Ultraviolet Disinfection



Ultraviolet Disinfection


Ernest R. Blatchley III

Thomas P. Coohill



Downes and Blunt1,2,3 conducted a series of logical, ordered experiments to examine the behavior of bacteria and fungi when exposed to sunlight. Their experiments, which were conducted in the absence of many of the analytical tools that are common among academic and industrial laboratories today, demonstrated several interesting conclusions:



  • (Sun)light is detrimental to the development of bacterial and fungal communities.


  • The extent of exposure to sunlight appears to be related to the extent to which growth of these microbial communities is suppressed.


  • The actinic (ie, ultraviolet [UV]) portion of the solar spectrum is primarily responsible for this behavior.

This classic work may be viewed as the first demonstration of the effects of UV radiation on microbial communities. In the time since these basic discoveries were reported, the effects of UV radiation on microorganisms and microbial communities have been elaborated in detail. Today, UV-based disinfection systems are commonly applied in many settings. Numerous important contributions to our current understanding of the mechanisms of action of UV radiation on microorganisms and critical biomolecules were presented in the 1950s and 1960s. Subsequent expressions of concern with common forms of disinfection, such as chlorination, promoted research to incorporate these findings with developments of fundamental photochemical reactor theory, which in turn facilitated rapid increases in the usage of UV radiation for disinfection of water after roughly 1980. Today, UV disinfection is among the most commonly applied disinfectants for water; it is also being used for disinfection and preservation of solids, surfaces, and air. The goal of this chapter is to describe the fundamental concepts that are needed to understand the design, behavior, and validation of UV-based microbial inactivation systems used in water treatment. Similar principles apply with applications of UV disinfection to other media. Because the process relies on photochemistry, the basic principles of photochemistry are presented first, followed by a development of some of the practical issues related to UV-based microbial inactivation processes.


BASIC PRINCIPLES OF PHOTOCHEMISTRY

Photochemical processes are initiated by absorption of electromagnetic radiation. Two basic laws of photochemistry have been presented. The first law of photochemistry, also known as the Grotthus-Draper law, states that light (electromagnetic radiation) must be absorbed for a photochemical reaction to take place. The second law of photochemistry, also known as the Stark-Einstein law, states that for each photon of radiation absorbed, only one molecule will be excited. The second law also forms the basis for the concept of a quantum yield (Φ), which defines the ratio of molecules that react to photons absorbed. In turn, this allows for assignment of 1:1 equivalence between photons and molecules. The laws of photochemistry also indicate that two conditions must be met for a photochemical reaction to take place. First, a photon must be absorbed by a target molecule, or in some cases by a neighboring molecule, which may act as a photosensitizer. Second, the absorbed photon must have sufficient energy to break an existing bond or form a new one. If these two conditions are met, it is possible, although not certain, that a photochemical reaction may take place. If either condition is not met, a photochemical reaction will not take place.



Photon Energy

The German physicist Max Planck is credited with the development of the basic mathematical expression used to define photon energy:


Where:

E = photon energy

h = Planck constant

c = speed of light

λ = wavelength

Because reacting molecules and photons normally react on a 1:1 basis (as described by the Stark-Einstein law), it is often useful to express Planck law in an alternative form (note that 1 einstein is defined as being equal to Avogadro’s number of photons):


Where: QE = energy per einstein

A = Avogadro’s number

Much in the same way that molecules (or atoms) may be viewed as the smallest discrete units of a chemical compound, the photon may be viewed as the smallest discrete unit of radiation energy. Across the electromagnetic spectrum, there exists a wide range of wavelengths and corresponding photon energies (Figure 9.1). The focus of this chapter is on photochemical reactions that take place within biomolecules. It is impractical to develop a comprehensive summary of bond energies associated with all moieties that are relevant in these molecules because of the dependence on molecular substituent groups and other factors. It is possible, however, to summarize representative values of the bond energies that characterize some of the common bonds in biological systems (see Figure 9.1). From this summary, it is evident that the bond energies of interest in microbial molecules are generally coincident with photon energies in the UV portion of the spectrum. Radiation with wavelengths less than approximately 320 nm is often sufficiently energetic to promote reactions in biomolecules. Radiation in the vacuum UV (VUV) portion of the spectrum is highly energetic but also tends to be absorbed strongly. Because of this, VUV radiation does not penetrate far into liquid or solid media; it is strongly absorbed by molecular oxygen (O2), which limits its ability to be transmitted through gases. As such, most UV photoreactors are based on radiation in the UVB and UVC portions of the electromagnetic spectrum. UV radiation is defined as the portion of the electromagnetic spectrum that lies between visible light (ie, electromagnetic radiation that can be detected by a healthy human eye) and X-radiation. The UV spectrum is further subdivided into several subregions. Table 9.1 provides a summary of wavelength cutoffs for these regions (see Figure 9.1) as well as distinguishing features of each wavelength range.






