# Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment
## 2 Model Development
### Peak Concentration  
Obtaining direct information about the concentration PDF is currently only possible by using extremely complex modelling techniques such as DNS (Direct Numerical Simulation) [71, 72] and LES (Large Eddy Simulation) [73, 74] modelling. However, these techniques, together with the complex techniques known as PDF methods [11], can only be applied to limited spatial domains and time intervals, far from the needs of applications of practical interest. When, therefore, one needs to tackle the olfactory annoyance generatedby an industrial source to the surrounding population over a whole year, it is necessary to choose different methods. As it is well-known, full knowledge of the instantaneous concentration PDF would be achievable if its infinite relative statistical moments were all known. Realistically, only the budget equations for some of the statistical moments can be written and (numerically) integrated, and therefore it is necessary to introduce some empirical elements in the PDF identification process. In the Eulerian budget equation for a generic \(n^{-th}\) moment (e.g., the second one) there are terms dependent on the \(n+T^{-th}\) moment [8], and therefore also simple integration of this equation needs the introduction of appropriate closures that parameterize these terms. Normally, when the interest is on the PDF of instantaneous concentrations \(c\), as primary closure the hypothesis of a known a priori PDF, statistically describing \(c\), is introduced. For example, if the PDF is a two-parameter distribution, it will be sufficient to know the field of the mean value and the concentration variance at each point of the calculation domain and at each moment. Different studies [41; 46; 63; 70; 75; 76; 77; 78] tackled the problem of identifying the type of PDF on the basis of various experimental measurements carried out in laboratory studies and in the atmosphere, but without reaching a decisive conclusion. The most common are the exponential, the clipped normal, Log-normal, Gamma, and Weibull. The Log-Normal is simple, but it is the stiffest [79]. In other works [80; 81], exponential and clipped normal PDF are also supported. Currently, two of the most considered PDFs were used to calculate the peak concentration: the Gamma [24] and the Modified Weibull [82; 83]. The PDF Gamma [84] is very complex, but it is the most adaptable: due to this, it is probably the most suitable for dealing with the statistics of concentration fluctuations. Anyway, what emerges is that the actual PDF tends to be exponential near the emission source, while at great distances downwind its shape is much more symmetrical, therefore similar to a Log-Normal. Interesting are the observations made by [PERSON] et al. (2015) [55] analyzing the numerous measurements obtained in the wind tunnel. Authors state, \"these findings clearly support the existence of a universal function for the PDF of the concentration.\" Moreover, they show how the measures suggest the adoption, at least in first approximation, of a Gamma PDF type, precisely for its adaptability. It is worthy to remark, however, that also the Weibull distribution has similar characteristics. Known at (\(x\),\(y\)) the mean concentration \(\overline{C}\), its variance \(\sigma^{2}{}_{c}\) and so the concentration intensity \(i_{c}\), adopting as PDF a Gamma distribution or a Weibull distribution, it is then possible to obtain the \(n^{th}\) percentile concentration \(C_{n}\). According to the German legislation [36], the percentile of interest is the 90 th: in this way, the 90 th percentile of the sub-hourly concentration, \(C_{90}\), can be considered as the peak concentration, \(C_{p}\). Accordingly, the peak-to-mean can be defined as \(R_{90}~{}=~{}\frac{C_{p}}{\overline{C}}\). Even though the Gamma distribution seems to be the most realistic distribution, the fact that it does not allow the obtainment of the \(R_{90}\) in analytical form makes its practical use complex.