Recognition of Particle Size Changes in Fluidized Beds by

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Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Recognition of Particle Size Changes in Fluidized Beds by Recurrence and Cross Recurrence Quantification Analyses Hooman Ziaei-Halimejani, Reza Zarghami,* and Navid Mostoufi

Ind. Eng. Chem. Res. Downloaded from pubs.acs.org by KAOHSIUNG MEDICAL UNIV on 08/25/18. For personal use only.

Multiphase Systems Research Lab, School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran ABSTRACT: Pressure fluctuations were used for detecting the changes of mean particle size in fluidized beds by two online and accurate monitoring methods (recurrence and cross recurrence analysis (RA and CRA). In the qualitative approach, both methods contain specific structures changing by variation of the states of fluidized bed. In the quantitative approach, the recurrence rate is not suitable to use in detection of changes in particle size due to its high sensitivity to variations of gas velocity; however, determinism and entropy are suitable. Also, determinism of either recurrence plot (RP) or cross recurrence plot (CRP) is not able to identify small changes in particle size (about 10%), but can detect changes on the order of 20% or more. Moreover, entropy of RP is able to detect small changes of particle size in low velocities (up to 0.3 m/s), but entropy of CRP is sensitive to such a change in all velocities considered in this work.

1. INTRODUCTION The hydrodynamics of a fluidized bed can be altered because of an unexpected change of superficial gas velocity or mean particle size (sintering or agglomeration). Many researchers have investigated such malfunctions during fluidization;1−3 therefore, early detection of changes in the particle size can help prevent such undesired situations through appropriate change in the operating conditions. Various experimental methods can be utilized for measuring and investigating the hydrodynamics of fluidized beds. Among various methods, measurement of pressure fluctuations is very comfortable and also it includes a lot of information about different dynamic phenomena occurring in the bed (e.g., gas turbulence, bubble dynamics, and effect of operating conditions).4,5 Fluidized beds are complex unit operations because they involve two phase flow pattern which can be described by nonlinear dynamic relationships. Accordingly, a method with necessity of longterm data sampling, low time-consuming data treatment algorithms, low embedding parameters dependency, certainty in determination of embedding parameters, and distinctive result in high embedding dimensions is needed to investigate this nonlinear phenomenon.6 Recently, a recurrence plot (RP) has been proposed for analyzing nonlinear time series in which high-dimensional state space can be converted into a twodimensional (2D) plot; thus, results can be obtained without the above disadvantages.7−9 The concept of RP and recurrence quantification analysis (RQA) was introduced and developed to describe the behavior of the system in the phase space.7,10 Hydrodynamics of fluidized beds has been probed by RP and RQA showing that it is a powerful and easy method to detect flow pattern changes.11 Sedighikamal et al.,12 Tahmasebpoor et al.,13 and Babaei et al.9 investigated the hydrodynamics of fluidized beds by RP and © XXXX American Chemical Society

demonstrated its advantage in comparison to other methods. On the other hand, only a few research studies have been carried out about using a cross recurrence plot (CRP), as a bivariate extension of RP, for investigating the hydrodynamics of fluidized beds.14 They showed that CRP parameters are not sensitive to the superficial gas velocity variations, but are highly sensitive to changes in mean particle size. In the present research, two recent methods, i.e., RP and CRP (for the first time), are applied to recognize small changes in mean particle size in bubbling fluidized beds through measurement of bed pressure fluctuations. For this purpose, two qualitative and quantitative approaches were used. In the qualitative approach, RP and CRP structures were used for describing dynamic behaviors of different states of the fluidized bed. However, in the quantitative approach, different parameters (recurrence rate, determinism, and entropy of both RP and CRP) were compared based on their ability to identify changes in the particle size and the best parameter in each condition was determined.

