^{1}

^{*}

^{1}

^{1}

^{2}

Statistical analysis of the entrance and wall dynamics of a high-flux gas-solid riser was done using solid concentration time series data collected from a 76 mm internal diameter and 10 m high riser of a CFB system with a twin-riser 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 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 analysed using code prepared using FORTRAN 2008 to get statistical parameters and plot their profiles. Results obtained show that the gas-solid suspension flow in the riser is dominated by low solid concentration in the centre region and high solid concentration in the wall region which forms a core-annulus flow structure. The mean solid concentration in the wall region decreases with riser height from the dense bottom section to less dense in the fully developed flow section at the top of the riser. The gas-solid suspension flow in the centre region is dominated with uniform flow structure while the wall region is dominated with high fluctuations in solid concentration. Further, it was found that the entrance and developing flow sections of the riser exhibit high flow non-uniformities than the fully developed flow section of the riser. The flow non-uniformities in the entrance and developing flow section increase with increase in superficial gas velocity at constant solid flux. The wall region, from the entrance to the top sections of the riser along the axial direction exhibits both dilute and dense suspension flow.

The advantages of circulating fluidized bed (CFB) riser over other reactors in various industrial applications have resulted to the continued substantial growth of their usage [

Studies have shown that variations of solid concentrations in the gas-solids suspension flow systems in the CFB risers are one of the disadvantages that elevate gas-solid flow non-uniformities [

While various studies couple some statistical analysis with other approaches like spectral and or chaos analysis, this paper employs and presents a comprehensive statistical analysis of the temporal variations of solid concentration signals by basing the analysis in the wall region and entrance section in comparison to the centre region of the CFB riser. The paper has reviewed and used microscopic or time dependent and macroscopic or averaged parameter statistical methods related to CFB dynamics. Due to many concepts in use, more diverse results are presented in this paper because solid concentration data were collected across several radial positions (11) and axial elevations (8) which necessitated the use of taller high-flux CFB (10 m). Thus, results are presented in various subsections using Mean, Standard deviations, Skewness, Kurtosis, Intermittent Index, Average Absolute Deviation, Coefficient of Variation, Radial Non-uniformity Index and Probability Density Function.

Statistical analysis is one of the time-series analysis (TSA) techniques that employ various tools or parameters to investigate solid distribution in the gas-solid flow systems. Literatures show that solid particles distribution in the riser reactor during the gas-solid flow operations is highly influenced by solids backflows especially in the wall regions, formation of clusters and flow non-uniformities in the radial and axial profile [

Literatures present various studies that investigated the axial and radial distribution of solids in the gas-solid suspension flow system in circulating fluidized bed (CFB) risers and downers using statistical analysis. The study by Manyele et al. [

The solid distribution in the CFB riser and downers has been of interest in various studies since it affects various aspects of production process such as rate of heat and mass transfer which in turn affects the rate of chemical reactions and overall product yield. It is shown that the axial and radial flow structures in the CFB riser reactor is highly influenced by the operating conditions of the solid flux and superficial gas velocity [^{2}s, the radial solid distributions have been shown to be non-uniform with dilute or low solid concentration particularly in the centre region of the riser. Then solid concentration increases slowly in the middle region to dense concentration towards the wall of the riser reactor [_{s} ≥ 700 kg/m^{2}s, the flow structures changes to concave or parabolic curve form with higher concentration towards the wall [^{2}s, the solid concentration profiles are approximately exponential shaped [

The most used statistical tools or parameters in the analysis of solid concentration includes the Mean, Standard Deviations, Skewness, Kurtosis, Coefficient of Variation, Intermittent Index, Average Absolute Deviation (AAD), Radial Non-Uniformity Index (RNI) and Probability Density Functions (PDF).

In gas-solid CFB riser operation, the mean or average solid concentration gives the average distribution of solid particles in the gas-solid flow suspension around the sampling point [

Kurtosis is the fourth moment of the data signal which indicates the flatness of the distribution in comparison to the normal distribution. It shows whether data are peaked or flat when compared with a normal distribution [

Coefficient of variation is a measure of dispersion or scattering tendency of a probability distribution or frequency distribution [

The AAD describes the variability of data distribution. As far as the gas-solid flow is concerned the AAD is used to indicate the level of gas-solid flow fluctuations in the CFB riser [

Literature review shows that most of statistical analysis were developed and used separately. This paper combines all parameters to analyse the same data allowing for insight comparison of results and hence deeper understanding of the dynamics.

