Sensitivity study on quantitative methods by analysis of pressure

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Thermodynamics, Transport, and Fluid Mechanics

Sensitivity study on quantitative methods by analysis of pressure fluctuations in a fluidized bed Jianbin Wang, Xi Chen, and Wenqi Zhong Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b02937 • Publication Date (Web): 27 Aug 2018 Downloaded from http://pubs.acs.org on August 29, 2018

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Industrial & Engineering Chemistry Research

Sensitivity study on quantitative methods by analysis of pressure fluctuations in a fluidized bed Jianbin Wang†,‡, Xi Chen†,§, Wenqi Zhong†,§,*

†

Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of

Energy and Environment, Southeast University, Nanjing 210096, China

‡

Information Center, Zhaoqing University, Zhaoqing 526061, China

§

Centre for Simulation and Modelling of Particulate Systems, Southeast University – Monash

University Joint Research School, Suzhou 215123, China

ABSTRACT: The sensitivity and reliability of twenty-three data processing methods are studied by analyzing pressure fluctuations time series in a fluidized bed. Firstly, the concept of minimal stationary segment length is proposed to study signals stationarity. Secondly, three indexes estimating the sensitivity of methods are proposed and their algorithms are designed. The sensitivity problems of some methods are studied. The results show that the root causes of the SKEW (skewness) method sensitive to segment length include outliers and small value as denominator, and the STD (standard deviation) method is sensitive to data normalization because it depends on the range of time series. Finally, a common framework is proposed based on the 1

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three sensitive indexes and harnessed to evaluate the twenty-three methods. Two most reliable methods are picked and proved appropriate for identifying the flow regimes. Besides, the result also indicates that the time-domain methods have better reliability than frequency-domain and state-space methods.

KEYWORDS: fluidized beds; sensitivity study; sensitivity index; flow regime; pressure fluctuations

1. INTRODUCTION Flow regime is critical for the safe and efficient operating of the fluidized bed equipment as it dominates the mass and heat transfer and remarkably affects the chemical reaction in the fluidized bed. Time series analysis of pressure fluctuations is a common and useful way to identify flow regimes and reliable analysis methods are vital for successful flow regimes identification. Thus, this topic has attracted many researchers to contribute a large amount of works. van Ommen et al.1 reviewed at least 15 methods in time domain, frequency domain and state space to process the pressure fluctuations signals in order to characterize the flow dynamics and different flow regimes. The authors aimed to find out which methods can describe the relationship between the pressure characteristic and flow behavior best and gave valuable hints on how to use and interpret these methods.

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Numerous analysis methods are proposed or introduced to characterize the flow regimes so that they can be distinguished. Some basic methods have solid theoretical foundations, such as STD method from statistics, entropy method from informatics, wavelet analysis from mathematics and so on. Some methods are derived from the basic methods. Luo et al.

2

investigated the flow regimes and their transitions in an internal loop airlift reactor and concluded that HHT, Hurst exponent and chaotic analysis have great potential for the flow regime identification in airlift reactors. Sheikhi et al.

3

investigated a liquid-solid fluidized bed

with several analysis methods and concluded that the average power of frequency spectrum could predict two regime transition points successfully from pressure signals. Gao et al. 4 utilized multivariate multiscale sample entropy (MMSE) for different flow conditions and found that the MMSE method uncovered the dynamic flow behavior governing the transition from slug flow to churn flow. Llop et al. 5 investigated the gas-solid fluidization behavior in different flow regimes and found out that the flow regimes could be characterized by the dynamic moments method accurately. Briongos and Soler

6

used wide band energy analysis and state space analysis to

characterize the different fluidization regimes of a gas–solid fluidized bed, Gomez-Hernandez et al. 7 proposed the objective frequency division method and enhanced wide band energy analysis. Many researchers shared their creative ideas in this area. 8-15 Because gas-solid flow may exhibit many characteristics in different flow regimes, such as stochastic, broadband spectrum and so on. It is natural to use a variety of methods to analyze16, 17 3



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and compare to find which methods are more appropriate. Some researchers used statistical and frequency analysis methods18-20 to characterize the flow regimes and pointed out that some methods might be more appropriate. Li et al.

21

found three flow regimes and two transition

points are successfully identified by the methods of statistics, Hurst, Hilbert-Huang transform and Shannon entropy, they also found that the Hilbert-Huang transform and Shannon entropy analysis methods offered very high resolution in identifying the different flow regimes. Luo et al. 22

compared the effectiveness of some methods for flow regime identification, and ranked the

methods as follows: wavelet entropy > HOS > SPWVD > WT. Llop et al.

