Quantitative Microbiology 2, 249–270, 2000
`# 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.
`
`Mathematical Modeling of Ultraviolet Germicidal
`Irradiation for Air Disinfection
`
`W. J. KOWALSKI*
`Department of Architectural Engineering, The Pennsylvania State University, Engineering Unit A, University
`Park, PA 16802, USA
`*Corresponding author: e-mail: drKowalski@psu.edu
`
`W. P. BAHNFLETH
`Department of Architectural Engineering, The Pennsylvania State University, Engineering Unit A, University
`Park, PA 16802, USA
`
`D. L. WITHAM
`Ultraviolet Devices, Inc., 28220 Industry Drive, Valencia, CA 91355, USA
`
`B. F. SEVERIN
`M.B.I. International, P.O. Box 27609, 3900 Collins Road, Lansing, MI 48909, USA
`
`T. S. WHITTAM
`Department of Microbiology and Molecular Genetics, Michigan State University, East Lansing, MI 48824, USA
`
`Received January 12, 2001; Accepted October 4, 2001
`
`Abstract. A comprehensive treatment of the mathematical basis for modeling the disinfection process for air
`using ultraviolet germicidal irradiation (UVGI). A complete mathematical description of the survival curve is
`developed that incorporates both a two stage inactivation curve and a shoulder. A methodology for the evaluation
`of the three-dimensional intensity fields around UV lamps and within reflective enclosures is summarized that will
`enable determination of the UV dose absorbed by aerosolized microbes. The results of past UVGI studies on
`airborne pathogens are tabulated. The airborne rate constant for Bacillus subtilis is confirmed based on results of
`an independent test. A re-evaluation of data from several previous studies demonstrates the application of the
`shoulder and two-stage models. The methods presented here will enable accurate interpretation of experimental
`results involving aerosolized microorganisms exposed to UVGI and associated relative humidity effects
`
`Key words: UVGI, UV air disinfection, surface disinfection, survival curve, decay curve
`
`1. Introduction
`
`Ultraviolet radiation in the range 225–302 nm is lethal to microorganisms and is referred
`to as ultraviolet germicidal irradiation (UVGI). Water and surface disinfection with UVGI
`are proven and reliable technologies, but airstream disinfection systems have had varying
`and unpredictable performance in applications. In spite of the widespread use of UVGI
`today for air disinfection and microbial growth control, design information about the
`effects of UVGI on airborne pathogens lacks the detail necessary to guarantee predictable
`
`EXHIBIT 1012
`
`1
`
`

`

`250
`
`KOWALSKI ET AL.
`
`performance. In addition, few airborne rate constants are known with certainty due to the
`inherent difficulties of setting up an experiment and accurately interpreting test results.
`The methods described here will facilitate the experimental design and accurate
`interpretation of aerosol studies on the inactivation of airborne pathogens with UVGI,
`as well as assist the design of UVGI systems for specific applications. Two distinct
`components make up the complete model—a model of microbial decay under UVGI
`exposure that depends on the microorganism, and a model of the UV dose resulting from
`the UVGI system or test apparatus.
`
`2. Modeling Microbial Decay
`
`The classical exponential decay model treats microbial survival under the influence of any
`biocidal factor (Chick, 1908). The refinements presented here, the two-stage model and the
`shoulder model, extend its applicability. One alternative model, the multi-hit target model
`is also capable of accounting for the shoulder and two stages of inactivation. The latter has
`been adequately addressed elsewhere and is summarized here for comparison purposes at
`the end of this section.
`Microorganisms exposed to UVGI experience an exponential decrease in population
`similar to other methods of disinfection such as heating, ozonation, and exposure to
`ionizing radiation (Koch, 1995; Mitscherlich and Marth, 1984). The single stage
`exponential decay equation for microbes exposed to UV irradiation is as follows:
`S ¼ e
`kIt
`where S ¼ surviving fraction of
`constant (cm2=mJ), I ¼ UV intensity (mW=cm2),
`where 1 mJ ¼ 1 mW-s.
`
`ð1Þ
`k ¼ standard rate
`t ¼ time of exposure (seconds) and
`
`initial microbial population,
`
`The rate constant defines the sensitivity of a microorganism to UV exposure and is
`unique to each microbial species. Most published test results provide an overall rate
`constant that applies only at the test intensity. The standard rate constant k in equation (1)
`is the equivalent rate constant at an intensity of 1 mW=cm2 and is found by dividing any
`measured rate constant by the test intensity. The standard rate constant, therefore, is
`independent of intensity.
`The intensity in equation (1) can be considered to represent either the irradiance on a flat
`surface or the fluence rate through the outer surface of a solid (i.e. a spherical microbe). If
`the average intensity is constant, or can be calculated, then the standard rate constant can
`be computed as
`
`k ¼ ln S
`
`It
`
`:
`
`ð2Þ
`
`The value of the rate constant depends on whether the intensity is defined as irradiance
`or fluence rate, and can also depend on how it is measured. This matter is addressed in the
`section on UV dose. Table 1 lists some 30 pathogens and rate constants determined in
`
`2
`
`