Solid concentration data signals were collected from a 76 mm internal diameter riser of a CFB system with a twin-riser having 76 and 203 mm internal diameters and 10 m high shown in ^{2}s solids flux and 4.0 to 10.0 m/s gas velocity. Fluid catalytic cracking catalyst particles with 67 mm 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 centre to the wall of the riser. To each point, 29,100 data points of solid concentration were sampled in 30 seconds using optical fibre probe at 970 Hz.

The solid concentration data signals obtained from the experiment were statistically analysed using codes prepared in FORTRAN 2008 to determine the Mean ( ε s ¯ ), Standard deviation ( σ ( ε s ) ), Skewness (S_{k}), Kurtosis (K_{u}), Intermittent Index ( γ ), Average Absolute Deviation (AAD), Coefficient of Variation (CV), Radial Non-uniformity Index (RNI) and Probability Density Functions (PDF). The radial and axial profiles of the computed parameters were plotted and examined. Then interpretation was accomplished by relating various features of the profiles to the gas-solid flow dynamics in the circulating fluidized bed riser.

The mathematical formulations of the computed parameters are presented in

Where ε s is the solid concentration signal from i = 1 , 2 , 3 , ⋯ , N data; σ ( ε s ) max and ε s , m f are the maximum possible standard deviation for a particular cross-sectional average solids concentration and the local solids concentration at minimum fluidization respectively.

This study employed the mean ( ε s ¯ ) to evaluate the radial distribution of local average solid concentration along the riser height. The radial profiles of the local mean solid concentration at different axial levels of the riser for different operating conditions are shown in

Parameter | Microscopic/ macroscopic | Formula | Reference |
---|---|---|---|

Mean, ε s ¯ | Macroscopic | ε s ¯ = 1 N ∑ i = 1 N ε s ( i ) | [ |

Standard Deviation, σ ( ε s ) | Microscopic | σ ( ε s ) = 1 N − 1 ∑ i = 1 N ( ε s ( i ) − ε s ¯ ) 2 | [ |

Skewness, S k | Microscopic | S k = 1 N σ ( ε S ) 3 ∑ i = 1 N ( ε S ( i ) − ε S ¯ ) 3 | [ |

Kurtosis, K u | Microscopic | K u = 1 N σ ( ε S ) 4 ∑ i = 1 N ( ε S ( i ) − ε S ¯ ) 4 | [ |

Intermittent Index, γ | Macroscopic | γ = σ ( ε s ) σ ( ε s ) max = σ ( ε s ) ε s ¯ ( 1 − ε s ¯ − ε s , m f ) ¯ | [ |

Average Absolute Deviation (AAD) | Microscopic | A A D = 1 N ∑ i = 1 N ( | ε s ( i ) − ε s ¯ | ) | [ |

Radial Non-Uniformity Index (RNI) | Macroscopic | R N I ( ε s ) = σ ( ε s ) σ ( ε s ) max = σ ( ε s ) ε s ¯ ( ε s , m f − ε s ¯ ) | [ |

Probability Density Function (PDF) | Microscopic | f ( x ) = Pr [ ε s = x ] | [ |

Coefficient of Variation, CV | Macroscopic | C V = σ ( ε s ) ε s ¯ | [ |

and developing flow section (Z ≤ 3.96 m) in the wall region, profiles are irregular with relatively higher solid concentration and then become regular with relatively lower values in the upper section at Z > 5.13 m. This observation suggests relatively fluctuating flow structure with denser solid concentration compared to the fully developed flow section. Irregular profiles of solid concentration towards the wall reveals non-uniform flow structures in the wall region which might be attributed to the effects of non-uniform reintroduction of solid flux at the entrance and pulsation of gas from the air distributor. Such results concur with findings in other studies [