2. THEORY The recurrence plot technique (first introduced by Eckmann et al.10) and cross recurrence plot (an extension of RP8) are methods for visualizing the recurrences of a dynamic system or two dynamic systems, respectively, in a 2D plot. Highdimensional state space trajectories are too complicated to visualize; therefore, a solution to represent them in a 2D plot is Received: January 4, 2018 Revised: June 23, 2018 Accepted: July 2, 2018

A

DOI: 10.1021/acs.iecr.8b00054 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research Table 1. Definition of Quantitative Parameters parameter recurrence rate

equation

RR(ε) =

1 N2

CRR(ε) =

DET =

entropy

ENT = −

interpretation in RP

interpretation in CRP

∑ R i ,j(ε)

density of recurrence points

repeatability

similarity in different states

recurrence points that form diagonal lines

predictability (rule-obeying)

similarity in exact states

Shannon information entropy

complexity

dissimilarity

i ,j=1

1 N2

N

∑ CR i ,j(ε) i ,j

N ∑l = l lP(l) min N ∑l = 1 lP(l)

determinism

definition

N

N

∑

p(l) ln p(l)

l = l min

4. RESULTS AND DISCUSSION 4.1. Selection of Input Parameters. The first step in utilizing RP and CRP to investigate the effect of small changes in particle size on the hydrodynamics of a fluidized bed is determining the input parameters. Input parameters in both RP and CRP methods are epoch length, embedding dimension (m), delay time (τ), radius threshold (ε), and minimum length of the diagonal line (lmin) which should be set before recurrence and cross recurrence analyses. To make sure that a long enough part of a signal is considered in all cases, an epoch length of 2000 points was considered for both RP and CRP in the rest of the calculations of this work. Tahmasebpour et al.,13 Babaei et al.,11 and Savari et al.15 also used an epoch length as short as that considered in the present work. The embedding dimension (m) was fixed in the next step. Savari et al.15 investigated the effect of the embedding dimension on RP structures and RQA parameters and found that there is no important difference between results obtained with different embedding dimensions. Therefore, in the present study, an embedding dimension of 2 was selected in all RQA and CRQA parameters. Time delay (τ) is another input parameter needed to create RPs and CRPs. Many researchers have investigated the effect of time delay on RP structures and RQA and showed that this parameter has no effect on results,9,12,18,19 and in this research, it is selected equal to 1. The minimum length of the diagonal line (lmin) is typically considered 2 in both RQA and CRQA.9,13−15,20 For instance, Babaei et al.20 investigated the Lorenz system through its determinism and found that the proper value of lmin is 2 and used this value for investigating the hydrodynamics of fluidized beds. Therefore, the minimum length of vertical and diagonal lines was considered to be 2 in this work. Finally, the proper value of the radius threshold was determined in this work based on determinism and its standard deviation for both RP and CRP methods, and values of 0.12 and 0.08 were selected for RP and CRP, respectively. 4.2. Sensitivity to Particle Size Changes. 4.2.1. Qualitative Approach. Detection of changes in particle size in fluidized beds can be investigated based on visual inspection of RP and CRP structures. RP and CRP contain black and white points, creating various geometric structures related to the dynamic behavior of the system. These structures have different meanings in the RP than in the CRP. Figure 1 shows RPs of pressure fluctuations in fluidized beds at a superficial gas velocity of 0.3 m/s with a particle size of 150 μm (Figure 1a), by adding 10% 280 μm particles (Figure 1b) and 20% 280 μm particles (Figure 1c), and with a particle size of 280 μm (Figure 1d). As demonstrated in Figure 1, there are

of high importance, and RP and CRP have this ability. In other words, RP and CRP make qualitative and visual patterns of the time-series correlations in which nonstationary and short-term data is applicable.7,13−15 Moreover, there is no need for embedding parameters in this method and information can be extracted from the nonembedded signal.16 These 2D plots contain geometric structures which can be used in order to interpret qualitative parameters and introduce various quantitative parameters. In the present investigation, the recurrence rate, determinism, and entropy7,14 of both RP and CRP methods were utilized to detect changes in the particle size in fluidized beds. Table 1 shows equations and descriptions of these quantitative parameters for both RP and CRP methods.7,14 It should be mentioned that, in the present study, the signals normalized recurrence before carrying out cross recurrence analyses. Normalization reduces the sensitivity of the method to small changes in the superficial gas velocity.9,14,17 In this regard, the signals were normalized as follows: xi − μS xn = σS