23

used various

recurrence quantification analysis parameters (RQA) and found out that the most appropriate ones in characterizing and classifying the fluidization regimes were LAM and DET, and RET2. The comparison strategy is popular in other research areas as well. 24 Despite the success of these works, there is still room for improvement. For example, the methods investigated in the existing comparison research are still incomplete especially for some newly proposed ones, the categories of methods are not comprehensive, and so on. More importantly, the study on the properties of the method itself is neglected, such as reliability, which causes incomplete understanding and improper use of a method. Since the time series of pressure fluctuations in fluidized beds are almost unpredictable, the outcomes (i.e. characteristic numbers) obtained by the analytical methods are essentially estimates. The quality of these

4


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estimates may be uneven under the same condition. If we have a way to assess the reliability of this estimate, we can assess the reliability of the method. The main aim of this work is to pick the appropriate method set for flow regimes characterization by comparison. Based on the main aim and above considerations, some improvements are made. Firstly, twenty-three methods in time domain, frequency domain and state space are selected to make the comparison study sufficient and comprehensive. Secondly, the reliability of methods is investigated and discussed by the sensitivity analysis of data from different experimental factors like measuring positions, particle size and static bed heights. Finally, a framework for assessing and selecting the analysis methods is proposed. In the opinion of the authors, systemic sensitivity study of quantitative analysis methods has been rarely reported in terms of flow regimes characterization. However, Bartels et al. proposed a methodology25,

26

to compare the high selectivity of methods on agglomeration detection in

fluidized beds and found some suitable ones. The idea is undoubtedly enlightening for this article. This article does not address the following situations (not limited to these): the performance comparison of flow regimes recognition in multiphase flow system using classification models such as ANN, SVM and so on, 27-31 the sensitivity comparison for different devices, 32-35 different measurement technologies. 36-39

5



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2. EXPERIMENTS AND DATA The schematic diagram of the experimental system is shown in Figure 1. The system is composed of two parts: fluidized bed subsystem and absolute pressure (AP) measurement subsystem. For the sake of brevity, only one sensor wiring is drawn, in fact twenty-three sensors and three NI-9203 modules are used.

7

1

9 8 0 1 2 3 4 5 6 7 8 9 NI 9203

NI cDAQ-9174

6

3 5 2

+

4

_

10

1. Roots blower 2. Valve 3. Rotameters 4. Air chamber 5. Air distributor 6. Fluidized bed 7. Expansion section 8. Absolute-pressure sensor 9. DAQ device 10. DC power

Figure 1. Schematic diagram of the experimental system. The main body of the fluidized bed is a cylindrical column with internal diameter of 0.15 m and total height of 4.3 m. The air distributor is made of a stainless steel perforated plate with an open area of 2.45%. The experiment is carried out at room temperature and atmospheric pressure. The variable experimental conditions include the diameter of quartz sand particles, static bed height and gas velocity. The experimental conditions, measurement parameters and 6


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transition velocities of particles are listed in Table 1. The experiment data of six groups contains 2277 time series that consisting of about 150 million data points totally. Twenty-three AP measurement points are arranged at different heights that can be represented by a vector (i.e. [-5, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 60, 70, 80, 90, 100, 110, 130, 150, 170, 190, 210, 230] cm). -5 corresponding to the position in air chamber is 5 cm under the air distributor. Other positive values indicate the position heights based on the air distributor. Table 1. Experimental conditions, measurement parameters and transition velocities Group

1

2

3

4

5

Particle

P1

P2

Density (kg/m3)

2300

2300

Diameter (mm)

0.3~0.5

0.7~0.9

Geldart's classification

B

D

H0 (cm)

10

Gas velocity (m/s) [cases number]

20

30

10

20

6

30

1~2: 0.16~2.20[18]

4~5: 0.25~2.20[15]

3: 0.19~2.20[17]

6: 0.25~2.52[16]

Umf (m/s)

0.20

0.44

Uc (m/s)

1.19

1.63

Sampling frequency (Hz)

200

200

Data length

65536

65536

Number of time series

1219

1058

3. METHODS FOR ANALYSIS AND COMPARISON Twenty-three mathematical methods studied are selected from the literature. These methods can be categorized into nine time-domain methods, eight frequency-domain methods and six 7



state-space methods. The names, abbreviations, and sources of these methods are listed in Appendix SA. We try to cover as many methods as we can from the published works but there is still a possibility missing some. Besides, our focus is on how to compare and evaluate these methods. In the previous literature, it is difficult to avoid subjectivity factor on the comparison of the methods because of the lack of objective evaluation quantities. Then, how to evaluate these methods objectively? The first thing that comes to mind is the reliability or sensitivity of the method output (i.e. characteristics number). To answer these questions, three sensitivity indexes are proposed to make it feasible to compare sensitivity of methods objectively.

