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Chapter 5

Analysis of the Results

Even if the results have been analyzed at three different times (7600s, 9100s, 10600s), we have chosen to present here what happened at the last time, since not significant differences have been discovered passing from one solution to another.

Plotting the positions of the parcels for the three different times, it is

possible to observe that the situation does not change (see Fig.(5.1), Fig.(5.2),

Fig.(5.3)).

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Analysis of the Results 75

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104 0

500 1000 1500 2000 2500 3000 3500 4000 4500

x

z

t = 7600s

Figure 5.1: Plot of the parcels positions at time t=7600s. The input param- eters of this simulation are the mean ones: µ = 1φ, σ = 1.5φ, ψ = 0.7

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104 0

500 1000 1500 2000 2500 3000 3500 4000 4500

x

z

t = 9100s

Figure 5.2: Parcels positions at time 9100 s.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104 0

500 1000 1500 2000 2500 3000 3500 4000 4500

x

z

t = 10600s

Figure 5.3: Parcels positions at time 10600 s.

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Analysis of the Results 76

A more accurate confirmation can be found observing the size distribution of parcels in each stripe (both air and ground) at the three time instants (see Fig.(5.4)). In the considered domain, the parcels not on the ground have reached a steady regime both in the number than in the range of diameters.

The same result is valid on the ground for the size distribution only, while obviously the number of parcels deposited on the ground increases with time.

−10 0 0 10

500 1000 1500

φ

Number of parcels

Stripe 1−Air

−10 0 0 10

500 1000 1500

φ Stripe 2−Air

−10 0 0 10

500 1000 1500

φ Stripe 3−Air

−10 0 0 10

500 1000 1500

φ Stripe 4−Air

−10 0 0 10

2000 4000 6000

φ

Number of parcels

Stripe 1−Ground

−10 0 0 10

2000 4000 6000

φ Stripe 2−Ground

−10 0 0 10

2000 4000 6000

φ Stripe 3−Ground

−10 0 0 10

2000 4000 6000

φ Stripe 4−Ground

t = 7600s t = 9100s t = 10600s

Figure 5.4: Parcels size distribution plotted at three time instants. The simulation taken into account is still the one with the mean input values.

For these reasons in the next paragraphs the results of UQ and SA analysis

will be investigated only for the last instant of time (t = 10600s).

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5.1 - Some Deterministic Solutions 77

5.1 Some Deterministic Solutions

Without carry out the UQ analysis, some information about what happens by changing the initial conditions can be found simply considering the sim- ulations of LPAC concerning a particular input tern. Considering the 343 quadrature points, one for each triplet of input values, it is possible to extrap- olate the deterministic solutions corresponding to the following situations:

• input value: ¯ µ, ¯ σ, ¯ ψ;

• input value: µ max , ¯ σ, ¯ ψ;

• input value: µ min , ¯ σ, ¯ ψ;

• input value: ¯ µ, σ max , ¯ ψ;

• input value: ¯ µ, σ min , ¯ ψ;

• input value: ¯ µ, ¯ σ, ψ max ;

• input value: ¯ µ, ¯ σ, ψ min ;

¯

µ, ¯ σ, ¯ ψ are the mean values of the input PDFs and they are equal to 1φ, 1.5φ and 0.7 respectively. The maximum and the minimum values of the input parameters are reported in Tab.5.1.

µ max 2φ µ min 0φ σ max 1.7φ σ min 1.3φ ψ max 0.9 ψ min 0.5

Table 5.1: Maximum and minimum values of the initial distributions.

Each input value has been analyzed and the effect of its maximum vari-

ation has been considered.

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5.1 - Some Deterministic Solutions 78

5.1.1 Effect of the mean size

Fig.(5.5) shows the change in the parcels size distribution due to the variation of the mean value of the input diameters.

−10 0 0 10

500 1000 1500

φ

Number of parcels

Stripe 1−Air

−10 0 0 10

500 1000 1500

φ Stripe 2−Air

−10 0 0 10

500 1000 1500

φ Stripe 3−Air

−10 0 0 10

500 1000 1500

φ Stripe 4−Air

−10 0 0 10

2000 4000 6000 8000

φ

Number of parcels

Stripe 1−Ground

−10 0 0 10

2000 4000 6000 8000

φ Stripe 2−Ground

−10 0 0 10

2000 4000 6000 8000

φ Stripe 3−Ground

−10 0 0 10

2000 4000 6000 8000

φ Stripe 4−Ground

µ inlet = 0 µ inlet = 1 µ inlet = 2

Figure 5.5: Parcels size distribution in each region of the domain associated with the input value of σ and ψ equal to 1.5φ and 0.7 respectively, while µ varies from the minimum value of 0φ to the maximum one of 2φ. Also the mean solution associated to µ = 1φ is reported.

It is possible to observe that with the decreasing of the mean diameter of input distribution (in the Krumbein scale µ = 2φ corresponds to the smallest particles), a larger number of parcels is able to reach the right boundary of the domain, represented by Stripe 4.

Furthermore, we observe that the mean size observed in all the stripes,

except Stripe 1 in air, does not depend on the mean size of the particles

released in the atmosphere. Consequently, the mean size observed in the

air or on the ground in a particular stripe is not representative of the ini-

tial one. The statistical parameters that characterize the number of parcels

distribution are reported in tables: Tab.(5.2), Tab.(5.3), Tab.(5.4).

