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The study of the entrance and wall dynamics of a high-flux gas-solid riser was conducted using trajectory distances of the reconstructed attractors from solid concentration signals collected from a 76 mm internal diameters and 10 m high riser of a circulating fluidized bed (CFB) system. The riser was operated at 4.0 to 10.0 m/s gas velocity and 50 to 550 kg/m
^{2}s solids flux. Spent fluid catalytic cracking (FCC) catalyst particles with 67 μm mean diameter and density of 1500 kg/m
^{3} together with 70% to 80% humid air was used. Solid concentration data were analyzed using codes prepared in FORTRAN 2008 to get trajectories of the reconstructed attractors and their distances apart. Trajectory distances were found to increase from the centre towards the wall indicating the expansion of the attractor. The probability density function (PDF) of the trajectory distances changes from single peak at the centre to multiple peaked profiles in the wall region. Multiple peaked profiles indicate multifractal flow behaviours. Cumulative distribution functions (CDF) of the trajectory distances changes from single
*S*-shaped at the centre to multiple
*S*-shaped profiles in some locations of the wall region indicating multifractal flow behaviours. The PDF distribution of these distances at the entrance section and in the wall region forms different types of statistical distributions showing differences in gas-solid flow structures in various spatial locations of the wall region and the entrance sections. Most of the distributions at the centre fall under the Gumbel max distribution for all flow development sections of the riser, especially at air velocities of 5.5 m/s and 8 m/s showing uniform flow structures. Further, it was found that increase of the number of the phase space reconstruction embedding dimension increases the trajectory distances between the state vectors leading to the expansion of the attractor.

Fluidization technology is extensively used in various processes in which large contacts between the solid and fluid phases during reaction is required [

There various methods which are used to investigate the gas-solid flow dynamics in circulating fluidized bed risers such as statistical, frequency and chaos analysis methods [

Chaos analysis begins with reconstruction of attractors which is achieved by embedding the measured single variable time series using time delay and sufficient minimum embedding dimension. This step is very importance due to the fact that most of chaotic parameters used in characterizing the dynamical system depends on the correctness of the reconstruction process and the accuracy of the reconstructed attractors [

The use of statistical analysis of the vector pair distances referred to as trajectory distances has been used in describing the dynamics of high-flux gas-solid riser using pressure fluctuation signals [

The phase space or state space is a space spanned by dependent variables of a given dynamical system in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space [

The analysis of chaotic dynamic system makes use of the property that it must be sensitive to initial conditions [

Due to the fact that the phase space determine all states of a dynamical system, analysis of that system can be achieved in both identifying the system and predicting the future states via phase space representation [

The most challenging point in chaos analysis is the failure to understand the original phase space of the dynamical system. Thus the analysis achieved by using embedded or reconstructed phase space. The embedded space enables to draw out a multidimensional description of state space dynamics from time series data of a single dynamical variable, and generalizes the quantitative measures of a chaotic behaviour [

According to Takens’ embedding theorem it is possible to reconstruct phase space from a single time series so as to characterize a non-linear dynamical system through time delayed variables [_{N}) does indeed consist of scalar measurements of the state of a dynamical system, then under certain genericity assumptions, the time delay embedding provides a one-to-one image of the original set (X), provided the embedding dimension (m) is large enough” [

Further, it is possible to reconstruct the phase space attractor while preserving the topological characteristics from measurements of a single dynamical variable. This renders experimental systems, like CFB reactors with attractors of moderately low dimension, acceptable for analysis by chaos analysis tools. This idea was introduced by Packard et al. in 1980 [

The reconstructed attractor consists of orbits of points corresponding to different state vectors, X(i), which represents one point on the orbit of the attractor and is reconstructed from the data points of the time series. Reconstruction of the state vectors is achieved by the method of delays where a time delay, τ, of the time steps between the elements of the state vector and the number of elements, m, of the state vector equals to the embedding dimensions are chosen and transposed to create a multidimensional state vector [

X ( i ) = [ x ( i ) , x ( i + τ ) x ( i + 2 τ ) , ⋯ , x ( i + ( m − 1 ) ) ] T (1)

where the count, i = 1, 2, 3, …, N.

