Fractional solution of the puff model in three dimensional for pollutant dispersion in the atmosphere

doi:10.21203/rs.3.rs-3078327/v1

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Fractional solution of the puff model in three dimensional for pollutant dispersion in the atmosphere

https://doi.org/10.21203/rs.3.rs-3078327/v1

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An analytical solution of the three-dimensional fractional time-dependent advection-diffusion equation (ADE) in the puff model is solved. A simple model of the fractional differential equation for an instantaneous emission in non-homogeneous and time-dependent meteorological conditions has been studied. A Generalized Integral Laplace Transform Technique (GILTT) has been used to solve this problem. A comparison between the proposed model, previous work, and observed concentration from the experiment of Copenhagen is made.

Puff Model

Advection-Diffusion Equation

Laplace transform

GILTT

Proposed Model

Puff models are a practical approach, which describes the dispersion from a time-dependent emission point source. Most of the shape of the puffs is a Gaussian approach, Gaussian models are a realistic solution of the equation of advection-diffusion in the atmosphere where both wind and diffusion coefficients are constant with height. Hankel transform was employed to solve the 2-d steady state advection diffusion equation considering the wind speed and eddy diffusivity as variables [1]. Puff models are suitable to get the pollutant dispersion in the atmosphere over complex terrain. The pollutant diffusion in the puff models is modeled by the Gaussian formulation. However, for instance, Van Ulden proposed an approximate solution for describing the dispersion of a cloud of passive contaminants released from an instantaneous source near the ground [2]. Tirabassi and Rizza improved the puff model for a continuously emitting source by the superposition of a series of the cited cloud [3]. In the puff model Tirabassi and Rizza used the truncated Gram-Charlier expansion of the concentration field and the finite set of equations for the corresponding moments for solving the advection-diffusion equation [4]. Pereira et al. studied two puff models with the main feature that the horizontal dispersion is expressed by a Gaussian formula, meanwhile, the vertical puff shape is non-Gaussian and it is evaluated by two different techniques that allow obtaining an analytical solution of the one-dimensional advection-diffusion equation: the ADMM (Advection Diffusion Multilayer Method) and GILTT (Generalized Integral Laplace Transform Technique) techniques [5]. The first one is a model based on a discretization of the Atmospheric Boundary Layer (ABL) and the solution is given in the integral form. For more details see the works [6, 7, 8]. The second one skips the multilayer discretization of the height z appearing in the ADMM approach and is given in the series form. The idea of the GILTT approach comprehends the steps: expansion of the concentration in series of eigenfunctions attained from an auxiliary problem, replacing this equation in the advection-diffusion equation and taking moments, one comes out with a matrix ordinary differential equation that is then solved analytically by the Laplace Transform technique. For more information see the works [9, 10, 11, 12]. The solution of the puff model for pollutant dispersion in the atmospheric boundary layer by the GILTT method has been studied by [13]. A puff model using a three-dimensional analytical solution for the pollutant diffusion process has been estimated by [13]. Marrouf et al. introduced a steady-state one, two and three-dimensional mathematical model for the dispersion of pollutants from a point source in the atmosphere [14, 15, 16, 17]. In this work we present an analytical investigation for the three-dimensional fractional advectiondiffusion equation in puff model by using the Generalized Integral Laplace Transform Technique (GILTT), adopting Caputo’s formula of a fractional derivative. The solution is written as a product of onedimensional GILTT analytical solutions. At this point, is important to recall at the GILTT solution is derived for an eddy-diffusivity depending on the vertical direction. Therefore, this formulation is quite appropriate to simulate pollutant dispersion in a non-homogeneous atmosphere. Further, assuming in this treatment that the wind speed u and vertical eddy diffusivity Kz are functions only of the vertical height z, while the lateral eddy diffusivity Ky is taken to be dependent on the downwind distance x from the source and the vertical height z above ground. The proposed analytical formula has been compared with the Copenhagen experiment concentrations data set. Statistical measures have been utilized in the comparison between the observed and proposed concentrations by the new model with different values of the fractional-order a of the derivative. This study reveals that the concentrations obtained by the proposed model with α = 0:94 are in very good agreement with those measured than other values. The results of this study are discussed and presented in tables and illustrative figures.