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5.1 - Some Deterministic Solutions 79

µ=0φ

1-Air 2-Air 3-Air 4-Air

Mean [φ] 0.25 2.03 2.91 3.36

Std.Dev [φ] 1.37 0.74 0.53 0.51

Skewness 0.17 0.75 1.15 1.36

Kurtosis 0.13 0.72 0.63 4.07

Num. Par. 3445 859 297 158

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] -0.72 1.30 2.40 2.90

Std.Dev [φ] 1.06 0.46 0.24 0.15

Skewness -0.71 0.02 -0.05 0.01

Kurtosis 0.27 -0.59 -0.81 -0.67

Num. Par. 17334 5429 754 186

Table 5.2: Statistical parameters of parcels distribution referred to the lower input value of µ, (coarse particles).

µ=1φ

1-Air 2-Air 3-Air 4-Air

Mean [φ] 1.10 2.29 3.10 3.50

Std.Dev [φ] 1.40 0.86 0.64 0.60

Skewness 0.15 0.62 0.95 1.77

Kurtosis 0.10 0.77 1.30 2.63

Num. Par. 3582 1765 795 530

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] -0.72 1.30 2.40 2.90

Std.Dev [φ] 0.89 0.46 0.23 0.17

Skewness -0.89 -0.14 0.00 -0.03

Kurtosis 0.71 -0.63 -0.50 -0.59

Num. Par. 11146 7745 1635 504

Table 5.3: Statistical parameters of parcels distribution referred to the mean

input value of µ (1φ).

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5.1 - Some Deterministic Solutions 80

µ=2φ

1-Air 2-Air 3-Air 4-Air

Mean [φ] 1.99 2.65 3.29 3.71

Std.Dev [φ] 1.44 1.00 0.80 0.75

Skewness 0.15 0.62 0.95 1.77

Kurtosis 0.13 0.72 0.63 4.07

Num. Par. 3445 859 297 158

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] -0.72 1.30 2.40 2.90

Std.Dev [φ] 1.06 0.46 0.24 0.15

Skewness 0.14 0.71 1.11 1.20

Kurtosis 0.03 0.46 1.05 1.52

Num. Par. 3588 2768 1735 1209

Table 5.4: Statistical parameters of parcels distribution referring to the upper input value of µ (2φ).

5.1.2 Effect of the standard deviation

Keeping constant the mean diameter and the sphericity, Fig.(5.6) reveals

the effects of the extreme variation of the standard deviation of the initial

distribution. According to the classification proposed in Tab.(3.1), the initial

distribution is poorly sorted, which means that the variability of the parcels

dimensions around the mean value is high. The statistical values of the

frequency distributions are presented in the table below. (see Tab.(5.5),

Tab.(5.6), Tab.(5.7)).

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5.1 - Some Deterministic Solutions 81

−10 0 0 10

500 1000 1500

φ

Number of parcels

Stripe 1−Air

−10 0 0 10

500 1000 1500

φ Stripe 2−Air

−10 0 0 10

500 1000 1500

φ Stripe 3−Air

−10 0 0 10

500 1000 1500

φ Stripe 4−Air

−10 0 0 10

2000 4000 6000

φ

Number of parcels

Stripe 1−Ground

−10 0 0 10

2000 4000 6000

φ Stripe 2−Ground

−10 0 0 10

2000 4000 6000

φ Stripe 3−Ground

−10 0 0 10

2000 4000 6000

φ Stripe 4−Ground

σ inlet = 1.3 σ inlet = 1.5 σ inlet = 1.7

Figure 5.6: Parcels size distribution associated to µ = 1φ and ψ = 0.7 (mean input value), while σ varies from its minimum to its maximum value, which are 1.3ψ and 1.7ψ respectively.

σ=1.3φ

1-Air 2-Air 3-Air 4-Air

Mean [φ] 1.06 2.15 2.96 3.35

Std.Dev [φ] 1.23 0.77 0.54 0.50

Skewness 0.16 0.61 1.11 1.45

Kurtosis 0.12 0.58 1.21 3.08

Num. Par. 3621 1728 676 430

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] -0.15 1.39 2.41 2.89

Std.Dev [φ] 0.78 0.45 0.23 0.17

Skewness -0.86 -0.14 0.00 0.08

Kurtosis 0.67 -0.59 -0.60 -0.57

Num. Par. 10911 8629 1608 441

Table 5.5: Output results referred to the input triplet: µ = 1φ, σ = 1.3φ and

ψ = 0.7.

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5.1 - Some Deterministic Solutions 82

σ=1.5φ

1-Air 2-Air 3-Air 4-Air

Mean [φ] 1.10 2.29 3.10 3.50

Std.Dev [φ] 1.40 0.86 0.64 0.60

Skewness 0.15 0.62 0.95 1.77

Kurtosis 0.10 0.77 1.30 2.63

Num. Par. 3582 1765 795 530

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] -0.72 1.30 2.40 2.90

Std.Dev [φ] 0.89 0.46 0.23 0.17

Skewness -0.89 -0.14 0.00 -0.03

Kurtosis 0.71 -0.63 -0.50 -0.59

Num. Par. 11146 7745 1635 504

Table 5.6: Output results referred to the input triplet: µ = 1φ, σ = 1.5φ and ψ = 0.7.

