**5. Validating analytical solutions for air pollution dispersion modeling: analyzing experimental data**

This section offers a comprehensive analysis of the data generated by our model, which is founded on the Deaves and Harris wind velocity profile. The model is constructed using two different formulations of vertical turbulent diffusivity. The first formulation, with *L*<sup>1</sup> >0, factors in both the downwind distance and the vertical height, while the second formulation, with *L*<sup>1</sup> ! 0, considers only the vertical height. The performance of the model, as presented in Eq. (24), is assessed and confirmed through the use of datasets from the Copenhagen diffusion experiments in Denmark and the Prairie Grass experiments in the USA.

Atmospheric dispersion experiments were conducted in Copenhagen, Denmark, from June 27 to November 9, 1978, to study air pollutant transport in the lower atmosphere. The tracer gas used in these experiments was sulfur hexafluoride (SF6), which was released from a tower located at a height of 115 meters (*Hs*), with a roughness length of 0.6 meters (*z*0). Additional details of the parameters used in these experiments can be found in the publications of Gryning and Lyck [19] and Gryning et al. [20].

The Prairie Grass experiments, carried out near O'Neil, Nebraska, between March and August 1956, were a groundbreaking series of studies on atmospheric pollution behavior and transport. These experiments measured the dispersion of sulfur dioxide (SO2) released from a 1.5 meter tower at five downwind distances from the source: 50, 100, 200, 400, and 800 meters. The roughness length of the area was 0.006 meters, which is critical in determining the spread of air pollutants. The experiments were extensively documented by Barad et al. [21] and Nieuwstadt et al. [22], providing essential initial insights into atmospheric dispersion processes.

Overall, the Copenhagen and Prairie Grass experiments provided essential knowledge on the transport and dispersion of air pollutants in the lower atmosphere. The Copenhagen experiments used an inert tracer gas to examine general dispersion patterns, while the Prairie Grass experiments focused on the downwind transport of SO2, providing dispersion data across a range of distances. Both experiments relied on the roughness length of the terrain, a crucial parameter in determining how pollutants spread near the ground. The findings of these landmark experiments made significant contributions to scientists' understanding of atmospheric pollution behavior, particularly in the PBL near the Earth's surface.

It should be noted that the computations were carried out using two distinct programming environments, namely MATLAB and Wolfram Mathematica. Specifically, the function fzero, which utilizes a combination of the bisection, secant, and

*A Novel Development in Three-Dimensional Analytical Solutions for Air Pollution… DOI: http://dx.doi.org/10.5772/intechopen.112225*

inverse quadratic interpolation methods, was defined and executed in the MATLAB environment. Conversely, the FindRoot function, which combines Brent's and Newton's methods along with a method that approximates the Jacobian, was defined and executed in the Wolfram Mathematica environment.

To implement the analytical solution in a practical sense, we have employed a discretization technique for the atmospheric boundary layer. Specifically, we have discretized the layer into two and four sublayers, respectively. In the case of two sublayers, the boundary layer height is discretized using the following approach: For the two sub-layers case, the boundary layer height is divided into two sub-layers, and the discretization is taken as follows: *dz*<sup>1</sup> <sup>¼</sup> <sup>7</sup> <sup>13</sup> ð Þ *Hmix* � *<sup>z</sup>*<sup>0</sup> and *dz*<sup>2</sup> <sup>¼</sup> <sup>6</sup> <sup>13</sup> ð Þ *Hmix* � *z*<sup>0</sup> . On the other hand, for the four sub-layers case, the boundary layer height is divided into four sub-layers, and the discretization is taken as follows: *dz*<sup>1</sup> <sup>¼</sup> <sup>3</sup> <sup>15</sup> ð Þ *Hmix* � *z*<sup>0</sup> , *dz*<sup>2</sup> <sup>¼</sup> <sup>3</sup> <sup>15</sup> ð Þ *Hmix* � *<sup>z</sup>*<sup>0</sup> , *dz*<sup>3</sup> <sup>¼</sup> <sup>7</sup> <sup>15</sup> ð Þ *Hmix* � *<sup>z</sup>*<sup>0</sup> , and *dz*<sup>4</sup> <sup>¼</sup> <sup>2</sup> <sup>15</sup> ð Þ *Hmix* � *z*<sup>0</sup> . These formulas are used to accurately model the behavior of the boundary layer height in numerical simulations.

In this work, the results of the numerical convergence for concentration predictions of atmospheric dispersion models are indicated as a function of the number of eigenvalues used in the model for different downwind distances from the source. Reference is made to experiments conducted in Copenhagen (experiments 1 to 3) and Prairie Grass (experiments 1, 5, and 7).