Puff models are used to know the behavior of pollutants in non-homogeneous, non-stationary meteorological and emission coefficient [18]. This emission is discretized in temporal secession of puffs, separate transport and dispersion. Each puff is specified in accordance with the trajectory from its center, which is calculated for velocity of the local wind, while it is expanding in the time by of the dispersion coefficient describing the atmospheric turbulence. A mass of pollutant ${\Delta }m$ equals $Q{\Delta }t$, where $Q$ is the emission rate and ${\Delta }t$ is the time interval. Each puff has its trajectory described by its center which is defined as:

$${x}_{0}=\stackrel{-}{u}{\Delta }t {y}_{0}=\stackrel{-}{v}{\Delta }t {z}_{0}=\stackrel{-}{w}{\Delta }t$$

where $\stackrel{-}{u},\stackrel{-}{v}$ and $\stackrel{-}{w}$ are average of the wind speed in three dimensions.

The total concentration of pollutant in a given point is written as follows:

$${C}_{T}(x,y,z,t)=\sum _{\text{p}\text{u}\text{f}\text{f}=1}^{\text{t}\text{o}\text{t}\text{a}\text{l}\text{o}\text{f}\text{p}\text{u}\text{f}\text{f}{\prime }\text{s}}\text{‍}{\Delta }{M}_{\text{p}\text{u}\text{f}\text{f}}\left\{{\int }_{t=0}^{\infty }\text{‍}{c}_{\text{p}\text{u}\text{f}\text{f}}(x,y,z,t)H(t-{t}_{0})dt\right\}$$

where $H$ is the heaviside function and:

$${c}_{\text{p}\text{u}\text{f}\text{f}}(x,y,z,t)={c}_{1}(x,t){c}_{2}(y,t){c}_{3}(z,t)$$

As mentioned before the three functions ${c}_{1},{c}_{2}$ and ${c}_{3}$ for the Gaussian model gives as:

$${c}_{1}=\frac{1}{\sqrt{2\pi }{\sigma }_{x}}\text{e}\text{x}\text{p}\left[-\frac{1}{2}{\left(\frac{x-{x}_{0}}{{\sigma }_{x}}\right)}^{2}\right]$$

$${c}_{2}=\frac{1}{\sqrt{2\pi }{\sigma }_{y}}\text{e}\text{x}\text{p}\left[-\frac{1}{2}{\left(\frac{y-{y}_{0}}{{\sigma }_{y}}\right)}^{2}\right]$$

$${c}_{3}=\frac{1}{\sqrt{2\pi }{\sigma }_{z}}\text{e}\text{x}\text{p}\left[-\frac{1}{2}{\left(\frac{z-{z}_{0}}{{\sigma }_{z}}\right)}^{2}\right]$$

where ${\sigma }_{x},{\sigma }_{y}$ and ${\sigma }_{z}$ are empirical diffusion value [19].

The one-dimensional GILTT solution

$$\frac{\stackrel{c }{{\partial }^{\alpha }}c(z,t)}{\partial {t}^{\alpha }}={k}_{z}^{{\prime }}\frac{\partial c(z,t)}{\partial z}+{k}_{z}\frac{{\partial }^{2}c(z,t)}{\partial {z}^{2}}+Q\delta (z-{h}_{s})\delta (t-{t}_{0}), 0<\alpha \le 1$$

Eq. (7) is solved together with the following boundary condition:

$${k}_{z}\frac{\partial c}{\partial z}=0 at z=0,z={z}_{i}$$

$$c(z,t)=0 at t=0$$

supposing the solution of Eq. (7) in the form:

$$c(z,t)=\sum _{n=0}^{N}\text{‍}\frac{{c}_{n}\left(t\right){\zeta }_{n}\left(z\right)}{{N}_{n}^{\frac{1}{2}}}$$

with ${N}_{n}$ given by

$${N}_{n}={\int }_{\upsilon }\text{‍}{\zeta }_{n}^{2}\left(z\right)d\upsilon$$

$${\zeta }_{n}\left(z\right)=\text{c}\text{o}\text{s}\left({\lambda }_{n}z\right) , {\lambda }_{n}=\frac{n\pi }{{z}_{i}}$$

where ${\zeta }_{n}\left(z\right)$ is the eigen function and ${\lambda }_{n}$ is the eigen value,