Data were collected from a CFB system shown in ^{2}s solids flux and 4.0 to 10.0 m/s gas velocity. Fluid catalytic cracking catalyst particles with 67 μm mean diameter and density of 1500 kg/m^{3} were used. A 70% to 80% humid air was used for transporting the solid particles. Signals were sampled from eight (8) axial levels (i.e., Z = 0.98, 1.52, 2.73, 3.96, 5.13, 6.34, 8.74, and 9.42 m) and 11 radial points (i.e., r/R = 0.00, 0.16, 0.38, 0.50, 0.59, 0.67, 0.74, 0.81, 0.87, 0.92, and 0.98) at each level where r/R is the normalized radial distances from the center to the wall of the riser. To each point, 29,100 data points of solid concentration were sampled in 30 seconds using optical fiber probe at 970 Hz.

The trajectory distances of the state vectors from the reconstructed attractor with time delay, τ = 6 and embedding dimension, m = 15 were determined using a multi-dimensional Euclidian distances formula, used in chaos analysis. The

trajectory distances of the state vectors were calculated as per Equation (2) [

δ i j = ‖ X i − X j ‖ (2)

where ‖ X i − X j ‖ is the vector distance (Euclidean distance) between two reconstructed vectors, X_{i} and X_{j}. The Euclidian distance, δ, is the vector distance between vector pairs (X_{i}, X_{j}) of the trajectories. The vector distances (Euclidean distance), δ, was calculated using Equation (3) as follows [

δ i j = ∑ i = 1 N ∑ j = i + 1 N ( X i − X j ) 2 (3)

This quantity was determined on the reconstructed attractors using signals from different radial locations near the wall along the riser to study the chaotic behaviour of the high-flux riser. These distances from each signal were calculated and saved in the file. In this work the number of solid concentrations data points, N = 2000. The distances between the pairs (X_{i}, X_{j}) were computed by first keeping i constant at i = 1 and then navigating all points, j = i + 1 to N, and then i was changed from i = 1 to N [^{2} = 4 × 10^{6}, the computation which were achieved using prepared code in FORTARN 2008.

To establish the PDF of the trajectory distances, the data sets were first split into bins of equal intervals. This was achieved by subtracting the minimum value from the maximum value and dividing the obtained range by the set number of steps. The obtained interval was used to establish various bins starting with the one that included the minimum data value and get subsequent bins until the maximum data value was included.

Suppose δ is the distance separating any two points or trajectories on the attractor and N is the number of points counted on the attractor along the trajectories, then the PDF was established by setting equal intervals of δ, Δδ, between δ_{min} ≤ δ ≤ δ_{max} by choosing 50 steps to make 50 bins and counting number of points within the distance δ_{min} ≤ δ ≤ δ_{i}, that is, N_{i}.

The frequency for each interval, Δδ, was determined as per Equation (4)

f i = N i N t s × 100 % (4)

For count, i = 1, 2, 3, …

By changing to next interval δ_{i} ≤ δ ≤ (i + 1) Δδ until the last bin the frequency table was created and the frequency curve plotted which gives the PDF of the values of δ across the attractor. The cumulative distribution function was established where the cumulative frequency was determined by adding each frequency to the sum of its preceding frequency as per Equation (5):

c f i = f i − 1 + f i (5)

For count, i = 1, 2, 3, …

The shapes of PDF, CDF and their location along ln (δ)-axis were used to study the changes in the flow dynamics of the gas-solid riser.

The probability density functions (PDF) describe the relative likelihood for the random variable to take on a given value and the cumulative distribution function (CDF) is a function whose value is the probability that a variable takes a value less than or equal to the argument of the function. In this work the trajectory distances of the state vectors of the reconstructed attractors were determined and the PDF and the CDF of these distances were established and their profiles plotted.

_{g} = 5.5 m/s and solid flux, G_{s} = 300 kg/m^{2}s.

When different PDF from different signals are examined critically, their shapes like tails, height and number of peaks can be used get information on the behaviours of the gas-solid flow dynamics, spatial locations or operating conditions of the gas velocity and solid flux. A uniform gas-solid suspension flow gives a trajectory distances close to the mean distance and leads to the formation of single, narrow and tall PDF. Example of this is found in PDF profiles at r/R =

0.0 as shown in

Also different CDF generated from different signals display differences in shapes and number of S-shape in the CDF profiles. These features were used in this study to differentiate the behaviours of the gas-solid flow dynamics, spatial locations and or the operating conditions of the gas velocity. In

This study introduces another way of examining the gas-solid flow behaviours in the circulating fluidized bed riser using PDF and CDF of the trajectory distances from the reconstructed attractors. Further this work introduces new planes, that is, the frequency-ln (δ) plane and the cumulative frequency-ln (δ) plane.