σ=1.7φ

1-Air 2-Air 3-Air 4-Air

Mean [φ] 1.13 2.42 3.24 3.63

Std.Dev [φ] 1.57 0.95 0.76 0.71

Skewness 0.19 0.83 1.14 1.38

Kurtosis 0.08 0.94 1.16 2.52

Num. Par. 3540 1796 917 623

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] -0.47 1.41 2.43 2.91

Std.Dev [φ] 1.01 0.46 0.23 0.17

Skewness -0.91 -0.14 -0.07 -0.17

Kurtosis 0.73 -0.65 -0.48 -0.70

Num. Par. 11319 6997 1638 482

Table 5.7: Output results referred to the input triplet: µ = 1φ, σ = 1.7φ and

ψ = 0.7.

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5.1 - Some Deterministic Solutions 83

5.1.3 Effect of the sphericity

The last deterministic case concerns the variation of the sphericity (see Fig.(5.7)). For this analysis the statistical parameters are reported in Tab.(5.8), Tab.(5.9), Tab.(5.10).

−10 0 0 10

500 1000 1500

φ

Number of parcels

Stripe 1−Air

−10 0 0 10

500 1000 1500

φ Stripe 2−Air

−10 0 0 10

500 1000 1500

φ Stripe 3−Air

−10 0 0 10

500 1000 1500

φ Stripe 4−Air

−10 0 0 10

2000 4000 6000 8000

φ

Number of parcels

Stripe 1−Ground

−10 0 0 10

2000 4000 6000 8000

φ Stripe 2−Ground

−10 0 0 10

2000 4000 6000 8000

φ Stripe 3−Ground

−10 0 0 10

2000 4000 6000 8000

φ Stripe 4−Ground

ψ inlet = 0.5 ψ inlet = 0.7 ψ inlet = 0.9

Figure 5.7: Parcels size distribution in each region of the domain associated

with the input value of µ and σ equal to 1φ and 1.5φ respectively, while ψ

varies from the minimum value of 0.5 to the maximum one of 0.9.

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5.1 - Some Deterministic Solutions 84

ψ=0.5

1-Air 2-Air 3-Air 4-Air

Mean [φ] 1.01 1.76 2.77 3.26

Std.Dev [φ] 1.46 1.07 0.72 0.65

Skewness 0.15 0.26 1.02 1.30

Kurtosis -0.04 0.51 1.04 2.15

Num. Par. 3685 2645 1168 718

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] -1.02 0.661 2.05 2.59

Std.Dev [φ] 0.86 0.76 0.25 0.20

Skewness -0.75 -0.55 -0.08 -0.11

Kurtosis 0.16 -0.17 -0.50 -0.81

Num. Par. 4030 11934 2382 718

Table 5.8: Output values referred to the input triplet: µ = 1φ, σ = 1.5φ and ψ = 0.5.

ψ=0.7

1-Air 2-Air 3-Air 4-Air

Mean [φ] 1.10 2.29 3.10 3.50

Std.Dev [φ] 1.40 0.86 0.64 0.60

Skewness 0.15 0.62 0.95 1.77

Kurtosis 0.10 0.77 1.30 2.63

Num. Par. 3582 1765 795 530

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] -0.72 1.30 2.40 2.90

Std.Dev [φ] 0.89 0.46 0.23 0.17

Skewness -0.89 -0.14 0.00 -0.03

Kurtosis 0.71 -0.63 -0.50 -0.59

Num. Par. 11146 7745 1635 504

Table 5.9: Output values referred to the input triplet: µ = 1φ, σ = 1.5φ and

ψ = 0.7.

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5.1 - Some Deterministic Solutions 85

ψ=0.9

1-Air 2-Air 3-Air 4-Air

Mean [φ] 1.26 2.65 3.34 3.70

Std.Dev [φ] 1.37 0.76 0.60 0.57

Skewness 0.09 0.89 1.23 1.47

Kurtosis 0.19 1.00 1.31 3.02

Num. Par. 3236 1246 589 418

1-Ground 2-Ground 3-Ground 4-Ground

Mean [φ] 0.03 1.88 2.73 3.15

Std.Dev [φ] 0.98 0.35 0.20 0.15

Skewness -0.83 0.01 -0.08 0.05

Kurtosis 0.48 -0.64 -0.61 -0.68

Num. Par. 15556 5414 1131 364

Table 5.10: Output values referred to the input triplet: µ = 1φ, σ = 1.5φ and ψ = 0.9.

From the previous data it emerges that the characteristics of the initial

distribution are maintained only in Stripe 1-Air. The input parameters that

mainly influence the parcels distribution seem to be µ and ψ. With the chang-

ing in the mean diameter of the input distribution, it emerges that the total

number of parcels present on the domain varies a lot, since it pass from about

14000 units for µ = 0φ, to about 30000 parcels for µ = 2φ. It is interesting

to observe that the trend of the mean diameter of the parcels distribution

on the ground is approximately independent by the initial sorting, but i is

especially related to the sphericity. Also the standard deviation computed

in the stripes (both in air than on ground) is not particularly influenced by

the initial one. It is possible to observe that the increase of the sphericity

does not significantly affect the total number of parcels (30000 units), but it

is a leading parameter concerning the way in which the parcels are divided

on the domain.