$c(z,t)$ is the mean concentration in one dimension, $c$ is the caputo derivative, $\alpha$ is the order of the fractional, spatial operator ${k}_{z}$ is the vertical eddy diffusivity, $Q$ is the emission rate and $\delta (... )$ is the delta function

After introducing Eq. (10) in Eq. (7), multiplying with $\frac{{\zeta }_{m}\left(z\right)}{{N}_{m}^{\frac{1}{2}}}$ and integrating with respect to $z$ from $0\to {z}_{i}$, one gets:

$$\begin{array}{l}\left.\begin{array}{c}\sum _{n=0}^{N}\text{‍}\frac{{d}^{\alpha }{c}_{n}\left(t\right)}{d{t}^{\alpha }}{\int }_{0}^{{z}_{i}}\text{‍}\frac{{\zeta }_{n}\left(z\right)}{{N}_{n}^{\frac{1}{2}}}\frac{{\zeta }_{m}\left(z\right)}{{N}_{m}^{\frac{1}{2}}}dz=\sum _{n=0}^{N}\text{‍}{c}_{n}\left(t\right){\int }_{0}^{{z}_{i}}\text{‍}\left(\frac{{\zeta }_{m}\left(z\right)}{{N}_{m}^{\frac{1}{2}}}{k}_{z}^{{\prime }}\frac{d\left(\frac{{\zeta }_{n}\left(z\right)}{{N}_{n}^{\frac{1}{2}}}\right)}{dz}\right.\left.+\frac{{\zeta }_{m}\left(z\right)}{{N}_{m}^{\frac{1}{2}}}{k}_{z}\frac{{d}^{2}\left(\frac{{\zeta }_{n}\left(z\right)}{{N}_{n}^{\frac{1}{2}}}\right)}{d{z}^{2}}\right)dz\\ +Q\delta (t-{t}_{0}){\int }_{0}^{{z}_{i}}\text{‍}\frac{{\zeta }_{m}\left(z\right)}{{N}_{m}^{\frac{1}{2}}}\delta (z-{h}_{s})dz\end{array}\right\}\end{array}$$

Upon, employing the properties of the eigenfunction ${\zeta }_{m}\left(z\right)$ and the eigenvalues ${\lambda }_{n}$ as given in Eq. (12), and defined the matrices A, B as:

$$A=(a{)}_{n,m}={\int }_{0}^{{z}_{i}}\text{‍}\frac{{\zeta }_{m}\left(z\right)}{{N}_{m}^{\frac{1}{2}}}\frac{{\zeta }_{n}\left(z\right)}{{N}_{n}^{\frac{1}{2}}}dz$$

$$B=(b{)}_{n,m}$$

$$=-\frac{1}{{N}_{n}^{\frac{1}{2}}{N}_{m}^{\frac{1}{2}}}\left[{\int }_{0}^{{z}_{i}}\text{‍}{k}_{z}^{{\prime }}{\zeta }_{m}\left(z\right){\zeta }_{n}^{{\prime }}\left(z\right)dz+{\int }_{0}^{{z}_{i}}\text{‍}{k}_{z}{\zeta }_{m}\left(z\right){\zeta }_{n}^{{\prime }{\prime }}\left(z\right)dz\right]$$

$$=-\frac{1}{{N}_{n}^{\frac{1}{2}}{N}_{m}^{\frac{1}{2}}}\left[{\int }_{0}^{{z}_{i}}\text{‍}{k}_{z}^{{\prime }}{\zeta }_{m}\left(z\right){\zeta }_{n}^{{\prime }}\left(z\right)dz-{\lambda }_{n}^{2}{\int }_{0}^{{z}_{i}}\text{‍}{k}_{z}{\zeta }_{m}\left(z\right){\zeta }_{n}\left(z\right)dz\right]$$

Eq. (13) becomes:

$$\sum _{n=0}^{N}\text{‍}(a{)}_{n,m}\frac{{d}^{\alpha }{c}_{n}\left(t\right)}{d{t}^{\alpha }}=\sum _{n=0}^{N}\text{‍}-(b{)}_{n,m}{c}_{n}\left(t\right)+Q\frac{{\zeta }_{m}\left({h}_{s}\right)}{{N}_{m}^{\frac{1}{2}}}M\delta (t-{t}_{0})$$

where $M$ is the unit column matrix, stand for:

$$M=\left(\begin{array}{l}1\\ 1\\ ⋮\\ 1\\ \end{array}\right)$$

Eq. (16) can be written in matrix form as:

$$\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}Y\left(t\right)+FY\left(t\right)=\eta \delta (t-{t}_{0})$$