FIGURE 9.1 Photon energy (QE) as a function of wavelength. Superimposed on this figure are the wavelength ranges that define vacuum ultraviolet (UV), UVC, UVB, UVA, and visible radiation. Also included are typical bond energies (ΔH) for a number of bonds that are relevant to UV-based applications in water treatment and disinfection, as well as their corresponding wavelengths.

Although the preceding arguments suggest that essentially all photons within the UV range should be capable of promoting photobiochemical reactions, photons at wavelengths of less than 320 nm tend to be the most effective for this purpose. Solar UV disinfection processes are strongly influenced by the effects of UVA radiation, largely by mechanisms that involve heat and the formation of O2-based reactive intermediates. For some solar UV disinfection systems, incorporation of materials that allow for inclusion of UVB radiation can increase the rate of microbial inactivation; however, the vast majority of contemporary (nonsolar) UV disinfection systems employ “artificial” sources of UVC radiation to accomplish microbial inactivation.


Photochemical Kinetics

The rate of a photochemical reaction is directly related to the rate at which the target molecule absorbs incident radiation. For the case of monochromatic radiation, the rate of a photochemical process can be described as4:


Where:

r(λ) = rate of photochemical reaction

[B] = molar concentration (activity) of absorbing (reacting) compound

t = time

Ia(λ) = volumetric rate of photon (λ) absorption by the target molecule

Φ(λ) = quantum yield









TABLE 9.1 Summary of wavelength regions within the ultraviolet spectrum and some corresponding characteristics

























Type of Radiation


Nominal Wavelength Range (nm)


Characteristics


UVA


320-400


Comprises >99% of ambient solar UV radiation at sea level near mid-latitudes; promotes microbial inactivation through heat and formation of O2-based reactive intermediates


UVB


280-320


Comprises roughly 1% of ambient solar radiation at sea level near mid-latitudes; promotes microbial inactivation through photochemical damage to nucleic acids; promotes skin cancer and damage to other external tissues


UVC


200-280


Absent from ambient solar spectrum at sea level near mid-latitudes because of absorption by stratospheric ozone; must be generated locally to the application; causes damage to nucleic acids, proteins, and other biomolecules


Vacuum UV


10-200


Strongly absorbed by gas-phase O2 and to a lesser degree N2; highly energetic radiation that can be used to photolyze H2O and many other molecules that are normally stable


Abbreviations: H2O, water; N2, dinitrogen; O2, molecular oxygen; UV, ultraviolet.


Parenthetical terms (λ) are included in equation (3) to emphasize that these attributes of photochemical reactions may display wavelength dependence. From this point forward, the parenthetical terms will be dropped in the interest of simplifying the resulting mathematical expressions.

The volumetric rate of photon absorbance depends on the geometry and optical characteristics of the reaction vessel in which the reaction takes place. For highly transparent conditions, or for the situation where we examine the rate of a photochemical reaction at a point in a system, equation (3) reduces to the following:


Where:

E0 = local fluence rate

ε = molar absorption coefficient

QE = energy per einstein

Equation (4) also indicates that the local rate of a photochemical reaction is proportional to the local fluence rate. As described in the following text, the fluence rate within a photochemical reactor often displays strong spatial gradients; as such, we observe equally strong spatial gradients in local photochemical reaction rates in these systems. Because photochemical reactions tend to be rapid, mixing behavior within photochemical reactors tends to have a strong influence on reactor performance.