However, one of the peculiar finding is the sudden drop in solid concentration close to the wall at r/R = 0.98, especially at the level of Z = 2.73 m for all operating conditions as shown in _{g} = 8 m/s and G_{s} = 300 kg/m^{2}s the solid concentration increases to the maximum value of ε s ¯ = 0.499 at r/R = 0.92 and then drops to ε s ¯ = 0.0162 at r/R = 0.98. When U_{g} = 10 m/s and G_{s} = 300 kg/m^{2}s the solid concentration reaches a maximum value of ε s ¯ = 0.537 at r/R = 0.92 and then drops to ε s ¯ = 0.0418 at r/R = 0.98. Such observations are also found at Z = 1.52 m for U_{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s where the solid concentration reaches a maximum of ε s ¯ = 0.435 at r/R = 0.87 and then drops to ε s ¯ = 0.0333 at r/R = 0.98. This behaviour might be due to gas flow turbulences which lead to relatively high fluctuations between dilute and dense gas-solid suspensions flow.

The standard deviation ( σ ( ε s ) ) was used to examine the solid concentration fluctuations from the mean values when the riser is operated at different operating conditions.

wall of the riser. This shows that the centre and the middle regions are dominated with low fluctuations in solid concentrations compared to the wall region. This suggests a relatively higher uniform gas-solid flow structure in the central region or core-region than in the annulus region which is mostly the wall region. Also profiles in the entrance and developing flow sections (Z ≤ 3.96 m) are irregular with relatively higher σ ( ε s ) compared to the developed/upper sections of the riser. This shows high fluctuations of solid concentration in the entrance and developing flow sections compared to the top sections suggesting a non-uniform and highly fluctuating flow structures. Also profiles in the entrance and developing flow sections (Z ≤ 3.96 m) increases gradually towards the wall to maximum values and then drops suddenly to lower values. For instance, at Z = 1.52 m, for U_{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s, values of standard deviation increases to a maximum value of σ ( ε s ) = 0.241 at r/R = 0.81 and then drops to σ ( ε s ) = 0.102 at r/R = 0.98. Similar observation is seen at Z = 2.73 m where standard deviation increases to a maximum value of σ ( ε s ) = 0.265 at r/R = 0.81 and then drops to σ ( ε s ) = 0.142 at r/R = 0.98. The sudden drop in standard deviation suggests an abrupt decrease in fluctuations of the solid concentration in the gas-solid suspension flow. Similar observations are reported in other studies at Z = 1.94 m and Z = 5.84 m for the operating conditions of U_{g} = 7 m/s and G_{s} = 500 kg/m^{2}s, U_{g} = 9 m/s and G_{s} = 500 kg/m^{2}s and U_{g} = 9 m/s and G_{s} = 1000 kg/m^{2}s [_{g} = 8 m/s and G_{s} = 300 kg/m^{2}s profiles at the central region of the riser are closest to each other with relatively lower values of standard deviations. This might be due to relatively lower axial variations in the gas-solid fluctuations indicating a relatively uniform gas-solid suspension flow in the axial direction towards the top of the riser.

Results in

of the riser. Skewness profiles are relatively flat with relatively higher positive values in the centre and middle region which then decreases gradually towards the wall. However, some profiles in the entrance section, that is profiles at Z = 1.52 and 2.73 m decreases gradually in the wall region to negative values and then increases sharply to relatively higher positive values close to the wall. For instance when U_{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s the Skewness values decreases to S_{k} = −0.193 at r/R = 0.87 and then increases sharply to S_{k} = 5.32 at r/R = 0.87. Also when U_{g} = 10 m/s and G_{s} = 300 kg/m^{2}s the Skewness values decreases to S_{k} = −1.15 at r/R = 0.92 and then increases sharply to S_{k} = 4.24 at r/R = 0.98. These results indicate that solid concentration data signals in the wall region are non-symmetrical with positive or right skewed distribution which forms PDF with longer tails towards the right. This suggests that dense suspension flow dominates in the wall region of the riser.