The PDF of the trajectory distances of the reconstructed attractor reveals the scattering tendency of these distances in the phase space. For the trajectory distances data to be compared with the correlation integral data or curves, it was suggested to use same scale, that is ln (δ), which can be mapped on the ln (r), and hence locate the distances on such axis, originally developed by Grassberger and Procaccia [_{s} = 300 kg/m^{2}s for selected riser heights, z = 1.52, 3.96 and 9.42 m, corresponding to riser entrance, developing flow and fully developed flow sections.

Results shows that for U_{g} = 5.5 m/s the trajectory distances or the pair spacing between trajectories vector increases from the centre towards the wall especially in the developing section (Z = 3.96 m). Multiple peaks of the profiles for instances at Z = 9.42 m and r/R = 0.98 indicates the presence of multifractal flow structures.

When the gas velocity, U_{g} = 8.0 m/s,

in the developing section (Z = 3.96 m). Presence of more than one peak in some profiles for instances profiles at Z = 9.42 m, at Z = 3.96 and r/R = 0.81 indicates a multifractal flow structures, reported also in [

When the gas velocity, U_{g} = 10 m/s, _{g} = 5.5, 8 and 10 m/s. This tendency is more clear at low velocity, U_{g} = 5.5 m/s. At this velocity the center profiles (r/R = 0.0) in the developing flow and fully developed flow sections has single peaked PDF while at U_{g} = 8 m/s the PDF in the fully developed section (Z = 9.42 m) the profile is double peaked. At higher velocity, U_{g} = 10 m/s, the PDF profiles close to the wall, i.e., at r/R = 0.92 and 0.98 have higher tendency to form multiple peaks compared to the profiles at U_{g} = 8 and 5.5 m/s. As the axial elevation, Z, increase from the entrance section to the top section, the PDF profiles at r/R = 0.81 shifts towards the negative direction for all velocities, i.e., U_{g} = 5.5, 8 and 10 m/s. Further, it can be seen that the center profiles (r/R = 0.0) are located towards lower values of ln (δ) in the negative direction while profiles closer to the wall (i.e., r/R = 0.92 and 0.98) are located towards higher values of ln (δ) in the positive direction around the centre of the Frequency (%)-ln (δ) plane.

The dynamic study of the riser using PDF of the trajectory distances is reported in literature [

The probability density functions of the trajectory distances were further analysed to determine their distribution type for further characterization of gas-solid flow behaviours. The best fitting distribution were achieved using Easy Fit 5.3 Professional software. The software gives three tests of the goodness of fit, i.e., the Kolmogorov Smirnov, Anderson Darling and the Chi-square. The Kolmogorov Smirnov was chosen in this study where the statistics and the ranking for each of the fitted distribution are given. The first rank were taken as the best fit for the distribution and hence parameters were recorded as shown in _{g}, = 5.5, 8 and 10 m/s.

From _{g} = 10 m/s where the uniform distribution emerges. This indicates that the flow behaviours at the center have almost similar behaviour. In the wall region the trajectory distances gives different PDF distributions including the normal, beta, hypersecant and wake by distributions. The difference in distribution type indicates presence of different gas-solid flow behaviours in the wall region.

The cumulative distribution function (CDF) shows the probability that the variable takes a value less than or equal to the argument of the function. In this case, CDF shows the probability that the ln (δ) or δ takes a value less than (ogive) or equal to the frequency (%) indicated.

Z (m) | U_{g} (m/s) | r/R = 0.0 | r/R = 0.81 | r/R = 0.92 | r/R = 0.98 | ||||
---|---|---|---|---|---|---|---|---|---|

Distribution | Parameters | Distribution | Parameters | Distribution | Parameters | Distribution | Parameters | ||

9.42 | 5.5 | Gumbel Max | σ = 5.2693 μ = −1.0423 | Gumbel Max | σ = 2.8732 μ = 0.34164 | Normal | σ = 7.1255 μ = 2.0 | Normal | σ = 4.0424 μ = 2.0 |

8 | Gumbel Max | σ = 4.0105 μ = −0.31331 | Normal | σ = 4.8721 μ = 2.0024 | Power Function | α = 0.07565 a = 7.1852E-15 b = 43.293 | Logistic | σ = 4.9405 μ = 2.001 | |