4. RESULTS AND DISCUSSION In the first section, the concept of minimal stationary segment length is proposed. The second section is the core of the article, three sensitive indexes and related algorithms are designed, and the sensitivity problems of some methods are analyzed except the sensitivity index Sp because it is designed to be independent of experimental conditions. In the third section, an evaluation framework is designed and harnessed to identify the reliable methods. The last section introduces two applications of sensitivity study.

8


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4.1. Minimum of stationary segment length – SSLmin The pressure signals can exhibit different mean value stationarity under different gas velocities or flow regimes. The mean value obtained from each segment may deviate greatly if a very short segment length is used. The signal segment length increases continuously, the difference of mean values of segments keeps on decreasing until it is reduced to a reasonable range of deviation, then this length is defined as the minimum of stationary segment length (SSLmin). In this section, we first propose an algorithm to determine the SSLmin. And the flowchart of the algorithm is shown in Figure 2. Mean (mTs) and standard deviration (sTs)

A time series (ts) Segment length (segLen = 100, 200, ...)

1. The ts is divided into n segments

n segments of ts

2. The mean of each segment (mSeg) is calculated 3. Stability judgement of n segments

Repeat n times

|mSeg - mTs| ≤ ε * sTs

No, then next No = 0

Yes = 1

A n-dimensional vector of stability judgement (sjv)

4. All elements of sjv are 1 SSLmin = segLen

Yes

End

Figure 2. Flowchart of algorithm 1. Algorithm 1: Search the minimum of stationary segment length (SSLmin)

9



1. For a time series (ts) given a segment length (segLen), divide it into n non-overlapping segments (n = ⌊length (ts)/segLen⌋, ⌊⌋ is mathematical floor function). 2. The mean of each segment (mSeg) is calculated, and then the mean and standard deviation of the ts (mTs and sTs) are calculated. 3. The stationarity judgement of each segment is made using a discriminant (i.e. |mSeg - mTs| ≤ ε* sTs, ε is threshold coefficient). If true, the segment is stationary else nonstationary, a vector of stationarity judgement (sjv) with n dimensions will be obtained. 4. When all the elements of the n-dimensional vector are 1 (=stationary), then the ts is stationary for the given segLen, otherwise it is nonstationary. If the ts is nonstationary, the segLen is assigned a greater value and steps 1~4 are repeated. Otherwise, the SSLmin of the ts is found and the algorithm ends. Figure 3 displays the variation of SSLmin values. The X-axis is the fluidization number (fn) and the Y-axis is the AP probe height (h). The SSLmin can be regarded as a function of two variables here (i.e. SSLmin = f(fn, h)). The discrete form of SSLmin is a matrix. For an Uf, the time series sampled from m heights are processed by Algorithm 1 to generate an m-dimensional column vector. Then, an m-by-n SSLmin matrix is obtained from n cases of Uf and a figure can be created based on the matrix. The vertical solid line indicates the Umf and the vertical dashed line indicates the Uc.

10


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(a) 2.3

U mf

2.3

12000

1.7 1.5

10000

1.3 8000

1.1 0.9

6000

0.7 Region I

0.5 0.3 0.2 0.1 0

4000

400

2000

2500

III

The height of AP probe, h (m)

14000

1.9

Region II 1

2

3

4

5

6

7

8

9

10

U mf

2.1 Minimum of stable segment length - SSL min


2.1

(b)

16000

Uc

11

14000

1.9 12000

1.7 1.5

10000

1.3 8000

Region I

1.1 0.9

6000

Region II

0.7 400

0.5

4000

2500

0.3 0.2 0.1 0

Region III 2000

2500

1

The fluidization number, U f /Umf (-)

16000

Uc

2

3

4

5

6

7

8

9

10

11


(d)

Region I

Region I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Minimum of stable segment length - SSL min

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III

III

II Uf / Umf = 5.34

II Uf / Umf = 5.97

Uf / U mf = 6.6

Figure 3. SSLmin and stationary regions (ε = 0.1, data of group 1~3 is used) (a) H0 = 10cm (b) H0 = 20cm (c) H0 = 30cm (d) some flow patterns of (c) (0.2 seconds between two images). As shown in Figure 3, the stability (or instability) of gas-solid interaction with different fn and

h in the fluidized bed may be characterized by the SSLmin. That is, the smaller SSLmin (i.e. time window) means the signal is more stationary and also means the interaction is more stable, while bigger SSLmin has opposite meaning. Regions I, II and III represents different combination of operating conditions and bed positions according to the SSLmin value. Region I means SSLmin < 400, Region II means 400 < SSLmin < 2500, and Region III means SSLmin > 2500. In the fixed

