ+σz1+eµ+σz2))φ(z1)φ(z2) dz1dz2

=

∫ ∞−∞

∫ ∞−∞

((1− β)e−2kfαDmine−kfαe

µ+σz1e−kfαe

µ+σz2

+ βe−2kpαDmine−kpαeµ+σz1

e−kpαeµ+σz2


= (1− β)e−2kfαDmin∫ ∞−∞

e−kfαeµ+σz1

φ(z1) dz1

∫ ∞−∞

e−kfαeµ+σz2

φ(z2) dz2

+ βe−2kpαDmin∫ ∞−∞

e−kpαeµ+σz1

φ(z1) dz1

∫ ∞−∞

e−kpαeµ+σz2

φ(z2) dz2. (A.9)

Recognizing that the first and second integrals are equal, as well as the third and fourth

I1 =

∫ ∞−∞

e−kfαeµ+σz1

φ(z1) dz1 =

∫ ∞−∞

e−kfαeµ+σz2

φ(z2) dz2 (A.10a)

I2 =

∫ ∞−∞

e−kpαeµ+σz1

φ(z1) dz1 =

∫ ∞−∞

e−kpαeµ+σz2

φ(z2) dz2. (A.10b)

Substituting in (A.10a) and (A.10b) into (A.9) and equating to (A.1),

e−k2RED2 = (1− β)e−2kfαDminI21 + βe−2kpαDminI22 . (A.11)

54 Appendix A. Reactor Additivity Factor (RAF ) Proofs

Similarly, the same can be done for a single reactor

N

N0=

∫ ∞−∞

((1− β)e−kfα(Dmin+eµ+σz1) + βe−kpα(Dmin+eµ+σz1)

)φ(z1) dz1

= (1− β)e−kfαDmin∫ ∞−∞

e−kfαeµ+σz1

φ(z1) dz1 + βe−kpαDmin∫ ∞−∞

e−kpαeµ+σz1

φ(z1) dz1. (A.12)

e−k1RED1 = (1− β)e−kfαDminI1 + βe−kpαDminI2 (A.13)

Solving for RED1 and RED2 gives

RED1 = − 1

k1ln((1− β)e−kfαDminI1 + βe−kpαDminI2

)(A.14)

RED2 = − 1

k2ln((1− β)e−2kfαDminI21 + βe−2kpαDminI22

). (A.15)

RAF is defined as

RAF ≡ RED2

2RED1. (A.16)

To show that RAF cannot be equal to 1, suppose the contrary is true. Then,

1 =RED2

2RED1(A.17)

2RED1 = RED2. (A.18)

Also, if the reactor doses are perfectly additive, it can be shown that k1 = k2 = klin. Equating (A.1)

with (A.2) gives

e−kRED = (1− β)e−kfαRED + βe−kpαRED (A.19)

k = − 1

REDln((1− β)e−kfRED + βe−kpRED

). (A.20)

Substituting in k = k1, RED = 2RED1 for the single reactor and k = k2, RED = RED2 for the

reactors in series gives

k1 = − 1

2RED1ln((1− β)e−kf2RED1 + βe−kp2RED1

)(A.21)

k2 = − 1

RED2ln((1− β)e−kfRED2 + βe−kpRED2

). (A.22)

A.2. Proof of RAF < 1 for ρ = 0 55

However, it can be seen by inspection that k1 will equal k2 because of the equality in (A.18), so

k1 = k2 = klin. (A.23)

Substituting (A.18) and (A.23) into (A.14) gives

RED2 = − 2

klinln((1− β)e−kfαDminI1 + βe−kpαDminI2

)= − 1

klinln((

(1− β)e−kfαDminI1 + βe−kpαDminI2)2)

= − 1

klinln(

(1− β)2e−2kfαDminI21 + β2e−2kpαDminI22 + 2(1− β)βe−αDmin(kf+kp)I1I2

). (A.24)

Equating expressions (A.15) and (A.24), it is clear that

(1− β)e−2kfαDminI21 + βe−2kpαDminI22

6= (1− β)2e−2kfαDminI21 + β2e−2kpαDminI22 + 2(1− β)βe−αDmin(kf+kp)I1I2. (A.25)

Thus,

RAF =RED2

2RED16= 1, (A.26)

for ρ = 0.