For $t>0$, where $Y\left(t\right)$ is the column matrix, stand for:

$$Y\left(t\right)=\left(\begin{array}{l}{c}_{0}\left(t\right)\\ {c}_{1}\left(t\right)\\ .\\ .\\ {c}_{N}\left(t\right)\\ \end{array}\right)$$

and

$$F={A}^{-1}B , \eta ={A}^{-1}Q\frac{{\zeta }_{m}\left({h}_{s}\right)}{{N}_{m}^{\frac{1}{2}}}M$$

Eq. (17) is one dimensional matrix differential equation in $t$. upon, using Laplace transform on Eq. (17), which transform $t$ to $s$ and $Y\left(t\right)$ to $\stackrel{\sim}{Y}\left(s\right)$ namely:

$$\stackrel{\sim}{Y}\left(s\right)={\int }_{0}^{\infty }\text{‍}{e}^{-st}Y\left(t\right)dt$$

and by virtue of definition of Laplace transform of a fractional derivative of $\alpha$ order, which given by Caputo’s formula [20], namely:

$$\mathcal{L}\left(\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}Y\left(t\right)\right)={s}^{\alpha }\stackrel{\sim}{Y}\left(s\right)-\sum _{k=0}^{n-1}\text{‍}{s}^{\alpha -k-1}{Y}^{k}\left(0\right), n-1<\alpha \le n$$

which in our case is:

$$\mathcal{L}\left(\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}Y\left(t\right)\right)={s}^{\alpha }\stackrel{\sim}{Y}\left(s\right)-{s}^{\alpha -1}Y\left(0\right), 0<\alpha \le 1$$

Eq. (17) is transformed to:

$${s}^{\alpha }\stackrel{\sim}{Y}\left(s\right)-{s}^{\alpha -1}Y\left(0\right)+F\stackrel{\sim}{Y}\left(s\right)=\eta {e}^{-s{t}_{0}}$$

and after putting the matrix $F$ as:

$$F=XD{X}^{-1}$$

Eq. (23) can be rewritten as:

$$\stackrel{\sim}{Y}\left(s\right)=X\frac{1}{{s}^{\alpha }I+D}{X}^{-1}\left({s}^{\alpha -1}Y\left(0\right)+\eta {e}^{-s{t}_{0}}\right)$$

$$Y\left(t\right)={\mathcal{L}}^{-1}\stackrel{\sim}{Y}\left(s\right)$$

$$=X{\mathcal{L}}^{-1}\left(\frac{{s}^{\alpha -1}}{{s}^{\alpha }I+D}\right){X}^{-1}Y\left(0\right)+X{\mathcal{L}}^{-1}\left(\frac{{e}^{-s{t}_{0}}}{{s}^{\alpha }I+D}\right){X}^{-1}\eta$$

the inverse Laplace transform of the first term in the RHS (Right Hand Side) of the pervious Equation can be evaluated as:

$${\mathcal{L}}^{-1}\left(\frac{{s}^{\alpha -1}}{{s}^{\alpha }I+D}\right)=\left(\begin{array}{llllll}{E}_{\alpha }(-{d}_{1}{t}^{\alpha })& 0& & & & \\ 0& {E}_{\alpha }(-{d}_{2}{t}^{\alpha })& & & & \\ .& 0& {E}_{\alpha }(-{d}_{3}{t}^{\alpha })& & & \\ .& .& .& & & \\ .& .& & & & \\ 0& 0& & & {E}_{\alpha }(-{d}_{N}{t}^{\alpha })& \\ & & & & & \end{array}\right)$$

in which ${E}_{\alpha }(-{d}_{1}{t}^{\alpha })$ is Mittage-Leffer function defined as:

$${E}_{\alpha }(-{d}_{i}{t}^{\alpha })=\sum _{n=0}^{\infty }\text{‍}\frac{(-{d}_{i}{t}^{\alpha })}{{\Gamma }(n\alpha +1)} ,\alpha >0$$