`

`ULTRAVIOLET GERMICIDAL IRRADIATION
`
`251
`
`Table 1. UVGI rate constants for respiratory pathogens.
`
`Microorganism
`
`Adenovirus
`
`Vaccinia
`
`Coxsackievirus
`
`Type
`
`Virus
`
`Virus
`
`Virus
`
`Influenza A
`Echovirus
`Reovirus Type 1
`Staphylococcus aureus
`
`Virus
`Virus
`Virus
`
`Gramþ Bacteria
`
`Streptococcus pyogenes
`
`Gramþ Bacteria
`
`Mycobacterium tuberculosis
`
`Mycobacteria
`
`Mycobacteria
`Mycobacterium kansasii
`Mycobacterium avium-intra. Mycobacteria
`E. coli (reference only)
`
`Corynebacterium diptheriae
`Moraxella-Acinetobacter
`Haemophilus influenzae
`Pseudomonas aeruginosa
`
`Legionella pneumophila
`
`Serratia marcescens
`
`Gram Bacteria
`Gram Bacteria
`
`Coxiella burnetti
`Bacillus anthracis
`Bacillus anthracis
`Cryptococcus neoformans
`Fusarium oxysporum
`Fusarium solani
`Penicillium italicum
`Penicillium digitatum
`Rhizopus nigricans
`
`Rickettsiae
`Mixed spores
`Bacterial spore
`Fungal spore
`Fungal spore
`Fungal spore
`Fungal spore
`Fungal spore
`Fungal spore
`
`Gram Bacteria
`Gramþ Bacteria
`Gram Bacteria
`Gram Bacteria Mongold, 1992
`Gram Bacteria
`
`Reference
`
`Test medium
`Air=Plt=Wtr
`
`Jensen, 1964
`Rainbow, 1973
`Jensen, 1964
`Galasso, 1965
`Jensen, 1964
`Hill, 1970 (B-1)
`Hill, 1970 (A-9)
`Jensen, 1964
`Hill, 1970
`Hill, 1970
`Sharp, 1939
`Sharp, 1940
`Gates, 1929
`Abshire, 1981
`Luckiesh, 1946
`Lidwell, 1950
`Mitscherlich, 1984
`David, 1973
`Riley, 1976
`Collins, 1971
`David, 1973
`David, 1973
`Sharp, 1939
`Sharp, 1940
`Sharp, 1939
`Keller, 1982
`
`Collins, 1971
`Abshire, 1981
`Sharp, 1940
`Antopol, 1979
`Gilpin, 1984
`Antopol, 1979
`Collins, 1971
`Antopol, 1979
`Riley, 1972
`Sharp, 1940
`Sharp, 1939
`Rentschler, 1941
`Little, 1980
`Sharp, 1939
`Knudson, 1986
`Wang, 1994
`Asthana, 1992
`Asthana, 1992
`Asthana, 1992
`Asthana, 1992
`Luckiesh, 1946
`
`Air
`Plates
`Air
`Plates
`Air
`Water
`Water
`Air
`Water
`Water
`Plates
`Air
`Plates
`Plates
`Air
`Plates
`Air
`Air
`Air
`Air
`Air
`Air
`Plates
`Air
`Plates
`Water
`Plates
`Air
`Water
`Air
`Water
`Water
`Water
`Air
`Water
`Air
`Air
`Air
`Air
`Water
`Plates
`Plates
`Plates
`Plates
`Plates
`Plates
`Plates
`Air
`
`k ¼ Standard rate
`constant (cm2=mJ)
`
`0.000546
`0.000047
`0.001528
`0.001542
`0.001108
`0.000159
`0.000202
`0.001187
`0.000217
`0.000132
`0.000886
`0.003476
`0.001184
`0.000419
`0.009602
`0.006161
`0.001066
`0.000987
`0.004721
`0.002132
`0.000364
`0.000406
`0.000927
`0.003759
`0.000701
`0.0000021
`0.000599
`0.002375
`0.000640
`0.005721
`0.000419
`0.002047
`0.002503
`0.002208
`0.001047
`0.049900
`0.004449
`0.001047
`0.001225
`0.001535
`0.000509
`0.000031
`0.000102
`0.000112
`0.0000706
`0.0001259
`0.0000718
`0.0000861
`(continued )
`
`3
`
`