A.2 Proof of RAF < 1 for ρ = 0

Proof. If RAF < 1 for ρ = 0,

RAF =RED2

2RED1< 1 (A.27)

RED2 < 2RED1. (A.28)

Also, it can be shown that k1 > k2. Since RED2 < 2RED1,

(N

N0

)2

>

(N

N0

)1×2

(A.29)

e−k2RED2 > e−k12RED1 (A.30)

− k2RED2 > −k12RED1 (A.31)

k2RED2 < k12RED1 (A.32)


RED2

2RED1= RAF <

k1k2, (A.33)

where (N/N0)1×2 represents double the microbial inactivation of a single reactor. But since RAF < 1

RAF < 1 <k1k2

(A.34)

k1 > k2, (A.35)

or,

1 >k2k1. (A.36)

Substituting (A.14) and (A.15) into (A.28) gives

− 1


)< − 2

k1ln((1− β)e−kfαDminI1 + βe−kpαDminI2

), (A.37)

or,

1


)>

1

k1ln(

(1− β)2e−2kfαDminI21 + β2e−2kpαDminI22 + 2(1− β)βe−αDmin(kf+kp)I1I2

), (A.38)

or,

ln((1− β)e−2kfαDminI21 + βe−2kpαDminI22

)ln((1− β)2e−2kfαDminI21 + β2e−2kpαDminI22 + 2(1− β)βe−αDmin(kf+kp)I1I2

) > k2k1. (A.39)

But since 1 > k2/k1, the task is reduced to showing that the LHS of (A.39) is ≥ 1, which is accomplished

by


≥ (1− β)2e−2kfαDminI21 + β2e−2kpαDminI22 + 2(1− β)βe−αDmin(kf+kp)I1I2 (A.40)

A.3. Special cases where RAF = 1 for ρ = 0 57

(1− β)(e−kfαDminI1

)2+ β

(e−kpαDminI2

)2≥ (1− β)2

(e−kfαDminI1

)2+ β2

(e−kpαDminI2

)2+ 2(1− β)β

(e−kfαDminI1

) (e−kpαDminI2

)(A.41)

(e−kfαDminI1

)2 − β (e−kfαDminI1)2 + β(e−kpαDminI2

)2≥ (1− 2β + β2)

(e−kfαDminI1

)2+ β2

(e−kpαDminI2

)2+ 2β

(e−αDminkf I1

) (e−αDminkpI2

)− 2β2

(e−kfαDminI1

) (e−kpαDminI2

)(A.42)

− β(e−kfαDminI1

)2+ β

(e−kpαDminI2

)2≥ −2β

(e−kfαDminI1

)2+ β2

(e−kfαDminI1

)2+ β2

(e−kpαDminI2

)2+ 2β

(e−αDminkf I1

) (e−αDminkpI2

)− 2β2

(e−kfαDminI1

) (e−kpαDminI2

)(A.43)

β(e−kfαDminI1

)2+ β

(e−kpαDminI2

)2 − 2β(e−αDminkf I1

) (e−αDminkpI2

)≥ β2

(e−kfαDminI1

)2+ β2

(e−kpαDminI2

)2 − 2β2(e−kfαDminI1

) (e−kpαDminI2

)(A.44)

β((

e−kfαDminI1)2

+(e−kpαDminI2

)2 − 2(e−αDminkf I1

) (e−αDminkpI2

))≥ β2

((e−kfαDminI1

)2+(e−kpαDminI2

)2 − 2(e−kfαDminI1

) (e−kpαDminI2

))(A.45)

1 ≥ β. (A.46)

Since 0 ≤ β ≤ 1, (A.46) is true.