For the Copenhagen experiments, the models used four sub-layers in the vertical and tested two different formulations for calculating the vertical eddy diffusivity, a parameter that represents turbulence and mixing in the vertical direction. **Figure 1** shows the numerical convergence of normalized crosswind-integrated concentration as a function of the number of eigenvalues used in the model. The convergence is shown for four different downwind distances (200, 500, 1000, and 1500 m) from the source. The sentence indicates that for Prairie Grass, only one formulation of vertical diffusivity was used because the mixing height (*Hmix*) was much greater than the surface layer height (*Hs*), so the two formulations gave the same result.

**Figure 2** shows the numerical convergence of ground-level crosswind-integrated concentration for Prairie Grass experiments as a function of the number of eigenvalues at four downwind distances (50, 100, 200, and 800 m).

Numerical simulations were conducted to investigate the influence of the vertical eddy diffusivity formulation on the normalized crosswind-integrated concentration of air pollution near a point source. Specifically, the simulations were performed for Copenhagen experiments numbered 1, 2, and 3, and the results are presented in **Figure 3**.

The figure depicts the normalized crosswind-integrated concentration *<sup>c</sup> y <sup>H</sup>*ð Þ *<sup>x</sup>*, *<sup>z</sup>*<sup>0</sup> *<sup>Q</sup>* at ground level as a function of downwind distance, using two different vertical turbulence formulations. The results indicate that the choice of vertical eddy diffusivity formulation has a significant impact on the concentration near the point source. Notably, the figure shows that the concentration is higher when using the first formulation of the vertical eddy diffusivity, compared to the second one. Furthermore, the figure illustrates the expected decrease in crosswind-integrated concentration with increasing downwind distance due to the dispersion of air pollutants caused by turbulent mixing in the atmosphere. In conclusion, these simulations provide valuable insights into air pollution dispersion behavior and can inform future experiments and modeling efforts.

#### **Figure 1.**

*Copenhagen experiments number 1, 2, and 3: Numerical convergence of the proposed solution for two vertical turbulence formulations as a function of the number of eigenvalues at four distances using four sub-layers.*

The quality and performance of models are typically evaluated by comparing their predictions to observed (or measured) values. When observations are available, a common approach to evaluating model performance is to plot the predicted values against the observed values in a scatter diagram.

In this study, the performance of the models for the Copenhagen and Prairie Grass experiments was evaluated using scatter diagrams, which are shown in **Figures 4** and **5**, respectively. **Figure 4** compares the predicted and observed crosswind-integrated concentrations for two different formulations of the vertical diffusivity parameter and two different numbers of sub-layers used to represent the atmospheric boundary layer (with *H* ¼ 2 and *H* ¼ 4), while **Figure 5** compares the predicted and observed ground-level crosswind integrated concentrations for two different numbers of sub-layers in the atmospheric boundary layer (again with *H* ¼ 2 and *H* ¼ 4).

The scatter diagrams show good agreement between the predicted and observed values, indicating that the models are well-parameterized. However, a visual inspection of the figures also reveals that the observed concentrations tended to be slightly higher than the predicted concentrations.

While scatter diagrams provide a useful visual evaluation, a more rigorous evaluation of model performance is achieved through statistical metrics. In this study, the statistical indices used to evaluate the models' performance are defined in the reference [23]. These statistical metrics provide a more detailed and quantitative

*A Novel Development in Three-Dimensional Analytical Solutions for Air Pollution… DOI: http://dx.doi.org/10.5772/intechopen.112225*

**Figure 2.**

*Prairie grass experiment numbers 1, 5, and 7: Numerical convergence of the proposed solution as a function of the number of eigenvalues at four distances using four sub-layers.*

assessment of the models' performance, which can be used to identify any areas of weakness or sources of error in the models. We defined the statistical indices used in our study as:

• Normalized Mean Square Error (NMSE): A measure of the accuracy of a model or prediction, calculated as the mean square error divided by the variance of the observed data:

$$\text{NMSE} = \frac{\overline{\left(c\_o - c\_p\right)^2}}{\overline{c\_o} \times \overline{c\_p}} \quad \text{s}$$

• Mean Relative Square Error (MRSE): A measure of the accuracy of a model or prediction, calculated as the mean square error divided by the mean of the observed data squared:

$$\text{MRSE} = 4\overline{\left(\left(c\_o - c\_p\right)/\overline{\left(c\_o + c\_p\right)}\right)^2},$$

• Correlation Coefficient (COR): A measure of the linear relationship between two variables, usually the predicted and observed values of a model or prediction. It

#### **Figure 3.**

*Normalized concentration Eq. (24) as a function of distance using four sub-layers for Copenhagen experiment numbers 1, 2, and 3.*

*A Novel Development in Three-Dimensional Analytical Solutions for Air Pollution… DOI: http://dx.doi.org/10.5772/intechopen.112225*

#### **Figure 4.**

*Comparison of observed and predicted crosswind-integrated concentrations in Copenhagen (top panel) and prairie grass (bottom panel) experiments using scatter plots. The one-to-one line (y* ¼ *x) and factor-of-two lines (y* ¼ 0*:*5 � *x and y* ¼ 2 � *x) are shown.*

ranges from �1 to 1, with 1 indicating a perfect positive correlation and � 1 indicating a perfect negative correlation:

$$COR = \frac{\overline{(c\_p - \overline{c\_p})(c\_o - \overline{c\_o})}}{\left(\sigma\_o \sigma\_p\right)} \ , \ $$