In the case of UV disinfection, wherein the photochemical target is one or more biomolecules within a microbe, the physical interpretations of ε and [B] become nebulous. Moreover, data to describe these parameters in a disinfection context are rarely available. Because of this, the common practice is to resort to a mathematical expression to describe microbial inactivation kinetics (or UV dose-response behavior) based on the same logic that is used to describe the kinetics of purely photochemical reactions. Equation (5) illustrates the most basic form of a UV disinfection kinetic model:


Where:

N = concentration of viable microbial target organisms

k = inactivation rate constant

In effect, the inactivation constant (k) is a combined parameter that accounts for efficiency of photon absorbance (analog of ε), the efficiency of absorbed photon use to inactivate the microbial target (analog of Φ), and the energy of an einstein of photons (analog of QE). This modeling approach implicitly assumes that a 1:1 correspondence exists between a photochemical event and inactivation of an individual microbial target. In other words, this model suggests that a single photochemical event leads to inactivation of the microbial target. This approach is valid for some simple organisms, over a limited range of conditions. As will be demonstrated later, this simple model fails to describe the intrinsic kinetics of all UV disinfection processes; however, it does provide a starting point for a discussion of this topic.

Because equations (4) and (5) describe photochemical reaction rates at any point in space, they are broadly applicable for description of the behavior of photochemical reactor systems. Their application requires detailed information regarding the spatial distribution of radiant energy (ie, the fluence rate field) within a photochemical reactor system; for many UV photoreactors, additional information is also required to develop quantitatively accurate predictions or reactor behavior.



SOURCES OF ULTRAVIOLET RADIATION

UV radiation can be generated from natural and manufactured sources. Although the source of radiation does not influence the effects of photons that it generates, it is useful to understand the basic physics of radiation generation by these sources as well as their corresponding output spectra.

The sun represents perhaps the most basic source of UV radiation.5 The sun emits radiation as a result of incandescence, a process by which any hot object emits radiation. Planck’s law can be used to simulate the spectrum of radiation emitted from a blackbody at any given temperature; blackbody emission represents the maximum possible emission spectrum for a surface at a given temperature. The spectrum of solar radiation received outside the earth’s atmosphere has been measured by satellite-based instruments and high-altitude aircraft. The spectrum of solar radiation received at earth’s surface is influenced by time of day, time of year, latitude, and altitude, as well as atmospheric composition. Numerical models have been developed to simulate these effects and to allow simulation of ambient solar spectra at essentially any time and location. One model that have been developed for this purpose is the Simple Model of the Atmospheric Radiative Transfer of Sunshine (SMARTS).6

Figure 9.2 provides a summary of several reference solar spectra. Included in this figure are two spectra that are used to describe solar radiation that is imposed on earth’s atmosphere. The term air mass zero is used to describe these spectra. Air mass describes the path length of light through earth’s atmosphere relative to the path length that would be involved for radiation imposed at a zenith angle of zero (ie, perpendicular to earth’s surface).9 Therefore, air mass zero corresponds to solar radiation that has not yet encountered earth’s atmosphere (ie, the spectrum of radiation imposed on earth’s atmosphere). Another reference spectrum is included for air mass 1.5, which describes the spectrum of solar radiation imposed at an angle of 37 degrees. Also included in Figure 9.2 are spectra that were simulated using Planck’s law for blackbodies at 5777 K and 6000 K. These two temperatures span the nominal estimates of the surface temperature on the sun.






FIGURE 9.2 Reference solar spectra. Air mass zero spectra define radiation that is incident on the outside of earth’s atmosphere.7 The air mass 1.5 spectrum corresponds to solar radiation received at sea level imposed on earth’s atmosphere at a zenith angle of 37 degrees.8 The black-body spectra are presented for temperature estimates that have been used to describe the surface of the sun. For a color version of this art, please consult the eBook.

Solar radiation received at earth’s surface is substantially attenuated as compared to the air mass zero spectrum largely because of absorption by constituents of the atmosphere. Atmospheric ozone is a particularly strong absorber of UV radiation. The shortest wavelength of solar radiation received at near sea level elevations is roughly 290 nm. Earth’s atmosphere is essentially opaque to radiation at wavelengths shorter than 290 nm. This means that ambient solar radiation received at earth’s surface will include UVA and UVB radiation but essentially no UVC or VUV radiation. Although UVA and UVB radiation can be used to inactivate microbial pathogens, the most effective wavelengths for disinfection are found in the UVC range. Therefore, systems that employ UVC radiation for disinfection must rely on “artificial” sources of UV radiation. The most common artificial sources of UV radiation are mercury (Hg) lamps, of which there are two basic types: low pressure (LP) and medium pressure (MP).10 Hg lamps operate by striking an electrical arc across a space that contains Hg and an inert carrier gas, usually argon. Collisions between electrons and the atoms (or molecules) that occupy the space between the lamp electrodes result in electronic excitation of those atoms; electronically excited argon can transfer some of its energy to a nearby Hg atom. Relaxation of electronically excited Hg atoms to lower energy states results in photon emission. In addition to conventional LP lamps, there are also lamps that are based on an Hg amalgam rather than pure metallic Hg. The Hg amalgam lamps are characterized by essentially the same output spectrum as conventional LP Hg lamps (Figure 9.3), but they can generate more power than their conventional LP Hg lamp analogs. For practical purposes, LP Hg lamps are often considered to be monochromatic (λ = 253.7 nm) sources of UV radiation; however, LP Hg lamps are also characterized by several other, smaller lines in their output spectra, including a line at 185 nm and a few lines in the visible range that are responsible for the pale blue color that is evident when these lamps are illuminated.