11



bed regime (i.e. Uf0.3~0.5m) and insensitive elsewhere with fn being less than 1.8. After that, the sensitive region appears only in the middle of the bed close to the bed surface, and the area of sensitive region in turbulent bed is larger. In Figure 6c, the DWTENT method has more serious sensitivity problem and the insensitive regions almost appear only in the upper part of the bed. Taking the point A and B in Figure 6a as examples, the mechanism of sensitivity problem of SKEW method is discussed in detail. Figure 7a shows the pressure signal at point A, the outliers are clearly observed and distribute non-uniformly in the signal. The interval of outliers varies from 1000 to 10000, which overlaps with the range of segment length. Thus, the sensitivity 17



problem can be attributed to the outliers. On the other hand, few outliers can be observed in the signal at point B shown in Figure 7b, therefore the outliers cannot completely explain the sensitivity problem. But the data points in the signal distribute symmetrically on both sides of the mean. It means the skewness is close to zero, and the actual SKEW value is 0.002. Thus, tiny changes can be magnified by step 4 in algorithm 2 because small characteristic number acts as the denominator, and that is the root cause. (a)

2.3

2

1.5

1.1

0.5

A

0.9

0.5 1.5

0.1

1

0.7 0.5

X: 2.29 B Y: 0.2 S l = 1.35

Umf

0.5

Uc 0

1

2

3

4

1.7

2

1.5 1.3 1.1

0.7


0.5

1

0.5 0.1

0.3 0.2 0.1 0

Umf

0.5

Uc 0

1

2


2.1

1.5

0.9

5

2.3

0.5

3

4

5

The fluidization number, U f /Umf (-) (c)

Umf

3

Uc 2.5

1.9 1.7

2

1.5 1.3 1.1

1.5

0.1

0.9 0.5

0.7

1

0.5

2

0.5

0.3 0.2 0.1 0

Sensitivity index of segment length – Sl

1.3

X: 2 0.1 Y: 0.7 S l = 0.69

2.5

1.9


2.5

0. 1

0.5

1.7


2



3

2.1

1.9

0.3 0.2 0.1 0

(b)

2.3

3

2.1

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.5

0.1

0.1 0

1

2

3

4

5


Figure 6. Sensitive regions of Sl (data of group 6 is used) (a) SKEW method (b) PSDMF method (c) DWTENT method. 18


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The above discussion indicates that the segLen factor has a non-negligible effect on the outcome of some mathematical methods, and the sensitivity problem in some regions (called sensitivity regions) is especially serious. The sensitive regions are also closely related to the flow regimes. If treating the mathematical methods as statistical asymptotically unbiased estimation, then it may mean that the reliable methods of Sl approaches to the exact values quickly, while the sensitive methods are the opposite. (a)

0.08

(b)

4 3.5

Pressure amplititude (kPa)

0.06

Pressure amplititude (kPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.04 0.02 0 -0.02 -0.04 -0.06

3 2.5 2 1.5 1 0.5

-0.08

0 0

10

20

30

40

50

60

70

80

90

0

10

20

30

Time (s)

40

50

60

70

80

90

Time (s)

Figure 7. Two Signals segments at A and B (marked in Figure 6a). (a) Uf/Umf = 2, h = 0.7m, Sl = 0.69 (b) Uf/Umf = 2.29, h = 0.2m, Sl = 1.35.

4.2.2. The sensitivity index of data normalization – Sn Data normalization is beneficial to reduce the effect of signal amplitude on the characteristic number and facilitates the comparison between different results. Mostoufi et al. 45 mentioned that all pressure signals were normalized (by subtracting the mean value from the signal and then

19



dividing by the standard deviation) before applying STFT. He et al.,

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19

normalized data through

dividing the standard deviation of the pressure fluctuation by the time-averaged value. The disadvantage is that the physical meaning of results may become blurred. Therefore, data normalization should be used with caution. The normalization approach is described as follows: First, subtract the minimum from every number in the time series to obtain a new time series (i.e. ts1[k] = ts[k] - min (ts)), thus the minimum of ts1 is 0; Then divide ts1 by the maximum in it to form the final time series (i.e.

ts2[k] = ts1[k] / max (ts1)), making the maximum in the ts2 to be 1. Therefore, all the values in the ts2 are in the interval of [0, 1]. The definition of the Sn is very simple, S n = log10 |

method (normalized ts ) | , method(ts) method (ts )

generates a characteristic number estimated from the ts, when Sn is equal zero, the method is not sensitive to data normalization. The larger the Sn is, the more sensitive the method is to data normalization. There are some methods with sensitive problem of Sn in Table 3. The methods in time domain become a heavy disaster area. None of the methods in state space is sensitive. The STD method has the same Sn values with AAD and CDFFI methods. So do the PSDMP and PSDAI methods.