`

`252
`
`Table 1. (continued )
`
`Microorganism
`
`Type
`
`Reference
`
`KOWALSKI ET AL.
`
`Test medium
`Air=Plt=Wtr
`
`k¼ Standard rate
`constant (cm2=mJ)
`
`Cladosporium herbarum
`Scopulariopsis brevicaulis
`Mucor mucedo
`Penicillium chrysogenum
`Aspergillus amstelodami
`
`Fungal spore
`Fungal spore
`Fungal spore
`Fungal spore
`Fungal spore
`
`Luckiesh, 1946
`Luckiesh, 1946
`Luckiesh, 1946
`Luckiesh, 1946
`Luckiesh, 1946
`
`Air
`Air
`Air
`Air
`Air
`
`0.0000370
`0.0000344
`0.0000399
`0.0000434
`0.0000344
`
`various media. The wide variation in rate constants predicted reflects the differences in
`media, the test arrangements, and the methods of measuring the intensity. In general,
`aerosol studies yield moderately higher rate constants than plate studies. This could be
`expected since microbes tumbling in the air will receive exposure all around, while
`microbes on plates receive exposure in one plane only.
`
`In general, a small fraction of any microbial population is
`Two-stage survival curves.
`resistant to UVGI or other bactericidal factors (Cerf, 1977; Fujikawa and Itoh, 1996).
`Typically, over 99% of the microbial population will succumb to initial exposure but a
`remaining fraction will survive, sometimes for prolonged periods (Smerage and Teixeira,
`1993; Qualls and Johnson, 1983). This effect may be due to clumping (Moats et al., 1971;
`Davidovich and Kishchenko, 1991), dormancy (Koch, 1995), or other factors.
`The two-stage survival curve can be represented mathematically as the summed
`response of two separate microbial populations that have respective rate constants k1
`and k2. If we define f as the resistant fraction of the total initial population with rate
`constant k2, then ð1 f Þ is the fraction with rate constant k1. The total survival curve is
`
`therefore the sum of the rapid decay curve (the vulnerable majority) and the slow decay
`curve (the resistant minority).
`ð3Þ
`k1It þ f e
`Þe
`SðtÞ ¼ 1 fð
`
`k2It
`where k1 ¼ rate constant for fast decay population (cm2=mJ), k2 ¼ rate constant for resistant
`population (cm2=mJ), f ¼ resistant fraction.
`
`Figure 1 shows data for Streptococcus pyogenes that displays two-stage behavior. The
`resistant fraction of most microbial populations may be about 0.01–1% but some studies
`suggest
`it can be a large fraction for certain species (Riley and Kaufman, 1972;
`Gates, 1929).
`Values of the two-stage rate constants are summarized in Table 2 for the few microbes
`for which second stage data has been published. These parameters represent a re-
`interpretation of the original published results by the indicated researchers and in all
`cases an improved curve-fit resulted. The two-stage rate constants k1 and k2 listed in
`
`4
`
`

`

`ULTRAVIOLET GERMICIDAL IRRADIATION
`
`253
`
`Figure 1. Survival curve of Streptococcus pyogenes showing two stages, based on data from Lidwell (1950).
`
`Table 2. Two stage parameters (based on re-evaluation of original data).
`
`Airborne
`microorganism
`
`Reference
`
`k standard
`(cm2=mJ)
`
`Two stage curve
`
`k1
`(cm2=mJ)
`
`Pop.
`( f )
`
`k2
`(cm2=mJ)
`
`Pop.
`(17f )
`
`Adenovirus Type 2
`Coxsackievirus B-1
`Coxsackievirus A-9
`Staphylococcus aureus
`Streptococcus pyogenes
`E. coli (Reference only)
`Serratia marcescens
`Bacillus anthracis spores
`
`Rainbow, 1973
`Hill, 1970
`Hill, 1970
`Sharp, 1939
`Lidwell, 1950
`Sharp, 1939
`Riley, 1972
`Knudson, 1986
`
`0.000047
`0.000202
`0.000159
`0.000886
`0.000616
`0.000927
`0.049900
`0.000031
`
`0.00005
`0.000248
`0.00016
`0.01702
`0.00287
`0.008098
`0.0757
`0.000042
`
`0.99986
`0.9807
`0.7378
`0.914
`0.8516
`0.9174
`0.712
`0.9984
`
`0.00778
`8.81E-05
`0.000125
`0.0091
`0.000167
`0.003947
`0.0292
`0.000006
`
`0.00014
`0.0193
`0.2622
`0.086
`0.1484
`0.0826
`0.288
`0.0016
`
`Table 2 are overall rate constants that apply only at the intensity shown, which is the UV
`irradiation measured or given in the original test.
`
`The shoulder. The initiation of exponential decay in response to UVGI exposure, or any
`other biocidal factor, is often delayed for a brief period of time (Cerf, 1977; Munakata
`et al., 1991; Pruitt and Kamau, 1993). Figure 2 shows the survival curve for Staphylo-
`coccus aureus, where a shoulder is evident from the fact that the regression line intercepts
`the y axis above unity. Shoulder curves typically start out horizontally before developing
`full exponential decay slope.
`
`As shown in Figure 3, the initial part of the decay curve has zero slope at time t ¼ 0 and
`exponential decay is not fully manifested until time td. The intersection of the horizontal line
`
`5
`
`