A.3 Special cases where RAF = 1 for ρ = 0

Suppose that RAF = 1, then from (A.25)


= (1− β)2e−2kfαDminI21 + β2e−2kpαDminI22 + 2(1− β)βe−αDmin(kf+kp)I1I2 (A.47)

LHS = (1− β)e−2kfαDminI21 + βe−2kpαDminI22 (A.48)


RHS = (1− β)2e−2kfαDminI21 + β2e−2kpαDminI22 + 2(1− β)βe−αDmin(kf+kp)I1I2. (A.49)

A.3.1 Special Case: β = 1

Proof. Substituting β = 1 into (A.48) and (A.49) gives

LHS = e−2kpαDminI22 (A.50)

RHS = e−2kpαDminI22 . (A.51)

Since LHS = RHS, then RAF = 1 must be true.

A.3.2 Special Case: β = 0

Proof. Substituting β = 0 into (A.48) and (A.49) gives

LHS = e−2kfαDminI21 (A.52)

RHS = e−2kfαDminI21 . (A.53)


A.3.3 Special Case: kf = kp = k

Proof. If kf = kp = k then I1 = I2 = I since (A.10a) and (A.10b) differ by only the inactivation

constant. It follows that

LHS = (1− β)e−2kαDminI2 + βe−2kαDminI2

LHS = e−2kαDminI2 ((1− β) + β)

LHS = e−2kαDminI2 (A.54)

RHS = (1− β)2e−2kαDminI2 + β2e−2kαDminI2 + 2(1− β)βe−2αDminkI2

RHS = e−2kαDminI2((1− β)2 + β2 + 2(1− β)β

)RHS = e−2kαDminI2

(1− 2β + β2 + β2 + 2β − 2β2

)RHS = e−2kαDminI2. (A.55)

A.4. Proof that RAF monotonically decreases as ρ increases. 59


A.4 Proof that RAF monotonically decreases as ρ increases.

Proof. To show that RAF monotonically decreases as ρ increases from −1 → 1, it is sufficient to show

that RED2 monotonically decreases. RED2 can be written as

RED2 = − 1

klinln

(1− β)e−2kfαDmin

(∫ ∞−∞

e−kfα

(eµ+σz1+e

µ+σ(ρz1+√

1−ρ2z2))φ(z1) dz1

)2

+ βe−2kpαDmin

(∫ ∞−∞

e−kpα

(eµ+σz1+e

µ+σ(ρz1+√

1−ρ2z2))φ(z1) dz1

)2

= − 1

klinln

(1− β)e−2kfαDmin E

[e−kfα

(eµ+σz1+e

µ+σ(ρz1+√

1−ρ2z2))]2

+ βe−2kpαDmin E

[e−kpα

(eµ+σz1+e

µ+σ(ρz1+√

1−ρ2z2))]2 . (A.56)

To show that RED2 is monotonically decreasing, it will suffice to show that the inside of the logarithm

is monotonically increasing as ρ increases. Rewriting the first expectation as

Z1 = E

[e−kfα

(eµ+σz1+e

µ+σ(ρz1+√

1−ρ2z2))]

= E

[e−kfα(eµ+σz1)e

−kfα(eµ+σ(ρz1+

√1−ρ2z2)

)], (A.57)

and defining two new random variables V1 and V2 such that

V1 = e−kfα(eµ+σz1) (A.58a)

V2 = e−kfα

(eµ+σ(ρz1+

√1−ρ2z2)

). (A.58b)

Since eµ+σz1 and eµ+σ

(ρz1+√

1−ρ2z2)

are monotonic, V1 and V2 will be monotonic as well, and the

correlation between V1 and V2 will necessarily increase as ρ increases. Therefore, E [V1V2] or Z1 will

monotonically increase. A similar argument can be made for the second expectation of (A.56). Z12

and

Z22

will increase monotonically, and the sum of Z1 + Z2 times a constant will also increase monotonically,

thus guaranteeing that RED2 will decrease monotonically.