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5.2 - Polynomial Response Functions 86

5.2 Polynomial Response Functions

While the previous results gives an initial idea of the dependence of the parcels size distribution from the 3 investigated parameters at the inlet, to obtain a full picture of the variability of the results and their sensitivities we rely on the UQ analysis. Entering in the merits of the UQ analysis, first of all it is possible to plot the polynomials which actually are the output of the PCEg method. Fig.(5.8) shows the polynomial response function associated to the mean particles diameter in the Stripe 1-Air. Since the polynomial is a function of the three input variables, each plot has been made keeping con- stant one parameter and varying the other two in their range of definition. In this way it is possible to observe how the input uncertainty spreads through the output. In this particular case we find the previous results concerning the variation of the mean diameter in Stripe 1-Air, variation which accurately reproduces the input uncertainty.

A different situation is reported in Fig.(5.9) where the output parameter taken into account is the mean diameter of the parcels in Stripe 1-Ground.

Here the input variability referring to µ is not maintained and the sphericity

shows its leading role, while the initial standard deviation seems to not be

particularly influential.

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5.2 - Polynomial Response Functions 87

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1 0 0.5 1 1.5 2 2.5

σ ψ

Mean diameter Stripe 1−Air

(a) Polynomial response function plotted for µ = 1φ, while σ and ψ vary in their input range.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1 0 0.5 1 1.5 2 2.5

µ ψ

Mean diameter Stripe 1−Air

(b) Polynomial response function plotted for σ = 1.5φ, while µ and ψ vary in their input range.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5 1 0 0.5 1 1.5 2 2.5

µ σ

Mean diameter Stripe 1−Air

(c) Polynomial response function plotted for ψ = 0.7, while µ and σ vary in their input range.

Figure 5.8: Sketch of the polynomial response function associated to the

mean value of the parcels diameter in Stripe 1-Air. The domain reported on

x-y axis refers to the range of existence of Legendre polynomials[-1,+1].

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5.2 - Polynomial Response Functions 88

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5

−1.5 1

−1

−0.5 0 0.5

σ ψ

Mean diameter Stripe 1−Ground

(a) Polynomial response function plotted for µ = 1φ, while σ and ψ vary in their input range.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5

−1.5 1

−1

−0.5 0 0.5

µ ψ

Mean diameter Stripe 1−Ground

(b) Polynomial response function plotted for σ = 1.5φ, while µ and ψ vary in their input range.

−1 −0.5 0 0.5 1

−1

−0.5 0 0.5

−1.5 1

−1

−0.5 0 0.5

µ σ

Mean diameter Stripe 1−Ground

(c) Polynomial response function plotted for ψ = 0.7, while µ and σ vary in their input range.

Figure 5.9: Mean parcels diameter associated to Stripe 1-Ground. The

domain reported on x-y axis refers to the range of existence of Legendre

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5.3 - Statistical Analysis of the Response Functions 89

5.3 Statistical Analysis of the Response Func- tions

5.3.1 Cumulative and Probability Distribution Functions

The polynomial response functions can be also used to evaluate the variation of the output values (see Tab.(3.4)) for a wide number of initial conditions, without running the simulations again, but simply using the polynomials as

"emulators" of the Lagrangian code. In this way the computational time is extremely reduced. We have set DAKOTA in order to obtain 10000 sam- ples of the output quantities for a likewise number of random input triplets.

These samples are then used to compute the statistical parameters of each re- sponse value (i.e. mean value, standard deviation, skewness, kurtosis) and to evaluate the Cumulative Distribution Functions (CDFs) and the Probability Distribution Functions (PDFs) of the response values.

In the input file of DAKOTA it is possible to specify the probability levels at which the CDFs are evaluated: [0.01; 0.05; 0.25; 0.50; 0.75; 0.95; 0.99].

Fig.(5.10) shows the CDFs and the PDFs of the mean parcels diameter

computed in every stripes both in the air and on the ground. The red curve of

each plot represents the input distribution, which has been set as an uniform

distribution over the range of [0φ; 2φ].

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5.3 - Statistical Analysis of the Response Functions 90

−2 0 0 2 4

0.2 0.4 0.6 0.8 1

µ Stripe 1−Air

CDF

−2 0 0 2 4

0.2 0.4 0.6 0.8 1

µ Stripe 2−Air

CDF

−2 0 0 2 4

0.2 0.4 0.6 0.8 1

µ Stripe 3−Air

CDF

−2 0 0 2 4

0.2 0.4 0.6 0.8 1

µ Stripe 4−Air

CDF

−2 0 0 2 4

0.2 0.4 0.6 0.8 1

µ Stripe 1−Ground

CDF

−2 0 0 2 4

0.2 0.4 0.6 0.8 1

µ Stripe 2−Ground

CDF

−2 0 0 2 4

0.2 0.4 0.6 0.8 1

µ Stripe 3−Ground

CDF

−2 0 0 2 4

0.2 0.4 0.6 0.8 1

µ Stripe 4−Ground

CDF

(a)

0 2 4

0 0.5 1 1.5 2

Stripe 1−Air

µ

PDF

0 2 4

0 0.5 1 1.5 2

Stripe 2−Air

µ

PDF

0 2 4

0 0.5 1 1.5 2

Stripe 3−Air

µ

PDF

0 2 4

0 0.5 1 1.5 2

Stripe 4−Air

µ

PDF

−2 0 0 2 4

0.5 1 1.5 2 2.5

Stripe 1−Ground

µ

PDF

−2 0 0 2 4

0.5 1 1.5 2 2.5

Stripe 2−Ground

µ

PDF

−2 0 0 2 4

0.5 1 1.5 2 2.5

Stripe 3−Ground

µ

PDF

−2 0 0 2 4

0.5 1 1.5 2 2.5

Stripe 4−Ground

µ

PDF

(b)

Figure 5.10: CDFs (a) and PDFs (b) related to the mean parcels diameter.

