1 Introduction

When an effluent plume is released to the atmosphere, it is transported downwind as a result of the mean wind field and simultaneously, diffused in a domain, due to the fluctuating wind components. Consequently the plume is spread in both the horizontal and vertical dimensions. The nature of such dispersion is of interest, especially for its wide range of consequences for human health. Thus, during the last decades much effort has been made to understand the nature of the pollution dispersion. The Gaussian plume dispersion model which is mathematically drived from advection–diffusion equations is one of the most important methods that is widely used to estimate pollution levels. This model have been applied extensively in the study of emissions from large industrial operations as well as a variety of other applications including release of radionuclide in atmosphere [30]; seed, pollen and insect dispersal [35]; and odor propagation from livestock facilities [42, 50].

The most important parameters in this model which mainly impact the results are the standard deviation of vertical and horizontal dispersion of plume. The first considerable effort to estimate these parameters was carried out by Pasquill [38] who developed the stability categories from meteorological parameters. He presented a set of graphs to display these parameters as a function of downwind distance and stability class. Later, Gifford [20] utilized the same data as Pasquill, though changed the plume spreading parameters to the values of the concentration distribution. The result led to curves that we refer to as Pasquill and Gifford curves. These curves published in an EPA workbook by Turner [44], are based on the experimental data taken at distances of less than 1 km, then extrapolated out to greater distances. However with the advent of computers, it has become necessary to represent these observational curves with algebraic formula. Thus, much effort has been placed into best fit representations. Among them the most important ones are [9, 16, 23]. The more recent studies include [14, 17, 21, 22, 39, 49]. However, their complications and dependence on parameters like turbulence data, heat and momentum fluxes and so on made them difficult to be implemented, in spite of their theoretical basis.

A distinct advantage of the Gaussian plume model is its simplicity as well as its speed; also this model allows the differentiation among the amount of pollutants which come from static or dynamic sources. In this model, parameter values are related to readily measurement quantities such as stack gas exit velocity and temperature, the temperature of the surrounding air and wind speed which made this model popular and relatively easy to apply. It is also considered suitable where the end points of the calculations are long term average or time integrated concentration in air, as is normally the case in the assessment of relative concentration from continuous releases. In addition to significant merits of this method, there are some demerits resulting in the appearance of new generations, such as the models do not predict hourly observations at a specific time and the location beyond the immediate vicinity of the release, or model cannot track changing meteorological conditions.

The new generation models like UK-ADMS [12], ISC [47], Aeromode [48] and similar computerized packages are some instances which besides CFD modeling are used to simulate air pollution dispersion. One of the most significant works in this field is reported by AIJ group [1] in which they compared more than 60 sets of wind tunnel tests with numerical simulations results using a variety of numerical methods and turbulence models. They reported several important points and also showed that the skill in predicting flow varied widely. Latter Meroney et al. [37] extended their work by four additional sets of wind tunnel tests for air pollution dispersion and showed that Reynolds stress turbulent model (RSM) produces somewhat more realistic results than the Renormalization-Group (RNG) or k-\(\varepsilon \) models. However, the numerical simulations appear some disadvantages that causes Gaussian model to be still widely in use. First of all meteorological conditions are constantly changing like the speed and wind direction, sun radiation, ambient temperature, cloud density and so on that leads us not to have an absolute modeling. Secondly and the most important demerit is the requirement of much expertise, time to set up and large computing resources to run the model. Finally, changing geometry is inevitable that can completely deviate the estimation occurred by the model. On the contrary, Gaussian dispersion model can easily estimate the pollution concentration scale, especially with the precise value in a flat or slightly high terrain.

The main purpose of this paper is to increase the accuracy of Gaussian model by introducing new formulations for standard deviation of vertical and horizontal dispersion of plume for neutral atmospheric conditions. To achieve this, parallel Computational Fluid Dynamics simulations are carried out and the atmospheric boundary layer flows are simulated. The dispersion of a passive discharge from a single isolated stack for different input conditions is calculated which is a well understood case [5, 40, 41]. CFD results are then used as input data for Gaussian plume dispersion model. Then an inverse procedure is applied and standard deviations of horizontal and vertical dispersion are modified which results in new data. Then curves are fitted to these data so that new analytical formulations are concluded as well.