20


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Table 3. Some methods with sensitivity problem of Sn Method

Sn1

Sn2

Sn3

Sn4

Sn5

Sn6

Max of Sn

STD

0.896

0.638

0.715

1.213

1.028

0.822

2.086

AAD

0.896

0.638

0.715

1.213

1.028

0.822

2.086

SHANENT

0.5

0.382

0.355

0.576

0.442

0.375

3.953

CDFFI

0.896

0.638

0.715

1.213

1.028

0.822

2.086

NSTD

0.739

0.843

0.673

0.743

0.811

0.595

3.764

PSDMP

1.793

1.276

1.43

2.425

2.056

1.644

4.172

PSDAI

1.793

1.276

1.43

2.425

2.056

1.644

4.172

DWTELF

2.575

2.239

1.969

2.844

2.403

2.046

6.09

Sn1 ~Sn6: the mean of Sn in the group 1 ~ 6 Similarly, these methods with sensitivity problem have their own regions and causes. Figure 8 presents the identified sensitive regions of three methods: STD, PSDAI and DWTELF. Several regions are formed according to the Sn values. The region profiles in Figure 8a and 8b are almost the same, but the Sn values in Figure 8b is twice as that in Figure 8a. It also can be verified by comparing the Sn values of PSDAI and STD in Table 3. The theoretical explanation can be found in Appendix SC. In Figure 8a, the sensitive regions occupy most of the area in the figure. At low gas velocities, the sensitivity degree is higher and decreases with the increase of fn. In the fixed-bed regime and freeboard (>0.2m) with fn less than 2, the Sn values are basically larger than 1. After entering fluidization, the Sn values in dense phase bed is in the interval of [0.1, 0.3],

21



and change into [0.3, 1] when fn is larger than 2. The neighbor region on bed surface where Sn values are in the interval of [0.1, 0.3] expands upward rapidly near Uc. The insensitive region (white region) is narrow and long when fn is less than 10. In Figure 8c, the sensitivity regions cover almost the whole figure. There are two sensitivity regions (0.1< Sn frequency domain > state space. According to the total scores, the ACT and DFTENT methods are the most reliable methods, and the RPRR method is best in state space. Besides some less important factors are not taking into consideration. Taking the computational resource required by the methods as example, the RQA methods are far more consuming than others even we have accelerated the algorithm about 20 times faster by

28


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re-implementing it in C language. However, the performance and the resources consuming are only related to computing time and do not change the final result, thus this factor is neglected.

Table 6. The single item and total scores of twenty-three mathematical methods Method

Sl

Sn

Sp

Total

Method

Sn

Sl

Sp

score

Total Average score of a class

STD

3.43 1

4

8.43

SKEW

KURT

2.29 4

4

10.29 AAD

NSTD

2

1

4

CDFFI

1.71 1

ACT

3

DFTENT

1

4

4

9

3.86 1

4

8.86

7

SHANENT 1.29 1

3

5.29

3

5.71

HURST

1.14 3.71 4

8.85

4

4

11

4

4

3

11

PSDMF

1

4

3

8

PSDMP

1

1

3

5

PSDAF

2.86 4

3

9.86

PSDAI

2.14 1

3

6.14

DWTENT

1

4

3

8

DWTELF

2.71 1

2

5.71

DWTHI

1.29 4

2

7.29

CORRDIM 1.14 1

2

4.14

KOLENT

1.29 1

2

4.29

LLYAPE

1

1

1

3

RPRR

1.71 4

3

8.71

RPDET

2.57 4

2

8.57

RPLAM

1.29 4

2

7.29

29


8.27

7.63

6.76


Page 30 of 45

4.4. Applications 4.4.1. Improvement of methods One application of sensitivity study is to improve the mathematical methods with sensitivity problem. A simple example is given here to illustrate this. The SHANENT method has sensitivity problem of Sn before improvement (shown in Table 7). The reason is that the numerical range of time series needs to be linearly mapped to the range of [0, 1] when Sn is calculated, while the corresponding linear transformation is not performed on the parameter delta (listed in Table 4). When the SHANENT method is improved by dividing the range of old time series by delta, the new Sn values almost drop to zero and the sensitivity problem of SHANENT method is solved.