3. EXPERIMENTS Pressure fluctuations of a fluidized bed of Geldart B particles with different mean sizes were used in order to detect particle size changes. The measurements have been carried out by Zarghami el al.6 They used a Plexiglas pipe of 15 cm inner diameter and 2 m height as the fluidized bed. Air with ambient temperature was supplied to the column through a perforated plate distributor with 435 holes of 1.7 mm diameter, arranged in a 7 mm triangular pitch, and a mass flow controller (MFC) was utilized for measuring the gas flow rate. The initial aspect ratio of solids in all experiments was 1. In the experiment, 150 μm particles were used as the reference state and changes of particle size by adding 10 and 20% particles of 280 μm size (equivalent to mean sizes of 163 and 176 μm, respectively) and 280 μm particles were investigated. Furthermore, experiments were done at various superficial gas velocities, from 0.1 to 1.1 m/s, in order to investigate effect of gas velocity on the sensitivity of RP and CRP parameters. Properties of the particles are given in Table 2. Table 2. Properties of Particles particle size (μm)

density (kg/m3)

Umf (m/s)

Uc (m/s)

150 280

2640 2640

0.029 0.059

0.9 1.1 B


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Industrial & Engineering Chemistry Research

Figure 1. RP of pressure fluctuations of fluidized bed with particle sizes of (a) 150 μm, (b) 10% 280 μm + 90% 150 μm, (c) 20% 280 μm + 80% 150 μm, and (d) 280 μm at superficial gas velocity of 0.3 m/s. lmin = 2, m = 2, τ = 1, ε = 0.12, and epoch length = 2000.

many recurrence points creating horizontal, vertical, and diagonal lines, causing various patterns. There are both single points and regular patterns in Figure 1, from which it can be concluded that a fluidized bed follows a chaotic and nonlinear dynamic behavior, as also pointed out by other researchers.6,9,13,17,21−24 Formation of these various patterns is a result of phenomena such as formation, coalescence, and eruption of bubbles. Also, as can be seen in Figure 1b−d, by changing the particle size, observed structures in the RP also change. However, this change is recognized by observation only; changes in the number of single points as well as diagonal, horizontal, and vertical lines cannot be quantitatively identified. In fact, qualitative observation is not enough for extracting all the information from the RP and a quantitative approach is needed for interpreting the geometric structures of the plot. For this purpose, quantitative parameters such as determinism, recurrence rate, and entropy can be used. Figure 2 shows CRPs of pressure fluctuations in fluidized beds at superficial gas velocity of 0.3 m/s with a particle size of 150 μm (Figure 2a) and by adding 10% 280 μm particles (Figure 2b) and 20% 280 μm particles (Figure 2c), and with particle size of 280 μm (Figure 2d). In Figure 2, the bed of 150 μm particles was considered as the reference state. As can be seen in Figure 2, there is no main diagonal, or the line of identity (LOI), in a CRP. Thus, the main difference between CRP and RP at a glance is the absence of the LOI in the CRP. Furthermore, similar to the RP, patterns in the CRP change by changing the particle size in the bed. Larger local white zones represent a large distance between trajectories in the state space and suggest less similarity and attunement between the two systems. It can be seen that local