2 Methodology review

In the proposed procedure four major steps are carried out to increase the accuracy of Gaussian plume dispersion model. These steps are listed as below:

  1. (1)

    Numerical modelings of dispersion of a passive discharge from a single isolated stack for different input conditions (i.e., inlet velocity profile, dimensions of stack and temperature of air and outlet plume).

  2. (2)

    Determining relative concentration for at least 40 random points of the domain for each case.

  3. (3)

    Calculation of vertical and horizontal dispersion coefficients using relative concentration values obtained from step 2 and Gaussian model (inverse procedure).

  4. (4)

    Fitting curves to determine vertical and horizontal dispersion coefficients in order to obtain new formulations.

At the end the results are validated with the experimental data and well-known CFD cases. The following sections describe the above steps in detail.

2.1 Numerical modelings of dispersion from a single isolated stack

2.1.1 Governing flow equations and turbulence model

Governing equations of the flow field are the Reynolds’ Averaged Navier–Stokes (RANS) equations. The conservation of the mass and momentum equations are given as following:

$$\begin{aligned}&\frac{\partial }{\partial x_j }u_j =0\end{aligned}$$
(1)
$$\begin{aligned}&\frac{\partial }{\partial t}u_i +u_j \frac{\partial }{\partial x_j }u_i =-\frac{1}{\rho }\frac{\partial }{\partial x_i } P-\frac{\partial }{\partial x_j }\overline{{u}_i^{\prime } {u}_j^{\prime } } +\upsilon \frac{\partial ^{2}}{\partial x_i \partial x_j } u_i \end{aligned}$$
(2)

where \(u_{i,j }\)are the mean fluid velocities in the x and y directions respectivly, \(P\) the mean pressure, \(\rho \) the fluid density, \(\upsilon \) the kinematic viscosity and \({u}_{i,j}^{\prime } \) are the fluctuation velocities. In Eq. (2)  \(\overline{{u}_j^{\prime } {u}_j^{\prime }}\) is the Reynolds stress tensor divided by fluid density. In this study, Reynolds stress model with a linear pressure–strain and wall-reflection effect has been applied [19, 26, 27]. The wall-reflection term ensures redistribution of the normal stresses near the wall. It tends to damp the normal stress perpendicular to the wall while enhancing the stresses parallel to the wall. The energy equation is also given by the following:

$$\begin{aligned} \frac{\partial }{\partial t}(\rho E)+\frac{\partial }{\partial x_i}[u_i (\rho E+p)]=\frac{\partial }{\partial x_j }\left[\left(k+\frac{c_p \mu _t }{\Pr _t }\right)\frac{\partial T}{\partial x_j }+u_i (\tau _{ij})\right] \end{aligned}$$
(3)

where E is the total energy, k the molecular conductivity, c\(_{p}\) the constant specific heats at constant pressure, \(\mu _t\) the turbulent viscosity, \(\Pr _t \) the Prandtl number, T the temperature and \(\tau _{ij}\) is viscous stress tensor. For species transport the steady-state RANS equations supplemented with the Eulerian advection–diffusion equation are the governing equations which can be summarized in a tensor form as:

$$\begin{aligned} u_j \frac{\partial C}{\partial x_j }=\frac{\upsilon _t }{Sc_t }.\frac{\partial ^{2}C}{\partial x_i \partial x_{j } } \end{aligned}$$
(4)

where C is the mean mass fraction of the pollutant, u the mean velocity and \(\upsilon _{t}\)/Sc\(_{t}\) is turbulent diffusion. The turbulent viscosity, \(\upsilon _{t}\), obtained from RSM turbulence model and turbulent Schmidt number Sc\(_{t}\) (the ratio of momentum diffusivity to mass diffusivity) should be determined prior to the numerical simulation. In the present research following the works of Li and Stathopoulos [33] and Wang and McNamara [51] the value of 0.7 is chosen for this number. Results show that, this value lead to more accurate standard deviations of horizontal and vertical dispersion. The second-order upwind model [2] is used for the pressure terms and the QUICK scheme [31] is used for all other ones.