`

`254
`
`KOWALSKI ET AL.
`
`Figure 2. Survival curve of Staphylococcus aureus showing evidence of shoulder (Sharp, 1939).
`
`Figure 3. Development of shoulder curve, showing the effect of the time delay tc and relation to the tangent
`point d.
`
`6
`
`

`

`ULTRAVIOLET GERMICIDAL IRRADIATION
`
`255
`
`S ¼ 1 (100% Survival) at tc with the extension of the decay curve is known as the ‘‘quasi-
`threshold’’ in radiation biology (Casarett, 1968). The point td is tangent to both curves.
`The lag in response to the stimulus implies that either a threshold dose is necessary
`before measurable effects occur or that repair mechanisms actively deal with low-level
`damage (Casarett, 1968). The effect is species and intensity dependent. In many cases it
`can be neglected. However, for some species and sometimes for low intensity exposure, the
`shoulder can be significant and prolonged.
`Recovery due to growth during irradiation is assumed negligible and to be encompassed
`by the model—this should be at least partly true if the parameters are based on a broad
`range of empirical data. Recovery of spores, although not well understood, is recognized as
`a process associated with germination (Russell, 1982). The recovery of spores is, therefore,
`a self-limiting factor since a germinated spore invariably becomes less resistant to UVGI
`irradiation (Harm, 1980).
`An exponential decay curve with a shoulder will have an intercept greater than unity
`when the first stage rate constant is extrapolated to the y-axis. It is naturally assumed that a
`shoulder exhibited in the data is statistically significant and not an artifact of measurement
`uncertainty. Relative to a decay curve that intercepts at unity, the shouldered curve is
`shifted ahead by a time interval equal to tc, the quasi-threshold. The equation for the
`delayed single stage survival curve, when t  td is:
`Þ:
`ln SðtÞ ¼ kIðt tc
`
`ð4Þ
`
`The shoulder occurs during the time interval 0 < t < td. It is apparent that the shoulder
`portion is a non-linear function of ln S (see Figure 2). Insufficient data exist to precisely
`define the form of the relationship, but ln S cannot be simpler than a polynomial function
`of second order. The error resulting from this assumed mathematical relationship will be
`small as long as it provides a smooth transition between the horizontal and the delayed
`decay curve.
`Assuming a second order polynomial relationship between the dose (intensity times
`time) and ln S, we have:
`
`ln SðtÞ ¼ pðItÞ2
`
`ð5Þ
`
`where 0 < t < td, p ¼ a constant.
`between equations (4) and (5) at the tangent point t ¼ td. For any constant intensity I, the
`
`The constant p can be evaluated by requiring continuity through the first derivative
`
`slope of the exponential portion of the survival curve may be obtained by straightforward
`time differentiation of the right hand side of equation (4):
`
`ð6Þ
`
`ðln SÞ ¼ kI
`
`d d
`
`t
`
`7
`
`