(18)

5.3 - Statistical Analysis of the Response Functions 91

From the figure it is possible to observe that the input variability of µ is reproduced only in Stripe 1-Air, while, going away from the inlet, the mean diameter not only becomes smaller but also the original variability is lost (see (Tab.5.11)). This is evident since the CDFs tend to became steeper and they are localized around a particular value of µ, similarly the PDFs histograms of the mean diameter are shifted to higher values of µ (which means finer particles according to the Krumbein scale), and their dispersion decreases going away from the inlet. A quite good representation of the mean diam- eter input distribution can be obtained sampling in Stripe 2-Ground, which means from 20Km to 30Km from the inlet.

1-Air 2-Air 3-Air 4-Air

µ mean [φ] 1.12 2.27 3.08 3.51

µ std. dev [φ] 0.53 0.35 0.22 0.19

1-Ground 2-Ground 3-Ground 4-Ground

µ mean [φ] -0.39 1.33 2.41 2.90

µ std. dev. [φ] 0.38 0.40 0.20 0.16

Table 5.11: Mean value and standard deviation referred to the PDFs associ- ated to the mean parcels diameter in every cell.

In order to reconstruct the input variability of the ash deposit and to

study the ash dispersion, it is fundamental to investigate also the behavior

of the sorting of the parcels, in other words the dispersion of the parcels

dimensions around the mean value must be considered. In Fig.(5.11) are

reported the CDFs and the PDFs of the standard deviation of the parcels

frequency distributions in the domain. At the release location for σ it has

been assumed a uniform PDF over the range [1.3φ; 1.7φ] (red curves on the

figures). Concerning the air situation, the initial poorly sorted distribution

is conserved only in Stripe 1-Air, in the other cells the parcels distributions

tend to become moderately sorted (the standard deviation decreases), even if

it continues to have a good range of variation (see Fig.(5.11b)). Overall it is

possible to say that, going away from the inlet, the parcels are finer and more

uniform in their size distribution, even if a small dispersion around the mean

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5.3 - Statistical Analysis of the Response Functions 92

value is kept. On the ground the situation is quite different from the air, since

not only the input variability is immediately lost (the deposit is moderately

sorted in Stripe 1-Ground and becomes well sorted in Stripe 4-Ground), but

also, differently from the air, the slope of the CDFs curves for the standard

deviation changes and becomes steeper. This means that, while it is possible

to reconstruct the input mean diameter distribution by sampling in Stripe

2-Ground, the input standard deviation can not be detected in a particular

stripe. In order to retrieve the input standard deviation of the ash cloud

particles, it is fundamental to take into account the overall ash deposit and,

even in this case, the lack of a right information about the air situation can

compromises the measurement. The statistical parameters that describes the

trend of the standard deviation in the domain are reported in Tab.(5.12).

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5.3 - Statistical Analysis of the Response Functions 93

0 1 2

0 0.2 0.4 0.6 0.8 1

σ Stripe 1−Air

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

σ Stripe 2−Air

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

σ Stripe 3−Air

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

σ Stripe 4−Air

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

σ Stripe 1−Ground

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

σ Stripe 2−Ground

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

σ Stripe 3−Ground

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

σ Stripe 4−Ground

CDF

(a)

0 1 2

0 1 2 3 4 5

Stripe 1−Air

σ

PDF

0 1 2

0 1 2 3 4 5

Stripe 2−Air

σ

PDF

0 1 2

0 1 2 3 4 5

Stripe 3−Air

σ

PDF

0 1 2

0 1 2 3 4 5

Stripe 4−Air

σ

PDF

0 1 2

0 10 20 30

Stripe 1−Ground

σ

PDF

0 1 2

0 10 20 30

Stripe 2−Ground

σ

PDF

0 1 2

0 10 20 30

Stripe 3−Ground

σ

PDF

0 0.5 1 1.5 2

0 5 10 15 20 25 30

Stripe 4−Ground

σ

PDF

(b)

Figure 5.11: CDFs (a) and PDFs (b) of the standard deviation of the parcels

distribution.

(21)

5.3 - Statistical Analysis of the Response Functions 94

1-Air 2-Air 3-Air 4-Air

σmean [φ] 1.41 0.89 0.11 0.10

σstd. dev. [φ] 0.10 0.13 0.66 0.61

1-Ground 2-Ground 3-Ground 4-Ground

σmean [φ] 0.91 0.51 0.23 0.17

σmean [φ] 0.12 0.13 0.01 0.01

Table 5.12: Mean value and standard deviation referred to the PDFs associ- ated to the standard deviation of the parcels distribution in every cell.

Fig.(5.12) shows the CDFs and the PDFs associated to the skewness value

of the parcels frequency distributions. We have not set an uncertainty con-

cerning this parameter and we have supposed that the input distribution was

symmetrical. From Fig.(5.12) and from the data reported in Tab.(5.13) it is

possible to see that the initial symmetry disappears. In air the distributions

become fine and then strongly fine skewed (see Tab.(3.2)), while on ground

we can detect a coarse skewed distribution in Stripe 1-Air and, going away

from the inlet, the deposit recovers its symmetry. This situation is due to

the natural tendency of the fine particles to follow the carrier flow, while the

presence of the coarse particles is leading on ground.