A.5 Proof that RAF tends to 1 as α tends to 0 for ρ = 0

Proof. Starting from equations (3.13), (3.12), and (3.3), we have

RED12 = − 1

klinln

∞∫−∞

∞∫−∞

((1− β)e−kfD2(z1,z2) + βe−kpD2(z1,z2)


RED1 = − 1

klinln

∞∫−∞

((1− β)e−kfD1(z1) + βe−kpD1(z1)

)φ(z1) dz1

RAF ≡ RED12

2RED1.

Substituting equation (A.4) into (3.13) with ρ = 0, and (A.3) into (3.12) gives

RED12 = − 1

klinln

∞∫−∞

∞∫−∞



)(A.59)

RED1 = − 1

klinln

∞∫−∞

((1− β)e−kfα(Dmin+eµ+σz1) + βe−kpα(Dmin+eµ+σz1)

)φ(z1) dz1

. (A.60)

Substituting (A.59) and (A.60) into (3.3) gives

RAF =

− 1klin

ln

∞∫−∞

∞∫−∞



)− 2klin

ln

(∞∫−∞

((1− β)e−kfα(Dmin+eµ+σz1 ) + βe−kpα(Dmin+eµ+σz1 )

)φ(z1) dz1

) . (A.61)

A.5. Proof that RAF tends to 1 as α tends to 0 for ρ = 0 61

Taking the limit as α→ 0 and applying L’Hopital’s rule gives

limα→0

RAF

[ 00 ]=H

limα→0

∞∫−∞

∞∫−∞

((2Dmin + eµ+σz1 + eµ+σz2) (−kf )(1− β)e−kfα(2Dmin+eµ+σz1+eµ+σz2)

+ (2Dmin + eµ+σz1 + eµ+σz2) (−kp)βe−kpα(2Dmin+eµ+σz1+eµ+σz2))φ(z1)φ(z2) dz1dz2

∞∫−∞

∞∫−∞


+βe−kpα(2Dmin+eµ+σz1+eµ+σz2))φ(z1)φ(z2) dz1dz2

2

∞∫−∞

((Dmin + eµ+σz1) (−kf )(1− β)e−kfα(Dmin+eµ+σz1)

+ (Dmin + eµ+σz1) (−kp)βe−kpα(Dmin+eµ+σz1))φ(z1) dz1

∞∫−∞

((1−β)e−kfα(Dmin+eµ+σz1)+βe−kpα(Dmin+eµ+σz1)

)φ(z1) dz1

=

∞∫−∞

∞∫−∞

((2Dmin+eµ+σz1+eµ+σz2)(−kf )(1−β)+(2Dmin+eµ+σz1+eµ+σz2)(−kp)β)φ(z1)φ(z2) dz1dz2

1

2

∞∫−∞

((Dmin+eµ+σz1 )(−kf )(1−β) +(Dmin+eµ+σz1 )(−kp)β)φ(z1) dz1

1

=(−kf )(1− β) (2Dmin + E [eµ+σz1 + eµ+σz2 ]) + (−kp)(β) (2Dmin + E [eµ+σz1 + eµ+σz2 ])

2(−kf )(1− β) (Dmin + E [eµ+σz1 ]) + 2(−kp)(β) (Dmin + E [eµ+σz1 ]).

(A.62)

Recognizing that E [Dmin + eµ+σz1 ] = E [Dmin + eµ+σz2 ] results in

limα→0

RAF

=(−kf )(1− β) (2Dmin + 2 E [eµ+σz1 ]) + (−kp)(β) (2Dmin + 2 E [eµ+σz1 ])

2(−kf )(1− β) (Dmin + E [eµ+σz1 ]) + 2(−kp)(β) (Dmin + E [eµ+σz1 ])

=2(−kf )(1− β) (Dmin + E [eµ+σz1 ]) + 2(−kp)(β) (Dmin + E [eµ+σz1 ])

2(−kf )(1− β) (Dmin + E [eµ+σz1 ]) + 2(−kp)(β) (Dmin + E [eµ+σz1 ])

= 1. (A.63)

by Patrick C. Young - University of Toronto T-Space...Patrick C. Young Master of Applied Science...

Documents

Transcript of by Patrick C. Young - University of Toronto T-Space...Patrick C. Young Master of Applied Science...