Table 7. The Sn values of SHANENT method before and after improvement State

Sn1

Sn2

Sn3

Sn4

Sn5

Sn6

before improvement 0.500 0.382 0.355 0.576 0.442 0.375 after improvement

0.000 0.000 0.000 0.000 0.000 0.000

Max of Sn 3.953 0.010

It is possible to find solutions to correct the sensitivity problems of methods if the root causes are clear. However, finding solutions may need great effort, so the article has no plan to analyze every method with potential improvement possibilities and find their solutions. Several time-domain methods, such as STD, AAD and so on, have poor scores in Sn evaluation, but there are no ways to improve them, because these methods have no adjustable parameters, so this also 30


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reveals that the non-parametric method avoids complexity of parameters but loses adjustability. On the other hand, too many parameters will lead to complicated parameter estimation problems.

4.4.2. Test of the methods picked by the framework The ACT and DFTENT methods are picked by the sensitivity evaluation framework and tested to verify their effectiveness on characterizing flow regimes. The effect of measurement position and static bed height on methods performance are discussed. (1) Effects of measuring position The fluidized bed shows different dynamic behavior at different locations, such as the air chamber, dense-phase region, and dilute-phase region. Thus, the pressure features depend on the measurement positions. O.A. Jaiboon et al.

47

studied effect of flow pattern on power spectral

density (PSD) of pressure fluctuation in various flow regimes, found that PSD was depending on flow regimes and the location in the riser. There are disputes on the choice of measuring position and adequacy of characterizing the flow pattern or flow state in a fluidized bed by only one measuring point. 48, 49 Instead of trying to resolve these controversies, we check whether the two methods are capable of distinguishing the flow regimes at different measurement positions. Figure 10 shows the characteristic curves of the two methods at five heights. In Figure 10a, the Umf indicating the transition from fixed bed to fluidized bed is discriminated, the ACT has a sharp increase near Umf. In the bubbling bed, the ACT increases firstly and then decreases. When 31



the bubble bed turns into turbulent bed, the ACT enters the stationary state. The two transition points can be discerned clearly by all positions except AP12. (a) U mf

(b)

8

Uc

U mf

AP1(-5cm) AP2(5cm) AP3(10cm) AP6(25cm) AP12(60cm)

70 60

The DFT entropy of ts (bit)

80

The average cycle time of ts (-)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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50 40 30 20

Uc

AP1(-5cm) AP2(5cm) AP3(10cm) AP6(25cm) AP12(60cm)

7

6

5

4

3

10 0

2 0

0.5

1

1.5

2

2.5

0

The superficial gas velocity, Uf (m/s)

0.5

1

1.5

2

2.5


Figure 10. Characteristic numbers vs. Uf (i.e. characteristic curves) at different measurement height (data of group 2 is used) (a) ACT method (unit is sampling interval) (b) DFTENT method.

In Figure 10b, DFTENT is large in fixed bed because the signal is close to noise, the fluctuations are mainly caused by the blower and the signal energy is basically evenly distributed at each frequency. When entering the bubbling bed, the DFTENT decreases rapidly, and the Umf can be clearly distinguished. With the increase of Uf, the DFTENT decreases and reaches a minimum. At this time, the regularity of gas-solid flow is best, and energy concentrates in a narrow frequency band. With the further increase of Uf, the DFTENT increases gradually, and the motion regularity decreases and the complexity increases. When entering the turbulent bed, the trend of DFTENT becomes constant or rises slightly, and the turning point can be regarded as Uc. The motion complexity in the turbulence bed is higher than that in the bubbling bed, but the 32


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change rate of DFTENT is smaller. The good performance of characterizing and distinguishing flow regimes can be obtained at all positions except AP12. In evaluating or comparing the quality of the flow regimes characterization, a few questions may need to be answered, such as which method is better than other methods or which position is more suitable for the same method. Unfortunately, the answer depends mainly on experience rather than objective method. If we treat the classification result of flow regimes characteristic as a clustering, the question will be transformed into a problem of evaluating the clustering quality. That is, all the characteristic numbers in a characteristic curve constitute the elements of a clustering, and the characteristic numbers in the same flow regime constitutes a cluster. Therefore, three classification of flow regimes are three clusters and each characteristic curve in Figure 10 can be regarded as a 3-clustering. The Silhouette plot method proposed by Rousseeuw 50

is utilized to assess the quality. Figure 11 shows Silhouette plot of the AP3 curve in Figure 10a and the AP1 curve in Figure