The output spectra of MP Hg lamps are considerably more complex than those of LP Hg lamps. The MP lamp output spectra are characterized by numerous discrete lines superimposed on a continuum. The MP lamps tend to be much less efficient than LP Hg lamps in terms of
conversion of electrical power (input) into UV photons (output). The MP lamps can be much more powerful than LP lamps, and as such, they can be used to develop systems that are characterized by a smaller footprint and less complexity than equivalent systems based on LP Hg lamp technology. A normalized MP lamp spectrum is included in Figure 9.3. The spectra presented for LP (amalgam) and MP Hg lamps were normalized against the maximum values that were measured for each lamp type. Normalization was applied because the output spectrum measured for any lamp will depend on the location of the measuring device relative to the source as well as the composition of the media located between the source and the measurement location.






FIGURE 9.3 Normalized output spectra of Hg lamps. For both lamp types, spectra are normalized to the maximum value measured in the spectrum. Abbreviation: MP, medium pressure. Data courtesy of Trojan Technologies. For a color version of this art, please consult the eBook.

Historically, Hg lamps have been viewed as the default option for generation of germicidal UV radiation. However, the inclusion of Hg in these devices presents safety risks that motivate the development of alternative sources of UV radiation. From the regulatory perspective, some pressure to develop alternatives to Hg lamps can be found in the Minamata Convention on Mercury,11 an international treaty that was developed to protect human health and the environment from anthropogenic emissions of Hg. A similar piece of legislation in Europe has also resulted in restrictions on the use of Hg.12 As with the Minamata Convention, Hg-based lamps for general lighting are allowed, but there are restrictions on the mass of Hg that can be included in these lamps.

Excilamps represent a family of UV and VUV sources that include excimers (excited dimer) and exiplexes (exci ted complex). In both cases, a diatomic molecule or complex is electronically excited; relaxation back to a ground state results in photon emission at a characteristic wavelength. Excilamps are characterized by relatively narrow emission spectra, typically in the range of 2 to 30 nm at half maximum.10,13,14 Figure 9.4 illustrates the normalized spectral output of two common exciplex lamps: krypton chloride and xenon bromide. The output spectrum of an exciplex lamp can be adjusted through selection of the constituent gases that are used to form the exciplex. The most common configuration for these lamps is a so-called dielectric barrier discharge lamp,15,16,17 which usually is manufactured with a cylindrical geometry. Many other geometric configurations are possible with these lamps, including flat-panel displays. This geometric flexibility may provide opportunities to improve the optical characteristics of UV photoreactors.






FIGURE 9.4 Normalized output spectra of krypton chloride (KrCl) and xenon bromide (XeBr) exciplex lamps. For a color version of this art, please consult the eBook.

UV light-emitting diodes (LEDs) have emerged as important sources of antimicrobial UV radiation in recent years.18,19,20,21,22 Diodes are semiconductor materials that have been doped with chemicals to alter their ability to carry electrical current. Doping materials are added to semiconductors, such as silicon, to allow formation of charge carriers. A so-called p-n (positive-negative) junction is established by bringing into contact materials that have been doped to promote formation of positive charge carriers (holes) and negative charge carriers (electrons). Holes and electrons naturally migrate toward each other because of electrostatic attraction, but their migration can be influenced by application of electrical current. When applied properly, this current promotes migration of holes and electrons toward each other, resulting in their recombination. In turn, this causes the electron to experience a drop in energy, yielding a photon that corresponds to the bandgap of the dopant materials. Visible and infrared LEDs are commonly applied in electronic devices, including cell phones, televisions, room lighting, etc. The production of UV LEDs is possible through the use of group III nitrides as doping materials, especially including the aluminum-gallium nitride doping system. The spectral
output of these devices depends on the composition of the dopants. Figure 9.5 provides an illustration of the range of outputs observed among commercially available UV LEDs. Because of this, it is possible to select UV LEDs to have an output spectrum that is tailored to a given application.