white zones in Figure 2c are larger than those in Figure 2b and also larger than those in Figure 2a. However, visual observation only reveals qualitative information about changes of particle size in the bed and the fraction of diagonal lines and local white or black zones, which reveal the amount of similarity between two states, cannot be understood properly from the CRPs shown in Figure 2. Therefore, in the CRP method, similar to RP, a quantitative approach is needed in order to benefit from the ability of the plot to detect small changes in particle size. 4.2.2. Quantitative Approach. As mentioned before, patterns of RP and CRP reveal the dynamic behavior of a system or dynamic relation between two systems qualitatively, but they cannot provide detailed information. As a matter of fact, a quantitative approach is needed for extracting thorough information from RP and CRP to sense particle size changes by these methods. For this purpose, three RP and CRP parameters, i.e., recurrence rate (RR for RP and CRR for CRP), determinism, and entropy were utilized to detect changes in the hydrodynamics of a fluidized bed when the particle size was changed. Since the gas velocity often fluctuates in an industrial fluidized bed, a useful method for detecting change of particle size in the bed would be the one which is least sensitive to superficial gas velocity variations but sensitive enough to particle size change. It has been shown that normalization of the signal with respect to its standard deviation reduces its sensitivity to changes in the gas velocity;11,14,15 thus all signals were normalized before creating RPs and CRPs and extracting quantitative information from these plots by RQA and CRQA, respectively. C


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Figure 2. CRP of fluidized bed with particle size of (a) 150 μm and CRP for particle sizes of (b) 10% 280 μm + 90% 150 μm, (c) 20% 280 μm + 80% 150 μm, and (d) 280 μm and superficial gas velocity of 0.3 m/s. In CRP, fluidized bed with particle size of 150 μm and superficial gas velocity of 0.3 m/s was considered as reference. lmin = 2, m = 2, τ = 1, ε = 0.12 and ε = 0.08 for RP and CRP, respectively, and epoch length = 2000.

Figure 3. (a) Recurrence rate, (b) determinism, and (c) entropy of RP and CRP of pressure fluctuations of fluidized bed of 150 μm particles against superficial gas velocity. The bed of 150 μm particles at the superficial gas velocity of 0.1 m/s was considered as the reference state in CRP. lmin = 2, m = 2, τ = 1, and ε = 0.12 and ε = 0.08 for RP and CRP, respectively.

fluidized bed of 280 μm particles against superficial gas velocity. Pressure fluctuations of the fluidized bed at a superficial gas

Figure 3 shows variation of the recurrence rate, determinism, and entropy of RP and CRP of pressure fluctuations of the D


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Figure 4. Determinism of pressure fluctuation of fluidized bed for both RP and CRP as a function of particle size. lmin = 2, m = 2, τ = 1, and ε = 0.12 and ε = 0.08 for RP and CRP, respectively. Error bars are standard deviation of determinism.

Figure 5. Entropy of fluidized bed for RP and CRP as a function of particle size. lmin = 2, m = 2, τ = 1, and ε = 0.12 and ε = 0.08 for RP and CRP, respectively. Error bars are standard deviation of entropy.

velocity of 0.1 m/s were considered as the reference state for CRP. As illustrated in Figure 3b,c, determinism and entropy do not change considerably with gas velocity while Figure 3a demonstrates that the recurrence rate is sensitive to the superficial gas velocity. Consequently, both determinism and entropy can be proper candidates to detect changes in the particle size in a fluidized bed while the recurrence rate cannot be used for this purpose due to its sensitivity to the gas velocity. Since a suitable method in detection of particle size changes should not be sensitive to variations in the superficial gas velocity, it can be concluded that comprehensive information can be extracted by diagonal based parameters and single point parameters are not able to detect particle size changes.

Figure 4 shows variation of the determinism of both RP and CRP for pressure fluctuations of the fluidized bed as a function of particle size at various superficial gas velocities. Standard deviation of each parameter is shown in Figure 4. Changes of each parameter were compared with this range. In fact, if the determinism in larger particle size falls out of this range, the method will be sensitive to the changes in particle size. In Figure 4, a particle size of 150 μm was considered as the reference state. Figure 4 illustrates that determinism of the RP increases with increasing the particle size at a constant superficial gas velocity while determinism of the CRP exhibits a reverse trend against particle size. Increase in the determinism of the RP indicates that the behavior of the bed has become more periodic and E