• Fractional Bias (FB): A measure of the systematic bias in a model or prediction, calculated as the difference between the mean predicted and observed values divided by the mean observed value:

$$F \\ B = 2 \begin{array}{c} \left(\overline{c\_o} - \overline{c\_p}\right) \\ \left(\overline{c\_o} + \overline{c\_p}\right) \end{array},$$

#### **Figure 5.**

*Isolines of non-dimensional predicted crosswind-integrated concentration as a function of non-dimensional distance <sup>w</sup>*<sup>∗</sup> *<sup>x</sup> umeanHmix and non-dimensional depth <sup>z</sup> Hmix are presented for Copenhagen experiment number 2 (top panel) and prairie grass experiment number 7 (bottom panel) at different source heights: (a) Hs Hmix* <sup>¼</sup> <sup>0</sup>*:*1*, (b) Hs Hmix* ¼ 0*:*25*, (c) Hs Hmix* <sup>¼</sup> <sup>0</sup>*:*5*, and (d) Hs Hmix* ¼ 0*:*75*, using four sub-layers.*

*A Novel Development in Three-Dimensional Analytical Solutions for Air Pollution… DOI: http://dx.doi.org/10.5772/intechopen.112225*

• Fractional Standard Deviation (FS): A measure of the variability in a model or prediction, calculated as the standard deviation of the predicted values divided by the mean observed value:

$$F\mathbb{S} = 2 \begin{array}{c} \left(\sigma\_o - \sigma\_p\right) \\ \left(\sigma\_o + \sigma\_p\right) \end{array},$$

• Mean Geometric Ratio (MG): A measure of the logarithmic bias in a model or prediction, calculated as the exponential of the mean of the differences between the logarithms of the predicted and observed values:

$$MG = \exp\left(\overline{\log(c\_o)} - \overline{\log(c\_p)}\right) \dots$$

• Variance of Geometric Ratio (VG): A measure of the variability in the logarithmic bias of a model or prediction, calculated as the variance of the differences between the logarithms of the predicted and observed values:

$$VG = \exp\left[\overline{\left(\ln(c\_o) - \ln(c\_p)\right)^2}\right],$$

• Factor of Two (FAC2): A measure of the accuracy of a numerical weather prediction model, defined as the distance between the observed and predicted position of a weather system divided by the radius of the weather system. If the distance is less than or equal to twice the radius, the forecast is considered accurate and has a FAC2 score of 1. If the distance is greater than twice the radius, the forecast is considered inaccurate and has a FAC2 score of less than 1:

$$0.5 \le (c\_p/c\_o) \le 2\ \dots$$

The satisfactory numerical results presented in **Figure 4** are supported by the values obtained from the statistical performance measures. Upon examining these values, it is observed that they mostly fall within the range of acceptable performance for the two proposed formulations for vertical turbulent diffusivity. However, it should be noted that the simulated concentrations for experiments conducted with Prairie Grass tend to slightly overestimate the measured quantities. Specifically, when considering the turbulent diffusivity that depends on both x and z, the model produces results that are relatively more promising than those obtained with other methods.

**Figure 5** depicts the spatial distribution of pollutants for two different experiments —Copenhagen experiment number 2 and Prairie Grass experiment number 7, using contour lines that represent the adimensional transverse integrated concentration predicted based on the adimensional distance for experiment 2 and adimensional depth for experiment 7, where the adimensional distance for experiment 2 is defined as *w*<sup>∗</sup> *x=*ð Þ *u*mean*Hmix* .

### **6. Conclusions**

A solution has been developed for the 3-dimensional stationary state atmospheric diffusion equation, which incorporates more realistic formulations of wind velocity


#### **Table 1.**

*Statistical measures for the Copenhagen and Prairie grass experiments.*

profiles and two formulations of vertical turbulent diffusivity. This solution was achieved through a combination of an appropriate auxiliary eigenvalue problem with mathematical induction, while a transcendental equation was developed to determine the eigenvalues for any number of subdomains. The advection-diffusion equation was solved by dividing the planetary boundary layer into discrete layers and assuming average values of turbulent diffusivity and wind velocity in each subdomain. The solution was evaluated using the Copenhagen and Prairie Grass experiments for two and four layers, and the numerical convergence of the solution was verified based on the number of eigenvalues. The results indicated good agreement between predicted and observed values, and most of the calculated statistical indices were within an acceptable range for model performance. This current model could be a promising approach for accurately predicting atmospheric pollutant dispersion and may also be applicable to other continuous flows (**Table 1**).

*A Novel Development in Three-Dimensional Analytical Solutions for Air Pollution… DOI: http://dx.doi.org/10.5772/intechopen.112225*