(22)

5.3 - Statistical Analysis of the Response Functions 95

0 1 2

0 0.2 0.4 0.6 0.8 1

Skewness Stripe 1−Air

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

Skewness Stripe 2−Air

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

Skewness Stripe 3−Air

CDF

0 1 2

0 0.2 0.4 0.6 0.8 1

Skewness Stripe 4−Air

CDF

−1 −0.5 0

0 0.2 0.4 0.6 0.8 1

Skewness Stripe 1−Ground

CDF

−1 −0.5 0

0 0.2 0.4 0.6 0.8 1

Skewness Stripe 2−Ground

CDF

−1 −0.5 0

0 0.2 0.4 0.6 0.8 1

Skewness Stripe 3−Ground

CDF

−1 −0.5 0

0 0.2 0.4 0.6 0.8 1

Skewness Stripe 4−Ground

CDF

(a)

0 1 2

0 2 4 6 8 10 12

Stripe 1−Air

Skewness

PDF

0 1 2

0 2 4 6 8 10 12

Stripe 2−Air

Skewness

PDF

0 1 2

0 2 4 6 8 10 12

Stripe 3−Air

Skewness

PDF

0 1 2

0 2 4 6 8 10 12

Stripe 4−Air

Skewness

PDF

−1 −0.5 0

0 2 4 6

Stripe 1−Ground

Skewness

PDF

−1 −0.5 0

0 2 4 6

Stripe 2−Ground

Skewness

PDF

−1 −0.5 0

0 2 4 6

Stripe 3−Ground

Skewness

PDF

−1 −0.5 0

0 2 4 6

Stripe 4−Ground

Skewness

PDF

(b)

Figure 5.12: Scketchs of the CDFs (a) and PDFs (b) associated with the

skewness value in each cell.

(23)

5.3 - Statistical Analysis of the Response Functions 96

1-Air 2-Air 3-Air 4-Air

Sk mean 0.15 0.64 1.09 1.42

Sk std. dev. 0.04 0.23 0.09 0.17

1-Ground 2-Ground 3-Ground 4-Ground

Sk mean -0.85 -0.29 -0.05 -0.08

Sk std. dev. 0.10 0.30 0.06 0.10

Table 5.13: Mean value and standard deviation referred to the PDFs associ- ated to the skewness value of the parcels distribution in every cell.

The last statistic parameter taken into account is the kurtosis. As in the previous case, we have not considered the kurtosis as an uncertain input parameter and we have chosen its initial value equal to zero. Fig.(??) and Tab.(5.14) show the trend of the CDFs and PDFs of the kurtosis computed in every stripe. The particles still in air and those fallen on the ground show an opposite behavior. The distributions become leptokurtic in air (positive values of kurtosis) and platykurtic on the ground (negative values of kurtosis).

This means that in air the dispersion around the mean value is due also by the

particles whose dimensions are far from the mean value, while on the ground

there is a major concentration of the particles around the mean value.

(24)

5.3 - Statistical Analysis of the Response Functions 97

0 2 4

0 0.2 0.4 0.6 0.8 1

Kurtosis Stripe 1−Air

CDF

0 2 4

0 0.2 0.4 0.6 0.8 1

Kurtosis Stripe 2−Air

CDF

0 2 4

0 0.2 0.4 0.6 0.8 1

Kurtosis Stripe 3−Air

CDF

0 2 4

0 0.2 0.4 0.6 0.8 1

Kurtosis Stripe 4−Air

CDF

−1 0 0 1

0.2 0.4 0.6 0.8 1

Kurtosis Stripe 1−Ground

CDF

−1 0 0 1

0.2 0.4 0.6 0.8 1

Kurtosis Stripe 2−Ground

CDF

−1 0 0 1

0.2 0.4 0.6 0.8 1

Kurtosis Stripe 3−Ground

CDF

−1 0 0 1

0.2 0.4 0.6 0.8 1

Kurtosis Stripe 4−Ground

CDF

(a)

0 2 4

0 2 4 6

Stripe 1−Air

Kurtosis

PDF

0 2 4

0 2 4 6

Stripe 2−Air

Kurtosis

PDF

0 2 4

0 2 4 6

Stripe 3−Air

Kurtosis

PDF

0 2 4

0 2 4 6

Stripe 4−Air

Kurtosis

PDF

−1 0 0 1

2 4 6

Stripe 1−Ground

Kurtosis

PDF

−1 0 0 1

2 4 6

Stripe 2−Ground

Kurtosis

PDF

−1 0 0 1

2 4 6

Stripe 3−Ground

Kurtosis

PDF

−1 0 0 1

2 4 6

Stripe 4−Ground

Kurtosis

PDF

(b)

Figure 5.13: CDFs (a) and PDFs (b) which show the trend of the kurtosis

value on the domain.

(25)

5.3 - Statistical Analysis of the Response Functions 98

1-Air 2-Air 3-Air 4-Air

Ku mean 0.08 0.73 1.04 2.77

Ku std.dev. 0.07 0.21 0.23 0.86

1-Ground 2-Ground 3-Ground 4-Ground

Ku mean 0.56 -0.26 -0.56 -0.63

Ku std.dev. 0.27 0.5 0.07 0.14

Table 5.14: Mean value and standard deviation referred to the PDFs associ-

ated to the kurtosis value of the parcels distribution in every cell.

(26)

5.3 - Statistical Analysis of the Response Functions 99

The number of the parcels present both in air than on ground decreases and loses variability moving away from the inlet (see Fig.(5.15)).