$Y\left(0\right)$ in Eq. (26) can be evaluated by introducing Eq. (10) in Eq. (3), one gets:

$$Y\left(0\right)=0$$

Then Eq. (26) becomes:

$$Y\left(t\right)=X{\mathcal{L}}^{-1}\left(\frac{{e}^{-s{t}_{0}}}{{s}^{\alpha }I+D}\right){X}^{-1}\eta$$

$$=X\left(\sum _{n=0}^{\infty }\text{‍}\frac{(-{t}_{0}{)}^{n}}{n!}{\mathcal{L}}^{-1}\left(\frac{{s}^{n}}{{s}^{\alpha }I+D}\right)\right){X}^{-1}\eta$$

$$=XG\left(t\right){X}^{-1}\eta$$

where $G\left(t\right)$ is a diagonal matrix which elements are series in the form:

$$\sum _{n=0}^{\infty }\text{‍}\frac{(-{t}_{0}{)}^{n}}{n!}{t}^{\alpha -n}{E}_{\alpha ,\alpha -n}(-{\lambda }_{i}{t}^{\alpha })$$

$${E}_{\alpha ,\beta }\left(x\right)=\sum _{k=0}^{\infty }\text{‍}\frac{{x}^{k}}{{\Gamma }\left(k\alpha +\beta \right)} \alpha ,\beta >0$$

Then we get the solution $c(z,t)$ by substituting Eq. (12) and Eq. (??) in Eq. (10). Similarly, we will find $c(x,t)$ and $c(y,t)$ as previously in section (3).

The Copenhagen experiment was held in the Northern part of Copenhagen, Denmark, in unstable and neutral conditions. A non-buoyant tracer of Sulfur Hexafluoride ($S{F}_{6}$) was released from a $115m$ high stack. The sampling units were located at a distance of $2$ to $6km$ from the point of release. The roughness length was reported to be $0.6m$ [21, 22].

The meteorological parameters during the Copenhagen experiment are taking from [13]. The values of ground-level center line normalized concentrations measured during Copenhagen experiment of SulphurhexaFluoried ($S{F}_{6}$) and the corresponding values of fractional proposed model by the new model given by Eq. (2) and previous work [23] are presented in Table (1). In this work, the vertical eddy diffusivity coefficient expressed as the formulations proposed by [24] and written as:

$${k}_{z}=k{w}_{\text{*}}z\left(1-\frac{z}{{z}_{i}}\right)$$

where $k$ is the von-Karman constant, ${w}_{\text{*}}$ is the convective velocity, $z$ is the vertical height and $h$ is the boundary layer height. While the lateral eddy diffusivity ${k}_{x}$ and ${K}_{y}$ are taken as given by [25]:

$${k}_{\vartheta }=0.1{w}_{\text{*}}{z}_{i}$$

where $\vartheta$ represents $x$ and $y$ respectively.

Table 1

Normalized observed and predicted center line ground-level concentrations $\left(1{0}^{-7}s{m}^{-3}\right)$ for Copenhagen experiment considering different fractional parameters $\left(\alpha \right)$.
Run	Distance (m)	Observed	Predicted			previous work*
Run	Distance (m)	Observed	$\alpha$=0.93	$\alpha$=0.94	$\alpha$=0.95	previous work*
1	1900	10.5	4.29	9.34	20.58	5.34
1	3700	2.14	0.75	1.49	3.03	2.17
2	2100	9.85	4.15	8.37	17.11	7.67
2	4200	2.83	1.08	2.12	4.19	2.93
3	1900	16.33	8.15	14.19	24.83	13.74
3	3700	7.95	5.14	6.59	11.41	5.95
3	5400	3.76	1.53	2.63	4.57	3.72
4	4000	15.71	8.08	14.05	24.55	17.51
5	2100	12.11	6.14	10.90	19.39	20.94
5	4200	7.24	3.35	5.93	10.54	11.49
5	6100	4.75	1.99	3.52	6.24	7.52
7	2000	9.48	4.21	8.33	16.59	5.55
7	4100	2.62	0.75	1.44	2.77	2.03
7	5300	1.15	0.35	0.68	1.31	1.44
8	1900	9.76	4.67	8.45	15.37	8.43
8	3600	2.64	1.00	1.79	3.23	4.06
8	5300	0.98	0.38	0.66	1.14	2.59
9	2100	8.52	3.44	6.32	11.69	6.86
9	4200	2.66	1.07	1.95	3.56	2.55
9	6000	1.98	0.80	1.43	2.60	1.53
*[23]

One gets very closed agreement between the observed and predicted fractional ground level centerline normalized concentrations $\alpha =0.94$ compared with predicted $\alpha =0.93$, $\alpha =0.95$ and previous work as shown in the Fig. (1).