`

`256
`
`KOWALSKI ET AL.
`
`Similarly, the slope of the shoulder curve is obtained by differentiation of the right hand
`side of equation (5):
`
`ð7Þ
`
`ðln SÞ ¼ 2pI 2t
`
`d d
`
`t
`
`The constant p is determined by equating (6) and (7) at time td:
`
`p ¼ k
`2Itd
`Substitution of this expression for p into equation (5) and equating (6) and (7) at t ¼ td
`
`ð8Þ
`
`yields the relation:
`
`¼ 2tc
`
`td
`
`ð9Þ
`
`Equation (9) is, in fact, a version of the result Appolonius of Perga arrived at in the 3rd
`century BC through lengthy geometry for the special case of ellipses, which are also
`described by second order polynomials (Elmer, 1989). The term p is now discarded, after
`substituting for equations (8) and (9), and equation (5) can be written in the form:
`
`ln S ¼ kI
`4tc
`
`t2
`
`ð10Þ
`
`In general, any data set describing single stage microbial decay can be easily fit to a
`
`single stage exponential decay curve. Normally, the y-intercept is fixed at S ¼ 1 when
`
`fitting data to a curve. If a shoulder is suspected, the constraint on the y-intercept should be
`removed and the coefficient of the exponential will then have some value greater than 1.
`This assumes, of course, that the shoulder is real and not a result of measurement
`uncertainty.
`The term Si, denotes the y-intercept of the shifted exponential portion of a survival curve
`with a shoulder, as shown in Figure 4. If Si is known, the value of tc can be determined by
`evaluating equation (4) at t ¼ 0:
`Þ
`¼ ln ðSi
`
`ð11Þ
`
`tc
`
`kI
`
`Note that this mathematical treatment of the shoulder requires transcending the dose
`term It since this must be separated into components. The dose may define a point on the
`shoulder but the intensity defines the shoulder itself. That is, the threshold tc is a function
`of the intensity only, not the dose. Furthermore, in two stage curves there is a separate
`shoulder for both stages, although the contribution due to the second stage (the resistant
`fraction) is typically small.
`
`8
`
`

`

`ULTRAVIOLET GERMICIDAL IRRADIATION
`
`257
`
`Illustration of generic shoulder model response to intensity, based on data for Aspergillus niger from
`Figure 4.
`UVDI (2000).
`
`The complete single stage survival curve can then be defined as the piecewise
`continuous function:
`
`8<
`ln SðtÞ ¼ kI
`:
`t2;
`kIðt tc
`
`4tc
`
`t  2tc
`t  2tc
`
`Þ;
`
`ð12Þ
`
`The time delay (the threshold tc) may approach zero at high intensities and it may be
`¼ 0 equation (12) reduces to equation (1).
`infinitely long for low intensities. For tc
`No studies exist that define the relationship between the threshold and the intensity but
`data from Riley and Kaufman (1972) suggests that a linear relation exists between
`intensity and the logarithm of the y-intercept Si. A theoretical basis can be found for
`the same relationship in the Arrhenius rate equation, which describes the influence of
`temperature or radiation on process rates as being that of simple exponential decay
`is an
`(Rohsenow and Hartnett, 1973). Assuming,
`therefore,
`that
`the threshold tc
`exponential function of the intensity I we can write:
`
`BI
`
`ð13Þ
`¼ Ae
`tc
`where A ¼ a constant defining the intercept at I ¼ 0, B ¼ a constant defining the slope of
`the plotted line of ln ðtc
`Þ vs. I.
`
`9
`
`