10 | Gumbel Max | σ = 4.3473 μ = −0.51067 | Gumbel Max | σ = 4.8334 μ = −0.79036 | Hypersecant | σ = 5.5092 μ = 2.0005 | Normal | σ = 4.723 μ = 2.0013 | |

3.96 | 5.5 | Gumbel Max | σ = 3.7447 μ = −0.16135 | Normal | σ = 5.7161 μ = 2.0016 | Wakeby | α = β = 0 γ = 0.45539 δ = 0.79149 ζ = −0.18323 | Wakeby | α = β = 0 γ = 0.22683 δ = 0.89205 ζ = −0.1013 |

8 | Gumbel Max | σ = 4.381 μ = −0.53154 | Wakeby | α = β =0 γ = 0.96022 δ = 0.58642 ζ = −0.32036 | Gumbel Max | σ = 4.5165 μ = −0.60692 | Wakeby | α = β = 0 γ = 0.54882 δ = 0.75212 ζ = −0.21321 | |

10 | Uniform | a = −6.7745 b = 10.776 | Uniform | a = −3.9335 b = 7.935 | Kumaraswamy | α_{1} = 0.02333 α_{2} = 0.44276 a = −3.9090E-15 b = 29.725 | Beta | α_{1} = 0.05168 α_{2} = 0.12406 a = 3.7075E-15 b = 19.7 | |

1.52 | 5.5 | Gumbel Max | σ = 3.7447 μ = −0.16135 | Gumbel Max | σ = 3.9172 μ = −0.26051 | Gen. Logistic | K = 0.84788 σ = 0.31815 μ = 0.20292 | Uniform | a = −6.3847 b = 10.389 |

8 | Gumbel Max | σ = 3.7447 μ = −0.16135 | Uniform | a = 7.9596 b = 11.96 | Normal | σ = 5.6584 μ = 1.9998 | Gumbel Max | σ = 4.1329 μ = −0.38528 | |

10 | Uniform | a = −8.7682 b = 12.769 | Uniform | a = −7.2731 b = 11.275 | Wakeby | α = β = 0 γ = 0.36001 δ = 0.83294 ζ = −0.15385 | Beta | α_{1} = 0.05168 α_{2} = 0.12406 a = 3.7075E-15 B = 19.7 |

function curves for ln (δ) for distances between points in the reconstructed attractor using solid concentration signals from selected radial positions for gas velocity, U_{g} = 5.5, 8.0 and 10 m/s when solid flux, G_{s} = 300 kg/m^{2}s for selected riser heights, Z = 1.52, 3.96 and 9.42 m, corresponding to riser entrance, developing flow and fully developed flow sections.

When U_{g} = 5.5 m/s most of the profiles are single S shaped with few exception at Z = 9.42 m and r/R = 0.98 where a double S shaped CDF was observed as shown in

For U_{g} = 8 m/s and G_{s} = 300 kg/m^{2}s most of the profiles are single S shaped with few exceptions with double S shaped like the profile at Z = 3.96 m and r/R = 0.81 also at Z = 9.42 and r/R = 0.81. Furthermore, _{g} = 10 m/s and G_{s} = 300 kg/m^{2}s also most of the profiles are single S shaped with few exceptions with double S shaped like the profile at Z = 9.42 and r/R = 0.81. Presence of the multiple S-shaped profiles indicates a distinct multifractal flow structure or mixed flow structure.

The CDF profiles shows features which are found for all velocities in various sections of the riser. For instance, the double S-shaped CDF is clearly observed at r/R = 0.98 in the developed section (Z = 9.42 m) when U_{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s. This kind of profiles is also found when U_{g} = 8 m/s and G_{s} = 300 kg/m^{2}s for instance in the developed section (at r/R = 0.81, Z = 9.42 m), developing section (at r/R = 0.81, Z = 3.96 m) and in the entrance section at r/R = 0.92 and Z = 1.52 m. Also for U_{g} = 10 m/s and G_{s} = 300 kg/m^{2}s double S-shaped profile is also found in the developed section (Z = 9.42 m) at r/R = 0.92, in the developing section (Z = 3.96) at r/R = 0.92 and in the entrance section (Z = 1.52 m) at r/R = 0.98. However, at this high velocity close to the wall in the developed section (Z = 9.42) the profile at r/R = 0.98 is multiple S-shaped.