2.1.2 Boundary conditions

For simulation of plume dispersion in atmosphere different types of boundary conditions are applied. The inlet plane and surface of stack outlet are specified as velocity inlet boundaries. Vertical and the top of the domain specify as symmetry (zero shear slip) and the outlet plane is specified as the pressure outlet plane.

For the ground surface, the standard wall function [28] with a roughness modification which is based on sand-grain is utilized. Such wall function in order to have the atmosphere boundary layer as input; need the equivalent sand-grain roughness height (\(\text{ K}_{\mathrm{S,ABL}}\)) which could be determined from roughness length (Z0) and roughness constant (CS). In this work the value of 0.1 and 0.5 are taken for roughness length and roughness constant, respectively [3]. The velocity profile in the inlet plane for a flat terrain is defined as following [11]:

$$\begin{aligned} u(z)=\frac{u^{*}}{\kappa }Ln\left(\frac{Z+Z_0 }{Z_0 }\right) \end{aligned}$$
(5)

where u(z) is the wind speed, u* the friction velocity, \(\kappa \) the von Karman’s constant taken to be 0.4 and Z\(_{0}\) the surface roughness length.

2.1.3 Computational grid

The characteristic of the domain for all CFD cases are illustrated in Fig. 1. For minimizing the influences of the boundaries on the results of numerical modelings; the domain sizes is taken equal or larger than the one suggested by the Cost Action 732 guideline [18]. As indicated in Fig. 2, unstructured high resolution grids around the discharge of stacks is applied which is gradually converted to a structured low resolution grid far enough from the stack. The grids also have greater resolution close to the ground near the stack.

Fig. 1
figure 1

Scheme of domain characteristics for all CFD cases

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Fig. 2
figure 2

The computational grid around the single stack and the ground surface vicinity

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To obtain a suitable computational grid in term of resolution of points, the grid sensitivity study is carried out. Thus, three computational grids with different resolutions are generated named as coarse, medium and fine grids, with the 3448210, 5342218 and 7260311 grid cells.

A sample case with the input parameters of; U \(=\) 5(m/s), H \(=\) 100(m), D \(=\) 3(m), Vs \(=\) 3(m/s) and Q \(=\) 1(g/s) is defined. The Relative Concentration (RC) values obtain for the coarse and fine grids compared with those of the medium grid in Fig. 3. As illustrated the medium and fine grid results are consistent with each other while the coarse grid results are different. Thus, in this work for all CFD cases the M grid is utilized for modeling plume dispersion.

Fig. 3
figure 3

Grid sensitivity analysis: concentration values obtained for a sample case [(U \(=\) 5(m/s), H \(=\) 100(m), D \(=\) 3(m), Vs \(=\) 3(m/s) and Q = 1(g/s)] with the Medium (M), coarse (C) and fine (F) grids

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2.2 Calculation of relative concentration

According to the procedure described in Sect. 2.1, numerical modelings of dispersion from a single isolated stack for different input conditions (i.e., inlet velocity profile, dimensions of stack and temperature of air and outlet plume) are performed. For each case at least 40 random points within the domain are created and the relative concentrations in these points are opted from CFD simulations. Figure 4, shows the profile of the velocity in the different distances of inlet plane both in the lower 40 m and full height of boundary layer for friction velocity of 0.5 as an example of neutral atmospheric condition. This figure also shows that the near ground velocity gradient gradually increases with the distances; however the overall mean velocity profile approximately keeps its shape.

Fig. 4
figure 4

The comparison between inlet velocity profile and its development of 3, 5 and 8 km downwind from inlet, a near ground, b full height of boundary layer for neutral atmosphere condition Dissipation (m\(^{2}\)  s-\(^3 \))

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As mentioned earlier, in this paper the RSM turbulence model is applied for simulating turbulent characteristics. Figure 5a, b shows the Turbulent Kinetic Energy (TKE) and the dissipation rate profile at different distances from the inlet plane for a sample case (U \(=\) 5(m/s), H \(=\) 100(m), D \(=\) 3(m), Vs \(=\) 3(m/s) and Q \(=\) 1(g/s)—see Sects. 2.3 and 2.4 for the definitions of the parameters). These figures indicate that the TKE levels decreases with the distance near ground; in contrast such parameter increases along the domain with the increase in height while Dissipation approximately keeps its rate and shows only some fluctuations throughout the length of the domain.