One finds that the most fractional predicted ground level centerline normalized concentration at $\alpha =0.94$ lie one to one line, while the data of previous work and predicted at $\alpha =0.93,\alpha =0.95$ inside a factor of two as shown in Fig. (2).

The statistical method calculating Normalized mean square error ($NMSE$), Fractional bias ($FB$), Correlation coefficient (COR) and factor of two ($FAC2$), The standard statistical performance measures that characterize the agreement between proposed (${C}_{p}={C}_{pred}$) and observations (${C}_{o}={C}_{obs}$), which gives the following results:

Table 2

Comparison between averages proposed, previous work for $S{F}_{6}$ and observed concentrations.
Statistical functions			$S{F}_{6}$
Statistical functions	NMSE	FB	COR	Fac2
previous work	0.19	-0.013	0.84	1.01
Predicated Conc. at \alpha = 0.94	0.044	0.19	0.997	0.83
Predicated Conc. at \alpha = 0.93	0.92	0.75	0.987	0.46
Predicated Conc. at \alpha = 0.95	0.36	-0.42	0.988	1.53

From the statistical method in Table (2), we find that the fractional ground level centerline normalized predicted concentrations at $\alpha =0.94$ are good agreement with its observed than previous work [23] and proposed at

$$\alpha =0.93, \alpha =0.95$$

An analytical three-dimensional fractional of the Advection Diffusion which describes a puff model is estimated. Comparison between a Proposed ground level centerline normalized concentration of puff model, previous plume model and it’s observed at the Northern part of Copenhagen experiment of $S{F}_{6}$ are done. One finds that the fractional proposed ground-level centerline normalized concentrations of puff model at $\alpha =0.94$ give well agreement with it’s observed of $S{F}_{6}$ than $\alpha =0.93,\alpha =0.95$ and previous plume work as shown from figures and statistics technique.

Ethical Approval

All Authors declare that they haven’t any financial or personal relationships that might bias the work being submitted, also they declare that their study not involving any human subjects.

Consent to Participate

All participants consent to participate of the present study.

Consent to Publish

Researchers should not generally publish or share identifiable individual data collected in the course of research without specific consent from the individual (or their representative).

Funding

Funding information is not applicable / No funding was received.

Conflict of interest

The authors declare that they have no conflict of interest.

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Fractional solution of the puff model in three dimensional for pollutant dispersion in the atmosphere

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Abstract

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1. Introduction

2. The puff models

3. Mathematical treatment

4. Evaluation of the model against Copenhagen data set

5. Results and discussions

6. Statistical Method

7. Conclusions

Declarations

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Funding

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References

Additional Declarations

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Run	Distance (m)	Observed	Predicted			previous work*
Run	Distance (m)	Observed	\(\alpha\)=0.93	\(\alpha\)=0.94	\(\alpha\)=0.95	previous work*
1	1900	10.5	4.29	9.34	20.58	5.34
1	3700	2.14	0.75	1.49	3.03	2.17
2	2100	9.85	4.15	8.37	17.11	7.67
2	4200	2.83	1.08	2.12	4.19	2.93
3	1900	16.33	8.15	14.19	24.83	13.74
3	3700	7.95	5.14	6.59	11.41	5.95
3	5400	3.76	1.53	2.63	4.57	3.72
4	4000	15.71	8.08	14.05	24.55	17.51
5	2100	12.11	6.14	10.90	19.39	20.94
5	4200	7.24	3.35	5.93	10.54	11.49
5	6100	4.75	1.99	3.52	6.24	7.52
7	2000	9.48	4.21	8.33	16.59	5.55
7	4100	2.62	0.75	1.44	2.77	2.03
7	5300	1.15	0.35	0.68	1.31	1.44
8	1900	9.76	4.67	8.45	15.37	8.43
8	3600	2.64	1.00	1.79	3.23	4.06
8	5300	0.98	0.38	0.66	1.14	2.59
9	2100	8.52	3.44	6.32	11.69	6.86
9	4200	2.66	1.07	1.95	3.56	2.55
9	6000	1.98	0.80	1.43	2.60	1.53
*[23]