Results show the location of some of the profiles to shift along the ln (δ) axis particularly in the profiles in wall region. The tendency to shift the location along the ln (δ) axis is seen for instance when U_{g} = 5.5 m/s, CDF at r/R = 0.81 shifts towards the negative direction i.e., towards lower values of ln (δ) from the entrance section towards the top sections of the riser. However, close to the wall at r/R = 0.98 the profile in almost similar location with the center profile (r/R = 0.0) while in the developing section (Z = 3.96 m) it shifts towards the positive direction, i.e., towards higher values of ln (δ). In the developed flow section it shifts back towards the negative direction at lower values of ln (δ). Also this behaviour is observed when U_{g} = 8 and 10 m/s, where the profiles at r/R = 0.81 shifts towards the negative direction from the entrance section towards the developed section.

Further observations show the span or width of a dense attractor in the sphere space for some of the radial positions to change. This is revealed through the change in the span of the rising part of the CDF. For instance, when the U_{g} = 5.5 m/s, the span of the profiles at r/R = 0.98 in the developed section (Z = 9.42 m) is higher compared to the span of the CDF in the same radial position in the entrance (Z = 1.52) and developing sections (Z = 3.96). This behaviour is also observed when U_{g} = 10 m/s for the profiles in the wall region at r/R = 0.92 and 0.98.

Results in _{g} = 10 m/s the wall region profiles (r/R = 0.81, 0.92 and 0.98) in the entrance section (Z = 1.52 m) are close together with nearly similar shape. This however is not seen in the same section of the riser for lower velocities. However, when U_{g} = 8 m/s the only close profiles with nearly similar shape are at r/R = 0.92 and 0.98. When U_{g} = 5.5 m/s the wall region profiles are apart with varying shapes. Secondly, in the entrance section (Z = 1.52 m) the center CDF profile (r/R = 0.0) has an S-shape with two linear part having different slopes. This behaviour is only observed for higher velocity, U_{g} = 10 m/s. Thirdly, with exception of the CDF profiles at r/R = 0.81 for U_{g} = 5.5 and 8 m/s the wall region CDF profiles (r/R = 0.81, 0.92 and 0.98) for all operating velocities are located to the higher values of ln (δ) relatively to the center profile (r/R = 0.0) which are located to the lower values of ln (δ) along the ln (δ) axis. Further, at higher velocities, i.e., U_{g} = 10 m/s, the formation of multiple S-shaped profile is seen close to the wall (r/R = 0.98) in the developed section (Z = 9.42 m). Multiple S-shaped CDF profiles are not seen for lower velocities.

Again use of the CDF of the trajectory distances is reported in literature which used pressure fluctuation signal that could not account for the dynamic variations along the radial direction [

The phase space dimension is the smallest dimension of the space in which a phase portrait is reconstructed where the trajectory does not cross itself. The number of elements of the state vector, which equals the number of coordinates in state space, is called embedding dimensions [

In

From

The entrance and wall dynamics of a high-flux gas-solid riser were studied using probability density functions and cumulative distributions functions of trajectory distances of the state vectors from the reconstructed attractors using solid concentration signals. This work has established the role of PDF and CDF in studying the gas-solid flow behaviour. The gas-solid flow behaviours at the wall region is well described and differentiated using PDF and CDF of the trajectory distances from the reconstructed attractor. The trajectory distances increase from the centre towards the wall indicating expansion of the attractor. The PDF of trajectory distances forms double and multiple peaks in wall region to signify

presence of bifractal and multifractal behaviours. These behaviours are presented by double and multiple S-shaped CDF profiles. The PDF distribution of these distances at the entrance section and in the wall region forms various types of statistical distribution showing differences in gas-solid flow structures in various spatial locations of the wall region and the entrance sections. At the centre most of the distributions fall under the Gumbel max distribution for all flow development sections of the riser, especially at lower velocities of 5.5 m/s and 8 m/s velocities showing uniform flow structures. In the wall region the PDF distributions shows different types in all flow development sections of the riser showing multifractal flow structures. Further, the increase in number of the phase space reconstruction dimension increases the trajectory distances between state vectors leading to the expansion of the attractor.

The authors declare no conflicts of interest regarding the publication of this paper.

Jeremiah, J.M., Manyele, S.V., Temu, A.K. and Zhu, J.-X. (2018) Analysis of Flow Dynamics in the High-Flux Gas-Solid Riser Using Trajectory Distances across Attractors Reconstructed from Solid Concentration Signals. Engineering, 10, 688-703. https://doi.org/10.4236/eng.2018.1010050