FIGURE 9.5 Normalized output spectra of ultraviolet light-emitting diodes as a function of aluminum-gallium composition (data from AquiSense Technologies). Parenthetical entries in the legend indicate peak width (nm) at half maximum. For a color version of this art, please consult the eBook.

UV LEDs have a number of other important advantages relative to conventional sources of UV radiation, such as Hg lamps. They contain no Hg, which is relevant in the context of the Minamata Convention and general public pressure. The UV LEDs are compact and durable and allow for instant on/off switching. Packaging of LEDs (including UV LEDs) allows for inclusion of optical elements that can provide for directional irradiation. As such, UV LEDs can function as nearly point sources of radiation, which offers considerable flexibility in reactor design. On the other hand, it is also important to recognize the drawbacks of UV LEDs. At present, the efficiency of these devices is lower than Hg lamps, and their costs are higher. Historic trends in LED development have indicated that as applications for these devices emerge, production processes improve to yield increasingly efficient and powerful LEDs.


MICROBIAL INACTIVATION AND REPAIR


Mechanisms of Inactivation

The mechanisms of microbial inactivation depend on the range of wavelengths imposed on the microbes. As illustrated in Figures 9.2, 9.3, 9.4 and 9.5, the spectra of radiation from natural and artificial sources of UV radiation vary widely. Similarly, the mechanisms of inactivation that result from UV exposure also demonstrate considerable variability.

Exposure to solar UV radiation is governed by time of day, time of year, position of the target, materials that exist between the target and the radiation source (eg, shade), and numerous microenvironmental parameters; however, ambient solar radiation is limited to the UVA and UVB portions of the electromagnetic spectrum. UVA radiation causes microbial inactivation largely through the formation of reactive O2 species (ROS), which then go on to react with microbial constituents.23,24,25,26 UVB radiation is known to cause damage to nucleic acids, including DNA.27 In general, the rates of microbial inactivation resulting from exposure to ambient solar UVA and UVB radiation are slow but can be relevant in some situations. Most UV disinfection systems employ artificial sources of UV radiation, usually in the UVC range. UVC radiation can alter nucleic acids, proteins, and even cellular structures like membranes, which are heavily involved in respiration. But DNA is the major target chromophore in much of the region where it absorbs most heavily, 240 to 310 nm, typically with a peak at 260 nm. The DNA also has the fewest copies of any major macromolecule and is the genetic material.28,29

Until the 1940s, most biologists believed that genetic material was protein, which UV radiation can alter. But in the 1920s, Gates30,31,32 showed that the action spectra (see the following text) for both cell inactivation and mutation closely followed the absorption spectrum for nucleic acids in bacteria; his results were not readily accepted. Later, the photobiological community began to study UV effects on nucleic acids, predominantly DNA, and were surprised to discover that the genetic material was easily altered by UV radiation and then, even more surprisingly, that these alterations could sometimes be repaired. These studies helped form the basis for a new field of research known as molecular biology. DNA can be modified by absorption of UV photons in a variety of ways.33 Individual bases can be altered by deamination, ring cleavage, or other direct effects. For example, adjacent bases can be covalently linked by UV into a cyclopyrimidine dimer (CPD). The CPDs are the major photoproduct near the DNA absorption peak (260 nm). Covalent bonds are stable at physiological temperatures. UV radiation can also form 6 to 4 pyrimidine photo-adducts (6-4 PPs, also referred to as 6-4 photoproducts) at a lower rate (about 30% of CPD formation). The 6-4 PPs have an absorption maximum near 320 nm.34 DNA replication is altered by UV photoproducts; as few as one lesion in the form of a CPD or 6-4 PP can stop it.