`

`258
`
`KOWALSKI ET AL.
`
`Given any two sets of data for tc and I, equation (13) can be used to determine the values
`of A and B. Prediction of tc for any arbitrary value of intensity I then becomes possible.
`Figure 4 shows hypothetical survival curves of spores subject to various intensities.
`The complete equation can be defined by combining equation (3) and equation (12),
`where a shoulder is considered to be present in both stages:
`SðtÞ ¼ f e
`k1It
`k2It
`0
`
`0 þ ð1 f Þe
`
`t  2tc
`t  2tc
`
`Þ;
`
`ð14Þ
`
`where
`
`t
`
`8<
`:
`0 ¼ t2
`ðt tc
`
`;
`
`4tc
`
`Parameters defining the shoulder characteristics of various microbes are summarized in
`Table 3. These were obtained by re-interpretation of the original published results by the
`indicated researchers and in all cases an improved curve-fit resulted. The parameters ‘A’
`and ‘B’, the threshold tc, and the intensity I cannot be established due to the paucity of data
`in the literature for different intensities.
`A few cases were found in the literature where the first stage intercept proved to be less
`than 1, which is probably due to experimental error and limited data sets. In all cases when
`shoulder parameters are evaluated, an error analysis should be performed to verify that the
`results defining the shoulder and second stage are meaningful.
`
`The multi-hit target model
`
`Alternate mathematical models have been proposed to account for the shoulder including
`the multi-hit model or multi-target model, recovery models, split-dose recovery models,
`
`Table 3. Shoulder parameters for classical and multi-hit models.
`
`Airborne microorganism
`(see Table 1 for
`References)
`
`Reference
`
`k standard
`(cm2=mJ)
`
`Classical model
`
`Multi-hit
`model (n)
`
`Intensity
`(mW=cm2)
`
`Intercept
`(Si)
`
`Threshold
`(tc)
`
`Reovirus Type 1
`Staphylococcus aureus
`
`Hill, 1970
`Sharp, 1939
`Gates, 1929
`Mycobacterium tuberculosis David, 1973
`Riley, 1961
`David, 1973
`Mycobacterium kansasii
`Mycobacterium avium-intra. David, 1973
`Mongold, 1992
`Haemophilus influenzae
`Abshire, 1981
`Pseudomonas aeruginosa
`Antopol, 1979
`Legionella pneumophila
`Rentschler, 1941
`Serratia marcescens
`Bacillus anthracis (mixed)
`Sharp, 1939
`Bacillus anthracis spores
`Knudson, 1986
`
`0.000132
`0.000886
`0.001184
`0.000987
`0.004720
`0.000364
`0.000406
`0.000599
`0.000640
`0.002503
`0.001225
`0.000509
`0.000031
`
`1160
`10
`110
`400
`85
`400
`400
`50
`100
`50
`1
`1
`90
`
`1.7237
`4.4246
`1.225
`2.6336
`1.7863
`6
`5.62
`1.0902
`1.3858
`1.288
`2.0824
`2.0806
`1.009
`
`3.1202
`87.38
`1.8432
`4.176
`3
`4.254
`9.227
`2.5703
`9.6818
`96.69
`190.5
`109.8
`215
`
`1.29
`4.92
`1.69
`2.34
`1.83
`6.11
`5.97
`1.18
`1.77
`1.67
`1.71
`2.63
`2.60
`
`10
`
`

`

`ULTRAVIOLET GERMICIDAL IRRADIATION
`
`259
`
`and empirical models (Russell, 1982; Harm, 1980; Casarett, 1968). The use of the
`multi-hit target model, for example, to determine shoulder characteristics is similar in
`form to the methods for the classical model (Anellis et al., 1965), and is addressed here for
`comparison purposes.
`
`The multi target model (Severin et al., 1983) can be written as follows:
`SðtÞ ¼ 1 1 e
`kIt
`
`ð15Þ
`
` n
`
`The parameter n represents the number of discrete critical sites that must be hit to
`inactivate the microorganism, and is unique for each species.
`In equation (15) the number of targets n must be unique to each population fraction in a
`two stage curve, since these behave as though they were independent. Therefore, by
`analogy to equation (14) we can write the complete two stage equation for the multi-hit
`model as follows:
`
`
`k1ItÞn1
`SðtÞ ¼ ð1 f Þ 1 ð1 e
`
`
`
` þ f 1 ð1 e
`
`
`
`k2ItÞn2
`
`ð16Þ
`
`In equation (16), n1 represents the number of targets for the species in population 1, the
`fast decay population, while n2 represents the number of targets in the resistant fraction.
`Figure 5 shows a comparison of shoulder curves generated by the classical model and the
`multi-hit model compared against test data on Staphylococcus aureus irradiated on petri
`dishes. The curves do not exactly coincide, but the question of which model is a more
`
`Figure 5. Comparison of classical shoulder model and multi-hit model with data for exposed plates of
`Staphylococcus aureus. Based on data from Sharp (1939) at an estimated test intensity of 1900 mW=cm2.
`
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`
`accurate predictor is indeterminate due to experimental error. It remains for future research
`to determine which model is a more accurate predictor of shoulder curves. Either model
`should suffice for basic analysis and design purposes.
`Table 3 includes the value of n, the number of targets, for the multi-hit model which
`have been derived from the original test data. These can be used to generate a single stage
`shoulder curve similar to the one for the listed shoulder parameters. In all cases the multi-
`hit model curve does not exactly coincide with the one produced by the classical model,
`yet the error is quite small.
`
`3. Modeling the UV Dose
`
`Two approaches can be taken to define the complete three-dimensional (3D) intensity field
`in any experimental apparatus involving airflow—measurement and calculation. Photo-
`sensors can provide a profile of the field but they have inherent problems in the near field
`(Severin and Roessler, 1998) and have difficulties when used inside reflective enclosures.
`The question of whether photosensors can be used to measure the fluence rate that an
`airborne microbe actually experiences is an unresolved one. Recent advances in the use of
`spherical actinometry (Rahn et al., 1999) may provide more realistic results since these
`sensors more closely resemble spherical microbes.
`The problems of photosensing and data interpretation can be avoided through analytical
`determination of the 3D intensity field. The use of radiation view factors to define the 3D
`intensity field for both the lamp and internal reflective surfaces has been detailed by
`Kowalski and Bahnfleth (2000) and is summarized here.
`Various models of the intensity field due to UV lamps have been proposed in the past,
`including point source, line source, integrated line source, and other models (Jacob and
`Dranoff, 1970; Qualls and Johnson, 1983; Beggs et al., 2000). The model used here is
`based on thermal radiation view factors (Modest, 1993), which define the amount of
`diffuse radiation transmitted from one surface to another.
`Figure 6 illustrates a lamp modeled as a cylinder where the planar area at which the UV
`intensity is to be determined is perpendicular to the axis and is at the edge of the cylinder.
`The fraction of radiative intensity that
`leaves the cylindrical body and arrives at a
`
`differential area (Modest, 1993) is:
`Lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
`p
`H 2 1
`þ X 2Hffiffiffiffiffiffiffi
`p
`XY
`
`
`
`ATAN
`
`1 L
`
`266664
`
`F ¼ L
`pH
`
`ð17Þ
`
`377775
`
` ATANðMÞ
`!
`r
`ffiffiffiffi
`
`X Y
`
`ATAN M
`
`The parameters in equation (17) are defined as follows:
`
`L ¼ l=r
`H ¼ x=r
`Y ¼ ð1 HÞ2 þ L2
`
`r
`ffiffiffiffiffiffiffiffiffiffiffiffiffi
`X ¼ ð1 þ HÞ2 þ L2
`H 1
`M ¼
`H þ 1
`
`12
`
`