0 2000 4000

0 0.2 0.4 0.6 0.8 1

Parcels Stripe 1−Air

CDF

0 2000 4000

0 0.2 0.4 0.6 0.8 1

Parcels Stripe 2−Air

CDF

0 2000 4000

0 0.2 0.4 0.6 0.8 1

Parcels Stripe 3−Air

CDF

0 2000 4000

0 0.2 0.4 0.6 0.8 1

Parcels Stripe 4−Air

CDF

0 1 2

x 10

4

0

0.2 0.4 0.6 0.8 1

Parcels Stripe 1−Ground

CDF

0 1 2

x 10

4

0

0.2 0.4 0.6 0.8 1

Parcels Stripe 2−Ground

CDF

0 1 2

x 10

4

0

0.2 0.4 0.6 0.8 1

Parcels Stripe 3−Ground

CDF

0 1 2

x 10

4

0

0.2 0.4 0.6 0.8 1

Parcels Stripe 4−Ground

CDF

(a)

0 2000 4000

0 1 2 3

x 10 Stripe 1−Air

−3

Parcels

PDF

0 2000 4000

0 1 2 3

x 10 Stripe 2−Air

−3

Parcels

PDF

0 2000 4000

0 1 2 3

x 10 Stripe 3−Air

−3

Parcels

PDF

0 2000 4000

0 1 2 3

x 10 Stripe 4−Air

−3

Parcels

PDF

0 1 2 3

x 10

4

0

0.5 1 1.5

2 Stripe 1−Ground x 10

−3

Parcels

PDF

0 1 2 3

x 10

4

0

0.5 1 1.5

2 Stripe 2−Ground x 10

−3

Parcels

PDF

0 1 2 3

x 10

4

0

0.5 1 1.5

2 x 10 Stripe 3−Ground

−3

Parcels

PDF

0 1 2 3

x 10

4

0

0.5 1 1.5

2 x 10 Stripe 4−Ground

−3

Parcels

PDF

(b)

Figure 5.14: CDFs (a) and PDFs (b) of the number of parcels distribution

over the domain.

(27)

5.3 - Statistical Analysis of the Response Functions 100

It is important to remark that this data do not refer to the effective number of the particles, but they refer to the number of the packages of particles representative of a particular diameter. By the way the distribution of the number of the parcels (see Tab.(5.15)) can be used to understand the way in which the total ash mass has been divided on the domain. Considering the total number of packages released during the simulation (27460) and having supposed that all the parcels have the same mass, Tab.(5.16) shows the mass of particles in each cell as percentage of the total mass (%wt).

1-Air 2-Air 3-Air 4-Air

Num parcels mean 3517 1843 907 593

Num parcels std. dev. 182 693 489 338

1-Ground 2-Ground 3-Ground 4-Ground

Num parcels mean 10693 7693 1689 525

Num parcels std. dev. 4807 2164 656 259

Table 5.15: Mean value and standard deviation referred to the PDFs associ- ated to the number of parcels present in every cell.

1-Air 2-Air 3-Air 4-Air Tot Air

wt% 12.8 6.7 3.4 2.1 25

1-Ground 2-Ground 3-Ground 4-Ground Tot Ground

wt% 38.9 28.0 6.1 1.9 75

Table 5.16: Values of the relative mass fractions computed in every cell. It is evident that the mass is mostly concentrated on the ground.

Even if the 75% of the total mass can be found on the ground, we have pre-

viously seen that the information about the size distributions of the parcels

still in air are fundamental in order to retrieve the input ash conditions.

(28)

5.4 - Sensitivity Analysis of the Output Values 101

5.4 Sensitivity Analysis of the Output Values

Using the 10000 samples performed on the output polynomials, DAKOTA is able to carry out the Sensitivity Analysis of the output values. As said in Chapter 2, we consider the Main and the Total Sobol Indices. The results of how the output variability can be apportioned to the different uncertain input parameters, are reported in the paragraphs below.

5.4.1 Main Sobol Indices

Fig.(5.15) and Fig.(5.16) show the Main Sobol Indices of the output parame-

ters up to the third order of interaction. This means that not only the weight

on the output variability of each input parameter is considered, but also all

the possible interactions between up to three input parameters are taken into

account. What emerges in a very clear way is the leading role of µ and ψ

and of the interaction between them. The contribution of σ is restricted to

the standard deviation of the frequency distribution in Stripe 1-Air, while its

presence is minimal in the other cells, especially in the ground ones. Taking

into account what happens on the ground, Fig.(5.16) reveals that µ is the

main factor that affects the number of parcels fallen in each stripe, while ψ

is the leading parameter for the other output values. This reveals us that

while the initial sphericity is decisive on the position of the parcels fallen on

the ground, the number of parcels, and so the mass on the ground, mainly

depends on the mean input diameter.

(29)

5.4 - Sensitivity Analysis of the Output Values 102

Figure 5.15: Main Sobol Indices of the output values concerning the air distributions.The variability of the output values (i.e. mean diameter of parcels distribution µ, standard deviation σ, skewness (Sk), kurtosis (Ku) and number of parcels (P)) is due to the variation of the input values of µ and ψ.