The distribution of PPs varies with wavelength. In vegetative cells, dimers predominate in the UVC. After exposure to UVB, other PPs such as single strand breaks (SSBs) and DNA-protein cross-links begin to accumulate.
Although UV does not directly cause strand breaks in DNA, some of the mechanisms that repair UV PPs can. This damage is often overlooked but can have major effects on cell survival.35 In the UVA, SSBs may be the predominant lesions.33 UV radiation can also promote the formation of DNA photohydrates. If the cell contains certain photosensitizers, UVA and visible radiation can also produce indirect DNA effects where the photons are absorbed by other compounds which, in turn, damage DNA.36 Unique PPs have been discovered in some studied cells. For example, the spores of Bacillus subtilis (and some other sporulating bacteria) produce a unique “spore PP” (5,6-dihydrothymine) and have a unique spore product photolyase for photoreactivation (PR, see the following text). In addition, the spore coat protein can act as an effective shield to UV. Dormant spores cannot readily repair UV damage, but vegetative cells can.37 Whether the cell survives after germination to the vegetative form depends on how much UV damage it contains and whether the vegetative cell had enough time to repair that damage. Spores irradiated under wet conditions are several orders of magnitude more sensitive to UV than spores exposed to UV when dry.38 As described previously, UVA is also associated with the production of ROS and damage to a variety of other endogenous chromophores.

Because DNA is closely associated with proteins in the chromatin structure of cells or in virus structures, it is not surprising that these close but separate moieties can crosslink after UV exposure. Some cells also contain pigments that can act in a similar manner. The replicating portion of DNA is most sensitive to UV-induced cross-linking with proteins. These cross-links cannot undergo PR but can be cleared by postreplication repair. In addition, Dewar valence isomers are formed by the photoisomerization of 6-4 PPs at wavelengths longer than 290 nm (peak production at 320 nm). They can be repaired by the (6-4 PP) photolyase.34


Physiological Responses Other Than Inactivation

Jagger39 showed that irradiation of amoeba cytoplasm can damage cells. He further showed this was due mainly to PP formation in mitochondrial DNA. Singlet O2 and free radicals can also be produced in some chromophores by UVA, and they can act as photosensitizers by destroying cellular components including membranes.34 For many microbes, loss of viability or infectivity takes place at modest UV doses as compared to the doses of UVC radiation required to bring about other observable physiological changes. For example, Blatchley et al40 observed roughly 6.8 log10 units of inactivation of Escherichia coli at a UV254 dose of 100 mJ/cm2; at the same dose, respiratory/metabolic activity was reduced by only 0.64 log10 units. Similar results were observed with Streptococcus faecalis. Redford and Myers41 reported inactivation of the algal species Chlorella pyrenoidosa to be approximately six times faster than reductions in photosynthesis as a result of UV254 exposure. Effective inactivation of the protozoan parasites Cryptosporidium parvum and Giardia lamblia (and related species) has been observed at remarkably low doses of UVC radiation, whereas interruption of other parts of their life cycle requires much larger UVC doses.42,43,44,45,46,47 These observations suggest that UVC irradiation is effective for preventing replication of waterborne microbes, but actual death of the organisms will require UVC doses that are substantially larger than those needed to accomplish loss of the ability to reproduce or infect.


Disinfection Kinetics

As indicated by equation (5), the rate of photochemical damage to a molecule at a local level is first order with respect to the concentration of the target molecule and the local fluence rate. If it is assumed that a microorganism will be inactivated when it accumulates a single unit of photochemically induced damage, then the rate of microbial inactivation will also follow these same principles, as indicated previously in equation (5) (repeated here for clarity):


Rearrangement of equation (6) to allow separation of variables, followed by integration yields the following form:


Where:

τ = period of exposure (s)

image E0,i · dt = UV dose (mJ/cm2)

Equation (7) indicates that a plot of ln image as a function of UV dose should yield a straight line with a slope of -k. In practice, it is common to present graphs of this nature in the form of log10 image versus dose, probably because most people are more comfortable with base 10 logarithms than they are with natural logarithms. In either form, equation (7) implies a straight line that passes through the origin on a semilogarithmic plot. When data from an experiment are fit to equation (7), the slope of the best-fit line through the origin will have a value of image.

Equations (6) and (7) describe the rate of inactivation for a microbe that is inactivated as a result of a single unit of damage, the so-called single-event model. Figure 9.6 illustrates the shape of the UV dose-response behavior that is predicted by the single-event model. Some simple organisms (eg, some viruses) follow dose-response behavior
that can be accurately simulated by the single-event model, at least for limited extents of microbial inactivation. Common deviations from single-event behavior have also been observed, including a shoulder in the limit of low doses and lag or tailing behavior that is often observed at large doses (see Figure 9.6). Models have been developed to account for these deviations. The series-event model48,49 was developed based on the assumption that microbes within a population would accumulate damage in a serial manner, as follows:

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