`

`ULTRAVIOLET GERMICIDAL IRRADIATION
`
`261
`
`Figure 6. View factor geometry for computing the intensity at a point some distance from the axis of a lamp
`modeled as a radiating cylinder.
`
`where l ¼ length of the lamp segment (arclength, cm), x ¼ distance from the lamp (cm),
`r ¼ radius of the lamp (cm).
`
`This equation applies to a differential element located at the edge of the lamp segment. In
`order to compute the view factor at any point along a lamp it must be divided into two
`segments. Equation (17) can be used to compute the intensity at any point beyond the ends of
`the lamp by applying it twice—once to compute the view factor for an imaginary lamp of the
`total length (distance between some point and the far end of the lamp) and then subtracting
`the view factor of the non-existent portion, or ghost portion. This method, known as view
`factor algebra, is detailed in Kowalski and Bahnfleth (2000) and elsewhere (Modest, 1993).
`Implicit in the use of this view factor is the spherical microbe assumption, or the
`assumption that microbes are spherical. In the view factor model, the cross-sectional area
`of a sphere is a flat disc that remains perpendicular to a line passing through the lamp axis,
`as shown in Figure 7. A source of error in this assumption is due to the fact that light rays
`coming from other parts of the lamp, as illustrated in the figure, are not always
`perpendicular to the disc surface. Analysis by the authors using a view factor model in
`
`Figure 7. Modeling of a spherical microbe as a flat disc (the cross-section of a sphere) that always faces the
`lamp axis.
`
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`
`which the intensity has been corrected for the cosines of the angles from non-perpendi-
`cular rays has established that this difference is quite small and can be neglected in most
`cases. This is due to the fact that when the disc element is close to the lamp surface the
`nearest sections of the lamp dominate the intensity field, while at large distances the
`cosines become small.
`The intensity field as a function of distance from the lamp axis is simply the product of
`the surface intensity and the view factor, where the surface intensity is computed by
`dividing the UV power output by the surface area of the lamp:
`
`I ¼ Euv
`Ftotal
`2prl
`where Euv ¼ UV power output of lamp, mW.
`
`ð18Þ
`
`Figure 8 illustrates photosensor data on a UV lamp compared against the predictions of
`the view factor model and a line source model. The line source model used in this figure is
`based on Beggs et al. (2000) and appears to underpredict the intensity field. Data from a
`number of other lamp models were reviewed and in all cases good agreement with the view
`factor model was obtained while other line source models typically deviate from perfect
`agreement with photosensor data (Kowalski and Bahnfleth, 2000).
`UV lamps have published ratings based on measurements taken at 1 m from the
`midpoint of the lamp axis (IES, 1981). The rated intensity for any tubular lamp can be
`computed by using equation (17) with two equal lamp segments. Figure 9 shows the view
`
`Figure 8. Comparison of view factor model and line source model predictions at the midpoint of typical UV
`lamps with photosensor data.
`
`14
`
`

