# Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment
## 2 Model Development
### Dispersion Parameters and Relative Concentration Intensity  
According to the hypotheses on which the Fluctuating Plume is based, the relationships between the parameters describing the absolute dispersion of the plume (\(y\), \(z\)), those that describe its relative dispersion (\(y_{\rm r}\), \(z_{\rm r}\)) and the parameters that describe the meandering of the centroid of the instant plume (\(y_{\rm m}\), \(z_{\rm m}\)) are  
\[\sigma_{y}^{2}\ =\ \sigma_{ym}^{2}+\sigma_{yr}^{2} \tag{22}\]  
\[\sigma_{z}^{2}\ =\ \sigma_{zm}^{2}+\sigma_{zr}^{2} \tag{23}\]  
Referring to [PERSON] et al. (2015) [54] for details and the Bibliography considered therein, we have that  
* given the flight time \(t\) (\(t=x/U_{\rm m}\)), the parameters \(\sigma_{v}\) and \(\sigma_{w}\), the Eulerian standard deviations of the transverse and vertical component of motion, can be obtained from the [PERSON] theory of dispersion [85] as \[\sigma_{y}^{2}\ =\ \frac{\sigma_{0}^{2}}{6}+2\sigma_{v}^{2}T_{Lv}\bigg{\{}t- T_{Lv}\bigg{[}1-exp\bigg{(}-\frac{t}{T_{Lv}}\bigg{)}\bigg{]}\bigg{\}}\] (24) \[\sigma_{z}^{2}\ =\ \frac{\sigma_{0}^{2}}{6}+2\sigma_{w}^{2}T_{Lw} \bigg{\{}t-T_{Lw}\bigg{[}1-exp\bigg{(}-\frac{t}{T_{Lw}}\bigg{)}\bigg{]}\bigg{\}}\] (25) where \(T_{Lv}\ =\ \frac{2\sigma_{0}^{2}}{C\varrho\varepsilon}\) and \(T_{Lw}\ =\ \frac{2\sigma_{w}^{2}}{C\varrho\varepsilon}\) are the Lagrangian decorrelation time scales, \(\sigma_{0}\) is the characteristic source dimension (\(\sigma_{0}\) is calculated as \(\sigma_{0}\ =\ \frac{D}{215}\)[86], where \(D\) is the internal diameter of the stack), \(C_{0}\) = 4.5 is the Kolmogorov constant and, finally, \(\varepsilon\) is the mean turbulent kinetic energy dissipation rate. * the longitudinal and vertical dispersion parameters for the relative dispersion can be expressed from the following relationships: \[\sigma_{yr}^{2}\ =\ \frac{\left(C_{r}/6\right)\cdot\varepsilon\cdot \left(t_{\rm s}+t\right)^{3}}{\bigg{\{}1+\Big{[}\frac{\left(C_{r}/6\right) \cdot\varepsilon\cdot\varepsilon^{2}}{2\sigma_{v}^{2}T_{Lv}}\Big{]}^{2/5} \bigg{\}}^{5/2}}\cdot exp\left[-\left(\frac{t}{T_{my}}\right)^{2}\right]+ \sigma_{y}^{2}\bigg{\{}1-exp\bigg{[}-\left(\frac{t}{T_{my}}\right)^{2}\bigg{]} \bigg{\}}\] (26) \[\sigma_{zr}^{2}\ =\ \frac{\left(C_{r}/6\right)\cdot\varepsilon\cdot \left(t_{\rm s}+t\right)^{3}}{\bigg{\{}1+\Big{[}\frac{\left(C_{r}/6\right) \cdot\varepsilon\cdot\varepsilon^{2}}{2\sigma_{v}^{2}T_{Lw}}\Big{]}^{2/5} \bigg{\}}^{5/2}}\cdot exp\left[-\left(\frac{t}{T_{mz}}\right)^{2}\right]+ \sigma_{z}^{2}\bigg{\{}1-exp\bigg{[}-\left(\frac{t}{T_{mz}}\right)^{2}\bigg{]} \bigg{\}}\] (27) with the characteristic times \(T_{my}\ =\ \alpha T_{Lv}\) and \(T_{mz}\ =\ \alpha T_{Lw}\) considering a constant value of \(\alpha\) equal to two. Taking the expression for \(\sigma^{2}_{yr}\) as example, it must respect two asymptotic conditions [51; 87]. The first one holds when \(t\)\(\rightarrow\)0; in this case, \(\sigma_{yr}\) takes the asymptotic form: \[t\to 0\ \ \ \ \ \ \ \ \ \sigma_{yr}^{2}\ =\ \left(\frac{C_{r}}{6}\right) \cdot\varepsilon\cdot\left(t+t_{\rm s}\right)^{3}\] (28) where \(C_{r}\) is the Richardson-Obukhov constant equal to 0.8 and \(t_{\rm s}\) is the time necessary for the plume to expand until it reaches dimension \(\sigma_{0}\), so \(t_{\rm s}\ =\ \big{[}\sigma_{0}^{2}/\left(C_{r}\varepsilon\right)\big{]}^{1/3}\)([PERSON] and [PERSON], 2007). The second one holds when \(t\)\(\rightarrow\)\(\infty\) and in this case: \[t\rightarrow\infty\ \ \ \ \ \ \ \ \ \ \sigma_{yr}^{2}\ =\ 2\cdot\sigma_{v}^{2}\cdot T_{Lv}\] (29) The intensity of the relative concentration \(i_{\rm cr}\) is a key parameter to fully define the Gamma probability distribution that characterizes the relative dispersion of the instantaneous plume, and therefore to actually obtain the various moments (including the average) through the relationship (14). However, this parameter has rarely been measured experimentally or numerically. Only few works available on the subject [55, 88].