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.
4 Chapter 1. Introduction
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.
6 Chapter 1. Introduction
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
10 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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
12 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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.
14 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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
16 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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
18 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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
2.4. Results and Discussion 19
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.
20 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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
2.4. Results and Discussion 21
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
22 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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.
24 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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.
Yuri A. Lawryshyn and Ron Hofmann. A Theoretical Look at Adding UV Reactors in Series. submitted,
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.
National Water Research Institute. Ultraviolet Disinfection Guidelines for Drinking Water and Water
Reuse. Third edit edition, 2012.
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.
United States Environmental Protection Agency. Ultraviolet Disinfection Guidance Manual for the Final
Long Term 2 Enhanced Surface Water Treatment Rule. 2006.
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.
26 Chapter 2. A CFD Analysis of Placing UV Reactors in Series
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
30 Chapter 3. A More Rigorous Look at Reactor Additivity
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)
32 Chapter 3. A More Rigorous Look at Reactor Additivity
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
∞∫−∞
((1− β)e−kfD1(z) + βe−kpD1(z)
)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
∞∫−∞
((1− β)e−kfD1(z) + βe−kpD1(z)
)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.
34 Chapter 3. A More Rigorous Look at Reactor Additivity
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
36 Chapter 3. A More Rigorous Look at Reactor Additivity
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
3.4. Results and Discussion 37
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.
38 Chapter 3. A More Rigorous Look at Reactor Additivity
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 =
3.4. Results and Discussion 39
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
40 Chapter 3. A More Rigorous Look at Reactor Additivity
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.
3.4. Results and Discussion 41
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.
42 Chapter 3. A More Rigorous Look at Reactor Additivity
(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
3.4. Results and Discussion 43
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
44 Chapter 3. A More Rigorous Look at Reactor Additivity
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
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
(b) good, β = 10−3
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
0.8
(c) good, β = 10−4
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
(d) worst, β = 10−2
kp (cm2/mJ)kf (cm2/mJ)
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
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
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 →∞)
3.4. Results and Discussion 45
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
46 Chapter 3. A More Rigorous Look at Reactor Additivity
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
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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.
Yuri A. Lawryshyn and Ron Hofmann. A Theoretical Look at Adding UV Reactors in Series. submitted,
pages 1–32, 2013.
48 Chapter 3. A More Rigorous Look at Reactor Additivity
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.
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.
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. Ultraviolet Disinfection Guidance Manual for the Final
Long Term 2 Enhanced Surface Water Treatment Rule. 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.
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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
)φ(z1)φ(z2) dz1dz2
= (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)
56 Appendix A. Reactor Additivity Factor (RAF ) Proofs
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
k2ln((1− β)e−2kfαDminI21 + βe−2kpαDminI22
)< − 2
k1ln((1− β)e−kfαDminI1 + βe−kpαDminI2
), (A.37)
or,
1
k2ln((1− β)e−2kfαDminI21 + βe−2kpαDminI22
)>
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)e−2kfαDminI21 + βe−2kpαDminI22
≥ (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)e−2kfαDminI21 + βe−2kpαDminI22
= (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)
58 Appendix A. Reactor Additivity Factor (RAF ) Proofs
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)
Since LHS = RHS, then RAF = 1 must be true.
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
Since LHS = RHS, then RAF = 1 must be true.
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.
60 Appendix A. Reactor Additivity Factor (RAF ) Proofs
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)
)φ(z1)φ(z2) dz1dz2
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
∞∫−∞
∞∫−∞
((1− β)e−kfα(2Dmin+eµ+σz1+eµ+σz2)
+ βe−kpα(2Dmin+eµ+σz1+eµ+σz2))φ(z1)φ(z2) dz1dz2
)(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
∞∫−∞
∞∫−∞
((1− β)e−kfα(2Dmin+eµ+σz1+eµ+σz2)
+ βe−kpα(2Dmin+eµ+σz1+eµ+σz2))φ(z1)φ(z2) dz1dz2
)− 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
∞∫−∞
∞∫−∞
((1− β)e−kfα(2Dmin+eµ+σz1+eµ+σz2)
+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)