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• In recurrence and cross recurrence analyses, investigating changes of particle size can be pursued by two qualitative and quantitative approaches. The qualitative approach cannot show the change accurately. • Both recurrence and cross recurrence plots contain geometric structures that change by variation of the state of fluidization. The RP and CRP structures can be divided into local white zones and local black zones. In the RP, local white zones increase with increasing particle size, corresponding to more and larger bubbles. In the CRP, these zones only show similarity and compare two states and the dynamic behavior of a single state cannot be obtained. • Three quantitative parametersrecurrence rate, determinism and entropywere calculated in order to investigate the detection of changes of particle size in the fluidized bed. It was found that the recurrence rate in both RP and CRP is highly sensitive to variations of the superficial gas velocity while entropy and determinism are not. This fact shows that the recurrence rate is not capable of detecting changes in particle size. • It was found that determinism of RP can detect small changes weakly (10% change in particle size) at lower velocities (up to 0.3 by 2% changes with particle size variation) and determinism of CRP is not able to sense small changes. However, both can detect greater changes (20% change in particle size or more). Moreover, the entropy of RP is able to detect small changes in particle size at low velocities (up to 0.3 m/s by 16% changes with particle size variation), but the entropy of CRP is sensitive to such changes in all velocities (with more than 16% changes). • Entropy is more sensitive to the change of particle size than determinism in both RP and CRP methods, and both parameters in RP are only applicable at low gas velocities.

predictable in a bed of larger particles. With increasing size of the particles, the minimum fluidization velocity of the bed increases, and bubbles fall in number and grow in size. The behavior of larger bubbles is more predictable than that of smaller ones; hence, the bed behavior becomes more predictable. In fact, bubbles in the beds of larger particles are formed at higher velocities compared to beds of smaller particles, meaning more predictable behavior at the same superficial gas velocity even by decreasing the fluidization number (U/Umf) with increasing particle size. On the other hand, determinism of CRP shows a similarity and attunement between two states, the decreasing of which shows less similarity between the two states. Determinism of CRP only indicates the relation of dynamic behavior of two systems (two signals) and cannot investigate information contained in a single signal. Consequently, both CRP and RP methods can be used to quantify changes in the dynamic behavior, which is the main purpose of this research. However, the RP method indicates dynamic behavior of a single state of the fluidized bed, which can be considered as a beneficial feature of RP compared to CRP. It is worth noting that, according to Figure 4, determinism of both RP and CRP cannot detect small changes in particle size (from 150 to 163 μm). Nevertheless, by increasing the size of particles to 176 μm, determinism of RP is able to detect this change in superficial gas velocities of 0.1 and 0.3 m/s and determinism of CRP can detect the change at all velocities. Moreover, the sensitivity of determinism of RP to particle size decreases with increasing superficial gas velocity. This decrease of sensitivity is less for CRP. This trend suggests that the effect of particle size on the hydrodynamics of a fluidized bed becomes weaker at higher gas velocities. In other words, the hydrodynamic status of the fluidized bed at high gas velocity is governed by the gas velocity rather than particle size. This fact can be explained by the difference between the superficial gas velocity and the minimum fluidization velocity. In fact, at low velocity (e.g., 0.3 m/s), fluidization of the bed with a particle size of 280 μm is less than the fluidization with a particle size of 150 μm (due to excess velocity difference, (U − Umf)280 < (U − Umf)150) forming enough bubbles. Figure 5 shows variation of the entropy of both RP and CRP of pressure fluctuations as a function of particle size at various superficial gas velocities. As shown in Figure 5, the entropy of RP decreases with increasing particle size, indicating that the complexity of fluidization is descending. In fact, the complexity of fluidization is related to the formation and size of bubbles, indicating that when particle size is decreased, lower stability of bed occurs and smaller bubbles are formed. In contrast, the entropy of CRP increases with increasing particle size, meaning that the two states have become less similar. Furthermore, Figure 5 demonstrates that the entropy of RP is able to detect small changes in particle size (from 150 to 163 μm) only at low velocities (less than 0.3 m/s) and this ability vanishes at higher velocities. In contrast, the entropy of CRP is able to detect small changes in particle size at all velocities. However, this ability is weaker at higher gas velocities.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +98 21 6696 7797. Fax: +98 21 6646 1024. E-mail: [email protected]. ORCID

Reza Zarghami: 0000-0001-7222-8838 Navid Mostoufi: 0000-0002-3285-6300 Notes

The authors declare no competing financial interest.