`

`ULTRAVIOLET GERMICIDAL IRRADIATION
`
`263
`
`factor model predictions of lamp ratings for almost one hundred commercially available
`UV lamps. Shown also are the predictions for a line source model, based on Beggs et al.
`(2000), in which it can be observed that good agreement is obtained for small lamps but
`that deviations progressively occur for larger lamps.
`Although the line source models tend to underpredict the intensity field, it should be
`noted that water-based disinfection with UVGI depends more heavily on near-field
`absorbance, and for such applications these models are capable of providing adequate
`results (Suidan and Severin, 1986). Air disinfection involves much larger distances and
`dimensions than water disinfection and so the accuracy of the model at these distances
`becomes more of a factor.
`
`In a rectangular duct, each of the four
`UV intensity field due to enclosure reflectivity.
`walls will reflect a fraction of the incident intensity it experiences at the surface. If the
`UV reflectivity is 75%, then 75% of the UV intensity that occurs at the surface (due to
`the UV lamp) will be reflected back into the enclosed space. The intensity at each
`surface is computed using equation (17), after which the intensity field due to the
`radiating flat rectangular surface can be computed by the following view factor equation
`(Modest, 1993):
`
`
`
`ATAN
`
`ATAN
`
`X
`
`p
`
`26664
`
`¼ 1
`2p
`
`Fh
`
`ð19Þ
`
`37775
`
`
`Yffiffiffiffiffiffiffiffiffiffiffiffiffiffi
`Xffiffiffiffiffiffiffiffiffiffiffiffiffiffi
`p
`
`
`1 þ X 2
`1 þ X 2
`Yffiffiffiffiffiffiffiffiffiffiffiffiffiffi
`p
`þ
`1 þ Y 2
`1 þ Y 2
`where X ¼ Height=x, Y ¼ Length=x, x ¼ perpendicular distance to the wall.
`
`Figure 9. Comparison of measured UV lamp ratings with predictions of the view factor model and a line source
`model (Beggs et al., 2000). The straight line (SL) represents perfect predictions.
`
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`
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`

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`
`This view factor is consistent with the previously stated spherical microbe assumption in
`that equation (19) is applied to all four walls as if the microbe faced each one individually.
`The intensity field due to each surface is then summed to obtain the total reflected intensity
`field. The error due to the fact that some portions of the rectangular surface are not
`perpendicular to the flat disc element that represents the spherical microbe is negligible for
`the same reasons discussed previously. In addition, the distances to the walls are, in general
`greater, and the surface intensities lower, rendering the error still more negligible.
`Subsequent
`reflections, called inter-reflections, can be accounted for with more
`sophisticated computational methods (Kowalski and Bahnfleth, 2000; Kowalski, 2001),
`but equation (19) may suffice as a reasonable first order approximation if the material has
`low reflectivity.
`
`Rate constant determination. Previously, no analytical method existed to enable
`researchers to accurately evaluate the intensity field and assess the dose received by
`airborne microbes in experimental setups. As a result, airborne rate constants could not
`easily be determined. In some airborne experiments the airstream was confined to an area
`of known intensity by directing airflow through a narrow slot (Riley and Kaufman, 1972).
`Given the methods and analytical tools described above, airborne rate constants can now be
`determined with improved accuracy. As mentioned previously, rate constants depend on the
`test apparatus and measurement methods and so are relative to the experiment from which
`they come. The analytical methods presented here enable easy resolution of the intensity field
`and can readily predict rate constants, but with the caveat that these rate constants are, in turn,
`dependent on these methods. That is to say, an airborne rate constant determined with the
`present methods can be used for predictive purposes, but only if these methods are also used
`for the predictions. Otherwise, what certainty there is in the prediction would be diminished.
`
`4. Air Mixing Effects
`
`In long ducts the velocity profile of a laminar airstream will approach a parabolic shape,
`with the velocity higher towards the center. However, fully developed laminar velocity
`profiles are unlikely to be achieved in laboratory or real-world installations. The design
`velocity of a typical UVGI system is about 2.54 m=s (500 fpm), producing a Reynolds
`number of approximately 150,000. Turbulent mixing is therefore more likely to be the
`norm. Even laminar flow involves mixing by diffusion and so actual operating conditions
`will lie somewhere between complete mixing and the idealized condition of completely
`unmixed flow. Real world conditions tend to approach those of complete mixing, and such
`was shown by Sever