e probability of the survival of a microorganism corresponds to the probability that the effective cross-section of the organism (a) escapes the incident photons. If N is the total number of incident photons over area A, then:

Or simply:

This corresponds to first order kinetics and is a typical representation of free microbe inactivation, where k is the inactivation constant and D is the ultraviolet dose.

An alternative picture for modeling the microbial inactivation is based on the presence of multiple "targets" in an organism. In this case, that is known as the multitarget model, all such targets must receive at least one hit for inactivation.14 Similar to the one-hit model, the inactivation of each target follows the negative exponential rule, therefore the probability of the inactivation of such an organism is:

Pr[inactivation] = Pr[1st target is hit] x Pr[2nd target is hit] x ••• x Pr[mth target is hit] = (1 - e-kD)(1 - e-kD) ■■■ (1 - e-kD) (18.3)

The probability of the survival of the organism in the multi-target model is:

In an alternative approach, the organism contains a single "target" that has to receive multiple "hits" before it is inactivated. This model is known as the multi-hit model14 or the series-event model.10 Both multi-target and multi-hit models successfully account for shouldered survival curves, but they do not predict the tailing phenomenon observed in wastewater disinfection processes.

A simple method to account for the tailing of dose-response curve is to consider the microbial population to consist of two subgroups.3 Both subgroups are inactivated in a one-hit fashion, but one is more resistant to ultraviolet irradiation than the other:

where p is the fraction of UV-resistant organisms (e.g., floc-associated microbes), and ki and k2(<k1) are the inactivation constants. This approach, known as the double-exponential model, predicts the tailing of dose-response curves, but it cannot create any "shoulder." To address this shortcoming, a simple variation of this model is suggested here, where the UV-sensitive subpopulation follows the multi-target model while the UV-resistant subgroup obeys the one-hit model:

A rigorous model to account for effects of flocs on the UV disinfection was proposed by Cairns et al.15 This approach considers the interaction of light with free microbes, floc size distribution, total number of microbial counts associated with flocs, and transmittance of the flocs to UV. Application of this model requires knowledge of size distribution of viable flocs. However, since such information is rarely available, this model has found limited use.

Most recently, Emerick et al.16 proposed that the inactivation of a microbial floc is controlled by the inactivation of a "critical" organism, and that the fraction of dose received by this organism is uniformly distributed. According to Emerick et al., flocs larger than a threshold diameter (about 20 microns) are not inactivated by ultraviolet irradiation. This model predicts that the survival rate at high doses of UV (D > 20) is inversely proportional to the UV dose, and cannot account for the shoulder.

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