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NOMENCLATURE CR = cross recurrence matrix between two phase space trajectories CRij = cross recurrence point CRP = cross recurrence plot CRQA = cross recurrence quantification analysis CRR = cross recurrence rate or recurrence rate of CRP DM = distance matrix between phase space vectors DET = determinism ENT = entropy i, j = indices lmin = minimum diagonal length m = embedding dimension N = number of data points Nl = number of diagonal RP = recurrence plot RQA = recurrence quantification analysis RR = recurrence rate of RP

5. CONCLUSIONS This study presents and compares the ability of two online monitoring methods in the detection of small changes in particle size in fluidized beds. In this regard, pressure fluctuations of the bed were measured at various superficial gas velocities and two recurrence and cross recurrence analysis methods were utilized. The following results were obtained through this work: F


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(16) March, T.; Chapman, S.; Dendy, R. Recurrence Plot Statistics and the Effect of Embedding. Phys. D 2005, 200 (1), 171. (17) van Ommen, J. R.; Coppens, M. O.; van den Bleek, C. M.; Schouten, J. C. Early Warning of Agglomeration in Fluidized Beds by Attractor Comparison. AIChE J. 2000, 46 (11), 2183. (18) Tahmasebpour, M.; Zarghami, R.; Sotudeh-Gharebagh, R.; Mostoufi, N. Study of Transition Velocity from Bubbling to Turbulent Fluidisation by Recurrence Plots Analysis on Pressure Fluctuations. Can. J. Chem. Eng. 2013, 91 (2), 368. (19) Fraser, A. M.; Swinney, H. L. Independent Coordinates for Strange Attractors from Mutual Information. Phys. Rev. A: At., Mol., Opt. Phys. 1986, 33 (2), 1134. (20) Babaei, B.; Zarghami, R.; Sedighikamal, H.; Sotudeh-Gharebagh, R.; Mostoufi, N. Selection of Minimal Length of Line in Recurrence Quantification Analysis. Phys. A 2014, 395, 112. (21) Fortes, A. F.; Joseph, D. D.; Lundgren, T. S. Nonlinear Mechanics of Fluidization of Beds of Spherical Particles. J. Fluid Mech. 1987, 177, 467. (22) van den Bleek, C. M.; Schouten, J. C. Deterministic Chaos: A New Tool in Fluidized Bed Design and Operation. Chemical Engineering Journal and the Biochemical Engineering Journal 1993, 53 (1), 75. (23) Schouten, J. C.; van den Bleek, C. M. Monitoring the Quality of Fluidization using the Short-Term Predictability of Pressure Fluctuations. AIChE J. 1998, 44 (1), 48. (24) Llauro, F.; Llop, M. Characterization and Classification of Fluidization Regimes by Non-Linear Analysis of Pressure Fluctuations. Int. J. Multiphase Flow 2006, 32 (12), 1397.

R = recurrence matrix between two phase space trajectories Rij = recurrence point P(l) = number of diagonal lines of length l P(l) = P(l)/Nl = probability distribution of diagonal lines of length l U = superficial gas velocity (m/s) Uc = velocity transition from bubbling to turbulent regime (m/s) Umf = minimum fluidization velocity (m/s) xi = first time series yi = second time series x⃗i = first trajectory in state space xn = normalized signal y⃗i = second trajectory in state space Greek Symbols

Θ = Heaviside function ε = threshold radius σ = standard deviation μ = mean of time series

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REFERENCES

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Recognition of Particle Size Changes in Fluidized Beds by

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