Fig. 5
figure 5

The comparison between a inlet TKE and b dissipation rate profile and its development of 3 and 5 km downwind of inlet (a) near ground, (b) full height of boundary layer for neutral atmosphere condition.

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2.3 Evaluation of dispersion coefficients

The Gaussian model is mathematically derived from advection–diffusion equations which for sources releasing continuously under steady-state conditions and ground reflection consideration is given as [45]:

$$\begin{aligned} C(x,y,z,H)=\frac{Qe^{\frac{-y^{2}}{2(\partial _y )^{2}}}}{2\pi \partial _y \partial _z U}\left[e^{(-\frac{(H-Z)^{2}}{2(\partial _z)^{2}})}+e^{(-\frac{(H+Z)^{2}}{2(\partial _z)^{2}})}\right] \end{aligned}$$
(6)

where C is the air pollutant concentration (g/m\(^{3})\), Q the source emission rate (g/s), U the wind speed at the height of effective plume rise (m/s), \(x\) the distance from the source in meter, H the effective plume rise (m) and \(\partial _y ,\partial _z \)are lateral and vertical dispersion coefficients. According to [45], this model should be able to predict concentration distributions within a factor of 3. The effective plume rise is calculated based on [25] formulation which defined as physical height of stack (\(\text{ H}_{\mathrm{s}}\)) plus the plume rise (\(\text{ H}_{\mathrm{p}}\)):

$$\begin{aligned} H_p =\frac{V_s d}{U}\left(1.5+2.68\times 10^{-3}p_0 \frac{\Delta T}{T_s}d\right) \end{aligned}$$
(7)

where \(\text{ V}_{\mathrm{s}}\) is the stacks outlet gas speed (m/s), d the inner stack diameter plume (m), p\(_{0}\) ambient pressure value (mbar), and \(\text{ T}_{\mathrm{s}}\) outlet gas temperature (\({^\circ }\)k) and \(\Delta \text{ T}\) the difference of \(\text{ T}_{\mathrm{s}}\) with ambient temperature. For momentum ratios \(\text{ M}_{\mathrm{r}}=\text{ Vs/U}>1.5\) the maximum plume rise is 3d\(\times \text{ M}_{\mathrm{r}}\) [8]. Note that the wind speed, U, is at height H.

2.3.1 Briggs model

Briggs [7] proposed a series of algebraic interpolation formulas for both rural and urban terrain based on various types of data [10, 43] which have become widely known. The formulation and its coefficients for neutral atmospheric conditions are given in Eq. 8 and Table 1, respectively.

Table 1 Dispersion parameters for neutral atmosphere conditions (Briggs model)
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$$\begin{aligned} \partial _y\, or\, \partial _z =ax(1+bx)^{c} \end{aligned}$$
(8)

2.3.2 Doury model

Out of a review of experimental data [6, 15, 16, 29] derived a relationship between dispersion coefficients and the travel time, t, as:

$$\begin{aligned} \partial _y\, or\, \partial _z =A(t)^{k} \end{aligned}$$
(9)

In this formula two variations of dispersion coefficients are considered and categorized as normal and weak diffusion. The value of A and k coefficients for a variety of time travel in normal atmospheric conditions considered in this research is given in Table 2.

Table 2 Dispersion parameter for neutral atmosphere conditions (Doury model)
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2.3.3 Hanna et al. model

Hanna and Drivas [23] suggested an analytical formulation on Pasquill–Gifford experimental data, which is mentioned in the following,

$$\begin{aligned} \partial _y\, or\, \partial _z =ax^{b}+c \end{aligned}$$
(10)

The coefficients a, b and c for neutral atmospheric conditions are given in Table 3. Although better algebraic fits are possible, this model is still widely in use.