Figure 5.16: Main Sobol Indices of the output values concerning the ash

deposit on the ground. In this case the contribution of ψ variation is leader

(30)

5.4 - Sensitivity Analysis of the Output Values 103

5.4.2 Total Sobol Indices

As said in Chapter 2, the usefulness of the Total Indices lies in their capability to quantify the amount of the output variability due to the variation of a particular input parameter considering its interactions with the other input values. Fig.(5.17) and Fig.(5.18) show the plot of the Total Sobol Indices associated to the output values.

Figure 5.17: Total Sobol Indices related to the air parcels distributions.

Figure 5.18: Total Sobol Indices related to the air parcels distributions.

(31)

5.4 - Sensitivity Analysis of the Output Values 104

The best use of the Total Sobol Indices consists in the calculation of the difference between them and the first order of the Main Sobol Indices. In this way we obtain clear information about the weight of the interaction between the input parameter analyzed and all the other input variables. There are not interactions between the input parameters if, for a particular output value, the total and the main indices have the same value. Figures from Fig.(5.19) to Fig.(5.26) report, for each cell, the comparison between the main and the total indices computed for every output value.

Figure 5.19: Main Sobol Indices and Total Sobol Indices computed for the

output values of Stripe 1-Air. The three graphs represent the contribution

to the output variability due to µ, σ and ψ respectively

(32)

5.4 - Sensitivity Analysis of the Output Values 105

Figure 5.20: Comparison between Main Sobol Indices and Total Sobol Indices computed for the output values of Stripe 2-Air.

Figure 5.21: Comparison between Main Sobol Indices and Total Sobol Indices

computed for the output values of Stripe 3-Air.

(33)

5.4 - Sensitivity Analysis of the Output Values 106

Figure 5.22: Comparison between Main Sobol Indices and Total Sobol Indices computed for the output values of Stripe 4-Air.

Figure 5.23: Comparison between Main Sobol Indices and Total Sobol Indices

computed for the output values of Stripe 1-Ground.

(34)

5.4 - Sensitivity Analysis of the Output Values 107

Figure 5.24: Comparison between Main Sobol Indices and Total Sobol Indices computed for the output values of Stripe 2-Ground.

Figure 5.25: Comparison between Main Sobol Indices and Total Sobol Indices

computed for the output values of Stripe 3-Ground.

(35)

5.5 - Analysis of the Drag Coefficient 108

Figure 5.26: Comparison between Main Sobol Indices and Total Sobol Indices computed for the output values of Stripe 4-Ground.

5.5 Analysis of the Drag Coefficient

The Sensitivity Analysis has shown that the output values are mainly af- fected by the input variability in µ and ψ. This can be more deeply inves- tigated by the analysis of the drag coefficient for particles in the sphericity and size ranges investigated. Recalling the expression of the drag coefficient (Eq.(1.10)), it is possible to study the effects of the input uncertainty on the value of C D . Fig.(5.27) reports the trend of C D as a function of the particles diameter expressed in meters for three different values of initial sphericity (the maximum, the minimum and the mean value). C D value is a function of the wind field, we have considered the values of drag coefficient computed at t = 10600s. There is more than one C D associated to a certain diameter;

this because particles with the same diameter can have different velocity and

so the drag coefficient changes. Comparing this graph with the one reported

in Fig.(5.28), it is possible to see that ψ is the main factor concerning the

variation of the C D . For instance, for particles with the same diameter (in

the range 5 − 6 ∗ 10 −3 ), but different sphericity, the C D jumps from C D ' 1

(36)

5.5 - Analysis of the Drag Coefficient 109

there is not a significant change in the value of the C D .

0 1 2 3 4 5 6

x 10

−3

1

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Particles Diameter (m)

Cd

ψ=0.5 ψ=0.7 ψ=0.9

Figure 5.27: Trend of C D as a function of the particles diameter for the three different values of sphericity.

0 1 2 3 4 5 6

x 10

−3

0

1 2 3 4 5 6

Particles Diameter (m)

Cd

µ = 0 φ µ = 1 φ µ = 2 φ

Figure 5.28: Trend of C D as a function of the particles diameter varying the

mean input diameter.

(37)

5.5 - Analysis of the Drag Coefficient 110

Fig.(5.29) shows the variation of the C D as a function of the Reynolds number. We can see that at high Reynolds number the sphericity is again the main parameter affecting the drag coefficient.

0 5000 10000 15000

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Re

Cd

ψ=0.5 ψ=0.7 ψ=0.9

(a) Drag coefficient as a function of Reynolds number for three different sphericity values:ψ = 0.5, ψ = 0.5, ψ = 0.7, ψ = 0.9.

0 500 1000 1500 2000 2500 3000

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

Re

Cd

ψ=0.5 ψ=0.7 ψ=0.9

(b) Trend of C D considering the lowest Reynolds numbers until Re = 3000.

Figure 5.29: Drag coefficient as a function of Reynolds number.

(38)

5.5 - Analysis of the Drag Coefficient 111

However SA has indicated that also µ plays a fundamental role in ash

dispersion and deposition. Particles with different diameter present a differ-

ent inertia and, hence, different interaction with the fluid flow. While the

larger fragments tend to have an independent behavior from the flow, the

finer ones more closely follow background flow motion. Recalling Eq.(1.22)

it is possible to observe that, having the finer particles a larger velocity than

the coarser ones, considering the relative velocity between the particles and

the flow, the drag force can become a thrust force. So, even if the smaller

particles have a greater drag coefficient, their relative velocity respect to the

carrier flow allow them to have a major acceleration phase and so to reach

the the largest distances from the vent (40Km).

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