10b. To better understand the relationship between the Silhouette value and the characteristic number, the characteristic curve is also included. The X-axis is point index in the clustering and the curve, the left Y-axis is the Silhouette value and the right Y-axis is the characteristic number. The Silhouette Value (or Index) corresponding to each characteristic number is defined as s (i ) =

b(i ) − a (i ) , max{a (i ), b(i） }

33



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where a(i) is the average distance (i.e. absolute difference) from ith characteristic number to the other characteristic numbers in the same regime (or cluster), and b(i) is the average absolute difference from ith characteristic number to the characteristic numbers in the closest regime. The s(i) ranges from -1 to 1. Closing to 1 means the point is in the correct cluster, closing to -1 means it belongs to another cluster but assigned to the wrong cluster. From Figure 11a and 11b, the quality of fixed bed cluster and turbulent bed cluster are very good. However, there is a decline in quality of the bubbling bed cluster because there are one small positive s(i) and four negative s(i) in Figure 11a, and there are three negative s(i) in Figure 11b. The decline in quality of bubbling bed cluster may be partially due to that the transition from bubbling bed to turbulent bed is gradual transition.

51

To evaluate the overall quality of the clustering, the average

Silhouette Values of each cluster and the clustering are defined as sC j =

sC =

1 nj

∑

nj i =1

s (i ), i ∈ C j and

1 m ∑ sC respectively, where Cj is the cluster j (j=1, 2, 3) which has nj elements. m j =1 j

Therefore, the quality of the clustering is better if the sC is higher. Generally, the two curves have good clustering quality because they have higher sC (0.67 and 0.75), that is, they do well in charactering the flow regimes.

34


fixed bed

DFTENT (bit)

Silhouette value (-)

ACT (-) fixed bed

Figure 11. Silhouette plot of the (a) AP3 curve in Figure 10a (b) AP1 curve in Figure 10b. (2) Effects of static bed height (a)

150

(b)

12 1(P1, H 0 = 10cm)

1(P1, H 0 = 10cm)

11

2(P1, H 0 = 20cm)

U mf

3(P1, H 0 = 30cm)

Uc

The DFT entropy of ts (bit)

The average cycle time of ts (-)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Silhouette value (-)

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4(P3, H 0 = 10cm) 5(P3, H 0 = 20cm)

100

6(P3, H 0 = 30cm)

50

Umf

Uc

U mf

2(P1, H 0 = 20cm)

Uc

3(P1, H 0 = 30cm)

10

4(P3, H 0 = 10cm)

9

5(P3, H 0 = 20cm) 6(P3, H 0 = 30cm)

8 7 6 5 4 3

0

Umf

2 0

0.5

1

1.5

2

2.5


3

0

Uc 0.5

1

1.5

2

2.5

3


Figure 12. Characteristic curves of two particles with different H0 (data of AP1 is used) (a) ACT method (b) DFTENT method (curve translation is performed).

In Figure 12, curve 1~3 are characteristic curves of particle 1 (P1 in Table 1) with different H0, and curve 4~6 are characteristic curves of particle 2 (P2 in Table 2). In Figure 12a, the flow regimes of P1 and P2 can be clearly distinguished by the ACT method. By calculation, the average Silhouette Values of the six curves are {0.19, 0.61, 0.71, 0.40, 0.83, 0.66}. It indicates 35



the curve 1 has low clustering quality and it does not discern Umf Clearly. In Figure 12b, the performance of DFTENT method is pretty good. The Umf and Uc can be determined from all of the curves except curve 1 because it has lowest sC (0.12). It can be seen that the H0 has more significant influence on the ACT method than on the DFTENT method. However, the two transition points seem not be affected. Overall, the ACT and DFTENT methods have good performance in identifying the flow regimes. It means that the framework helps us to pick the appropriate methods.

5. CONCLUSION This article focuses on the quantitative data analysis methods applied in characterizing the flow regimes in a fluidized bed. Based on analysis of multi-point measurements of pressure fluctuations, the sensitivity and reliability of twenty-three methods on three influence factors are studied. Then, two applications based on the sensitivity study are demonstrated and an objective method for evaluating the quality of identifying flow regimes is introduced. This research cannot only help discover the method defects but improve the method as well. This research also provides a practical framework for rapid method selection. Some important conclusions are as follows: 1) The signals of 23 AP measurement points in a fluidized bed are analyzed by using a search algorithm of SSLmin based on stationarity judgement. It shows that the features of SSLmin are 36