Table 3 Dispersion parameter of The Hanna model for neutral atmosphere conditions
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2.4 The proposed model

In this work, the aim is to derive new mathematical formulations for standard deviation of horizontal and vertical dispersion in neutral atmospheric conditions. Table 4 highlights the domain of validation of the proposed formulations based on Pasqual-Gifford stability class. The velocity is measured in 10 m height so in the majority of cases the present formulations are valid.

Table 4 Domain of validation of proposed formulations based on atmospheric stability classes for use with the Pasquill–Gifford dispersion model
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To derive new formulations (Fitting curves) for standard deviation of dispersion which accurately predicts pollution concentration in Gaussian plume model in comparison to other well-known models, an inverse procedure is applied.

In the numerical modeling part of this research, several cases are built in which for each case input conditions including inlet boundary layer velocity profile (0.5 \(<\) u* \(<\) 1.5), ambient temperature (5 \(<\) Ta \(<\) 27), height (20 \(<\) H \(<\) 100) and diameter (0.5 \(<\) D \(<\) 3) of the stack, outlet velocity (4 \(<\) Vs \(<\) 12) and the temperature (80 \(<\) Ts (c) \(<\) 140) of released gas are varied. Then concentrations for random distances from the stack in stream-wise, span-wise and vertical directions are determined. In each case, at least 40 points are chosen from those 10 points are selected for 100–300 m and 10 points for last 100 m of the domain, others are randomly chosen from other parts of the domain. Since the plume rise from a stack occurs over some distance downwind, Gaussian plume model should not be applied within the first hundred meters from the stack.

Then an inverse procedure is applied so that air pollutant concentration is the input and unknowns are only dispersion coefficients. Since there will be two unknown parameters of \(\sigma _{\mathrm{y}}\) and \(\sigma _{\mathrm{z}}\) in each equation for each point, there is not a unique solution for that. To obtain an approximation of these coefficients, least square method is applied, with initial value of Briggs model. MacKay et al. [36] used a similar least square method to estimate parameters such as surface mass transfer rate and Peclet number rather than emission rates. Then the new curves are fitted as:

$$\begin{aligned} \partial _y =ax^{b}+c x+d\end{aligned}$$
(11)
$$\begin{aligned} \ \partial _z =ax^{b}+c x^{d}+ex+f \end{aligned}$$
(12)

The coefficients of proposed formulations are given in Table 5.

Table 5 Dispersion parameter of The present model for neutral atmosphere conditions
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These curves are shown in Fig. 6 in comparison with the Briggs, Doury and Hanna results. As noted before in Gaussian model, the above equations should not be applied in the first 100 m from the stack because the plume rise occurs over some distance downwind. Table 6 shows the description of the method to calculate standard deviations of horizontal and vertical dispersions for three random points.

Fig. 6
figure 6

Dispersion coefficients for neutral atmospheric condition based on Pasqual–Gifford stability class a horizontal dispersion and b vertical dispersion

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Table 6 The description of the method to calculate standard deviations of horizontal and vertical dispersions for three random points
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3 Results

To assess the applicability and accuracy of the presented formulations, two cases are applied. The first case is an experimental study of releasing krypton-85 from 100-m-high stack above a complex coastal terrain at La Hague reported by Leroy et al. [32]. The second case is numerical pollutant dispersion from three 28-m-high stacks of Besat power plant near Tehran. To be able to evaluate the results independent of the volume of the pollutant flow from source(s), the parameter of Relative Concentration is applied as following:

$$\begin{aligned} RC=\frac{C(g/m^{3})}{Q(g/s)} \end{aligned}$$
(13)

where C and Q are the same as defined in Eq. 6. In order to compare the accuracy of the method, a statistical method of Hanna et al. [24] is applied. The following equations define the statistical performance measures, which include the fractional bias (FB), the geometric mean bias (MG), the normalized mean square error (NMSE), the geometric variance (VG) and the fraction of predictions within a factor of two of observations (FAC2):

$$\begin{aligned} \textit{FB}=\frac{(\overline{C_0 } -\overline{C_p } )}{0.5(\overline{C_0 } +\overline{C_p } )}\end{aligned}$$
(14)
$$\begin{aligned} \textit{MG}=\exp (LnC_o -LnC_p ) \end{aligned}$$
(15)
$$\begin{aligned} \textit{NMSE}=\frac{\overline{(C_o -C_p )} ^{2}}{\overline{C_o C_p } }\end{aligned}$$
(16)
$$\begin{aligned} \textit{VG}=\exp [(\overline{LnC_o -LnC_p } )^{2}] \end{aligned}$$
(17)
$$\begin{aligned} \textit{FAC2:} \frac{1}{2}\le \frac{C_p }{C_o }\le 2 \end{aligned}$$
(18)

where \(\overline{C}\) is the average over the data set, Cp is Gaussian model predictions and \(\text{ C}_{\mathrm{o}}\) is concentration measured by CFD. A perfect model has MG, VG, and FAC2 equal to 1.0; and FB and NMSE equal to 0.0 [13]. The FB indicates how well the computation produces the average values around the average values of observed variable. The ideal value of this measure is zero, but it can range from 2 to \(-\)2, The NMSE gives information on the strengths of deviations and not on the over or under prediction thus, always giving positive values. Differences on peak values have a higher weight on NMSE than differences on other values. For example, the linear measures FB and NMSE can be overly influenced by infrequently occurring high observed or predicted concentrations, whereas the logarithmic measures MG and VG may provide a more balanced treatment of extreme high values.

3.1 Release of krypton-85 from nuclear plant at La Hague

The French Institute for Radiological Protection and Nuclear Safety (IRSN) in 2001–2002 studied the behavior of atmospheric dispersion of 15 releases from the nuclear fuel reprocessing plant AREVA NC-La Hague, using the chemically inert gas krypton-85, as a tracer for plumes emitted from 100-m-high stack. Concentration measurements were carried out at neutral or slightly unstable atmospheric conditions and \(\sigma _{\mathrm{y}}\) were determined from these measurements. Table 7 shows the measured and calculated values of dispersion coefficients by the proposed and different other formulations [32]. The values of 13 campaigns which were measured in neutral conditions in comparison to calculated values of this work formulation shows a great capability of predicting \(\sigma _{\mathrm{y}}\)in different distances.

Table 7 The standard deviation of dispersion for different conditions
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The performance statistics for these data are shown in Table 8. This table shows that the new formulation on lateral standard deviation of dispersion outperforms the other well-known models of Briggs, Doury and Hanna. Positive FB in this table also indicates that all models under predict \(\sigma _{\mathrm{y}}\) specifically Doury and Hanna. However, the new formulation gives the best agreement with the observed mean and standard deviation. The new formulation also shows a better performance in the MG, NMSE, VG and FAC2 statistics. It should be noted that if FAC3 is used instead of FAC2, these values are 1 for all models which means that all data are approximately within the range.

Table 8 Statistical evaluations of the present work, Briggs, Doury, Hanna models and experimental data of nuclear fuel reprocessing plant La Hague
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3.2 Pollutant dispersion from Besat power plant

The second test is the calculation of the pollution dispersion from three 28-m height stack of Besat gas power plant located in the south of Tehran. The area is known as one of the most polluted parts of the Tehran. Due to the existence of several pollutant sources including rail station, gas refinery plant and also this old power plant, it is impossible to distinguish the portion of emission of pollutant gases from Besat’s Stacks with other sources. Then instead of using the measurement device, CFD simulation for modeling plum dispersion is applied which its specifications are described in the following.

The physical features of the stacks of the Besat power plant with its pollution emission are presented in Table 9. The site plant is 560 m long and 350 m wide in north-south and west-east directions, respectively.

Table 9 The physical features of the stacks of the Besat power plant with its pollution emission
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For the CFD simulations of gas dispersion from the stacks of Besat power plant, a domain size and model like single stack described in Sect. 2.1 are applied. Because almost all buildings in this area are short and dense tangle, a surface roughness of 3 m is applied for outside the site whereas buildings inside the site are exactly modeled. The domain is discretized using an unstructured hexahedral grid in the vicinity of stacks and buildings which is gradually converted into a structured low resolution grid in a distance from stacks. Figure 7, shows surface grid applied for the model. A high resolution grid is applied near the discharge positions because the pollutant concentrations are dependent upon the accuracy of the real plume spread rates.