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closely related with flow regimes. One or more regions in the bed are identified in different flow regimes. In fixed bed regime, all the bed is in Region I as the SSLmin < 400. After fluidization, the dilute phase or gas phase is mainly in Region I, the bottom bed is Region II and middle bed is in Region III which is more unstable than Region II. The sum of Region II and III may coincide with the major space of gas-solid interaction. The upper boundary of Region III may correlate with maximal height bed surface can reach, and the lower boundary may correlate with the dense bed height 2) Three sensitive indexes (Sl, Sn and Sp) assessing the methods sensitivity on segment length, data normalization and method parameters are proposed and their algorithms are designed. The sensitivity problems of some methods are studied. The results show that the root causes of SKEW method sensitive to segment length include outliers and small value as denominator, and the STD method is sensitive to data normalization because it depends on the range of time series. 3) Based on the three sensitive indexes, a common framework is proposed. The framework is in early stage and need more research on optimization of algorithms and parameters. However, the framework is scalable for small and large amount of data, is extensible to adding new sensitivity indexes, and is customizable to use your scoring functions. The framework provides a practical way suitable for methods selecting and can play a greater role if the database of time series and repository of methods code are established.

37



4) A systemic evaluation of twenty-three methods is performed by taking advantage of the previous framework. The ACT and DFTENT methods are more reliable than the others are and proved suitable for identifying the flow regimes. Besides, the result also indicates that the time domain methods have better performance than frequency domain and state space methods. It means there are improvement space for frequency domain and state space methods. Anyway, the sensitivity framework provides additional guarantees for reliable analysis which is vital for real-time monitoring of fluidization system.

ASSOCIATED CONTENT

Supporting Information

The following files are available free of charge. Tables S1~S4 and Appendix SA~SC. (PDF)

AUTHOR INFORMATION

Corresponding Author * E-mail: [email protected]. Tel.: +86-25-83794938. Address: Sipailou 2#, Nanjing 210096,

Jiangsu, P.R. China

Notes The authors declare no competing financial interest.

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ACKNOWLEDGMENT This work was supported by the Major Program of NSFC [No. 51390492] and Major research

program of Jiangsu Province [No. BE2017159].

NOMENCLATURE c = characteristic number cv = characteristic number vector fn = fluidization number fs = sampling frequency h = probe height or bed height H0 = static bed height mSeg = mean of segment mTs = mean of time series ncv = normalized cv segLen = segment length sjv = stability judgement vector sTs = standard deviation of time series Sl = sensitivity index of segment length Sn = sensitivity index of data normalization Sp = sensitivity index of method parameters 39



SSLmin = minimum of stationary segment length ts = time series Uc = transition velocity from bubbling bed to turbulent bed Uf = superficial gas velocity Umf = minimal fludization velocity ε = threshold coefficient ωi = weight of each parameter ABBREVIATIONS ACT, average cycle time; ANOVA, Analysis of Variance; AP, absolute pressure; DFTENT, discrete Fourier Transform entropy.

The abbreviations of method names are put into Appendix SA in Supporting Information.

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(49) van Ommen, J. R.; van der Schaaf, J.; Schouten, J. C.; van Wachem, B. G. M.; Coppens, M. O.; van den Bleek, C. M. Optimal placement of probes for dynamic pressure measurements in large-scale fluidized beds. Powder Technol. 2004, 139, (3), 264-276. (50) Rousseeuw, P. J. Silhouettes - A graphical aid to the interpretation and validation of cluster-analysis. J. Comput. Appl. Math. 1987, 20, 53-65. (51) Bi, H. T.; Grace, J. R.; Lim, K. S. Transition from bubbling to turbulent fluidization. Ind. Eng. Chem. 1995, 34, (11), 4003-4008.

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Table of Contents 1. INTRODUCTION 2. EXPERIMENTS AND DATA 3. METHODS FOR ANALYSIS AND COMPARISON 4. RESULTS AND DISCUSSION 4.1. Minimum of stationary segment length – SSLmin 4.2. Sensitivity study on influence factors 4.2.1. The sensitivity index of segment length – Sl 4.2.2. The sensitivity index of data normalization – Sn 4.2.3. The sensitivity index of method parameters – Sp 4.3. Sensitivity evaluation framework 4.4. Applications 4.4.1. Improvement of methods 4.4.2. Test of the methods picked by the framework 5. CONCLUSION Abstract graphic Sensitive problems of methods Methods in time domain, frequency domain, state space

Sl

Sensitivity study

Sensitive regions

Sn Three sensitivity indexes Sp

2 Applications

Causes of sensitivity

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Sensitivity study on quantitative methods by analysis of pressure

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