Fig. 7
figure 7

Surface grid around the stacks of Besat power plant and prevailing west-east wind direction

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Two flow conditions are applied for this case which, are within the range of the most possible neutral conditions in the area of based on the nearby meteorological station report. In the first case, the mean wind speed at 10-m-height is 7(m/s) and ambient and outlet gas temperatures are 300 (\(^{\circ }\)k) and 410 (\(^{\circ }\)k) respectively. In the second case the mean wind speed at 10-m-height is 5.5 (m/s) and ambient and outlet gas temperatures are 283 (\(^{\circ }\)k) and 383 (\(^{\circ }\)k). Table 10 shows the values of RC up to 10 km from central stacks of Besat power plant calculated by Briggs, Doury, Hanna and the proposed model in comparison to CFD results. The calculated and observed values of RC for two mentioned cases are plotted in Fig. 8, as a function of the distance to the central stack. This figure shows that, maximum concentration in neutral atmospheric conditions occur between 500 and 1,500 m of stacks in the prevailing wind directions.

Table 10 The value of RC up to 10 km from central stack of Besat power plant, wind speed at 10-m-height, ambient and outlet gas temperatures case (a) 7 (m/s), 300 (\(^{\circ }\)k) and 410 (\(^{\circ }\)k); case (b) 5.5 (m/s), 283 (\(^{\circ }\)k) and 383 (\(^{\circ }\)k)
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Fig. 8
figure 8

The calculated values of RC for Gaussian model (Briggs, Doury, Hanna, present work) in comparison to CFD. wind speed at 10-m-height, ambient and outlet gas temperatures a 7 (m/s), 300 (\(^{\circ }\)k) and 410 (\(^{\circ }\)k); b 5.5 (m/s), 283 (\(^{\circ }\)k) and 383 (\(^{\circ }\)k)

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Statistical evaluations on these data shown in Table 11 indicating that the formulations proposed in this work in all 5 factors achieve the best results where the worst predictions are for Doury and Hanna. Also the positive values of FB indicate that the RC is overestimated by all models. The number of calculated RC values which are within a factor of two for this work and Briggs are 13 of 15 (87 %) but checking the results reflects that the outputs of the present work are much closer to CFD values. The high values of the parameter VG for the Briggs, Doury and Hanna models in both cases indicate the low values of RC evaluated in near field for these models.

Table 11 Statistical evaluations of this work formulation, Briggs, Doury and Hanna models for RC at each point, wind speed at 10-m-height, ambient and outlet gas temperatures case a: 7 (m/s), 300 (\(^{\circ }\)k) and 410 (\(^{\circ }\)k); case b) 5.5 (m/s), 283 (\(^{\circ }\)k) and 383 (\(^{\circ }\)k)
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3.3 Parallelization

Since the domain size for modeling plume dispersion by CFD is very large and the problem is computationally intensive; parallelization is utilized. The grid is subdivided into several sub domains which are a function of the number of the processors with one-cell overlaps at the interfaces and RANS equations are solved for the sub domains. As illustrated in Fig. 9, the real speedup (defined as the computational time on 1 processor divided by the computational time on “N” processors) in comparison to ideal speedup decreases as the number of processors increases. In this work 32 processors are used.

Fig. 9
figure 9

Real and ideal speedup versus number of processors

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The parallel computer system of Computational Aerodynamics Laboratory at Amirkabir University of Technology (CALAUT) is used which has 40 CPUs comprising five nodes, each node packing four processors on one motherboard. The system configuration is summarized in Table 12.

Table 12 Configuration of parallel computer of CALAUT
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4 Conclusions

In order to improve the accuracy of Gaussian plume model, new analytical formulations for standard deviation of horizontal and vertical dispersion were suggested. To achieve this aim, Computational Fluid Dynamics simulations were set up for a single stack. Then ground level concentrations were determined at different distances. Inverse procedure was applied and standard deviations of horizontal and vertical dispersion were modified which resulted in new formulations. Using a statistical method the results of the presented approach are show its superiority to the models of Briggs, Doury and Hanna.