In this study Kurtosis (K_{u}) was used to examine the nature of the solid concentration signal distribution, whether peaked or flat PDF distribution. _{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s profiles at Z = 1.52 m decrease to the minimum values of K_{u} = −1.58 at r/R = 0.87 and then increases to K_{u} = 28.5 at r/R = 0.98. When the velocity, U_{g} = 8 m/s and G_{s} = 300 kg/m^{2}s, profiles at Z = 1.52 and Z = 2.73 m decrease to minimum values of K_{u} = −0.943 at r/R = 0.81 and K_{u} = −0.773 at r/R = 0.92 respectively and then increase to K_{u} = 19.9 and 23 respectively at the wall (r/R = 0.98). These observations indicates that the solid concentration data signals form a relatively flat and right skewed PDF in the wall region while at the wall, PDF profiles become peaked with short tails towards the right. This suggests that close to the wall (r/R = 0.98) there might be a slight uniformity due to formation of dense suspension or heavy clusters of solid particles.

This study employed Intermittent Index factor to examine both solid flow fluctuations and the extent of segregation in the gas-solid flow suspension; whereby higher values indicate segregated gas-solid flow behaviour and lower values show a more uniform gas-solid suspension.

profiles are flat and regular with slightly lower values in the centre and then becomes irregular and slightly increasing in the wall region especially in the entrance section for Z = 1.52 and 2.73 m. The relative increases of the Intermittent Index values in the wall region especially in the entrance and developing flow section (Z ≤ 3.96 m) shows slight segregated flow behaviour in these sections. This might be due to non-uniform reintroduction of solid and pulsation of the gas from the distributor which results in a more segregated flow. However, in the developed flow zone, i.e. Z = 6.23 m, values of γ are relatively lower with regular and slightly flat profiles which suggests a slightly uniform gas-solid suspension flow. Further, it can be observed that when U_{g} = 8 m/s and G_{s} = 300 kg/m^{2}s profiles are very close to each other compared to profiles in other process conditions. This might suggests a less segregated and non-fluctuating gas-solid flow which is more likely to occur for the dilute gas-solid flow in the central region of the riser.

The AAD was employed to examine the average value of the departure of instantaneous data values of solid concentration from the mean value. _{g} = 10 m/s and G_{s} = 300 kg/m^{2}s,

AAD profiles becomes wider spread in the central region which span to the maximum values of AAD = 2.22 × 10^{−5} for Z = 2.73 and 5.13 m. This shows higher average departure of the solid concentration data values from the mean. Such observation suggests presence of differences in the flow structures between the entrance and the transition zone with notable relative fluctuations between the less dilute and most dilute suspensions in the respective sections.

The coefficient of variation was used to measure the relative variability or the level of fluctuations of the gas-solid concentration distribution in the riser. _{g} = 8.0 m/s and G_{s} = 300 kg/m^{2}s, Z = 1.52 m for U_{g} = 10 m/s and G_{s} = 300 kg/m^{2}s and Z = 5.90 for U_{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s. The decreasing trend suggests that the variation of solid concentration in the gas-solid suspension flow decreases towards the wall which might be due to the increase in concentration from dilute to dense suspension flow.

However some of the profiles decreases progressively and then increase sharply close to the wall (r/R = 0.98). Such tendency is shown by the profiles at Z = 1.52 m when U_{g} = 8 m/s and G_{s} = 300 kg/m^{2}s where the CV decreases to a minimum value of CV = 0.448 and then increases to CV = 3.15. Also the same tendency is shown by the same profile when U_{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s where the profile decreases to CV = 0.554 and then increases to CV = 3.07. Further, this behaviour is also shown at Z = 2.73 m when U_{g} = 10 m/s and G_{s} = 300

kg/m^{2}s where the coefficient of variation decreases to CV = 0.421 and then increases to CV = 3.39. Higher values of CV occurs when the suspension flow has high variation in solid concentration leading to high standard deviation relative to its mean or in dilute suspension where the mean solid concentration is very low compared to its variations or fluctuations expressed in standard deviation. In this case higher values in the centre and middle region may suggests presence of low local mean solid concentration which is as shown in

The gas-solid flow non-uniformities in the riser were assessed using the Radial Non-uniformity Index (RNI).

The PDF was used to examine the nature of solid distribution in the gas-solid

suspension in the wall region, from the entrance section to the top section in comparison to the centre region. _{g} = 5.5, 8 and 10 m/s and G_{s} = 300 kg/m^{2}s. For each operating condition, the shapes of PDF at the centre region (r/R = 0.0) are tall in the left for low solid concentration with long tails towards the right (positively skewed) for all axial elevations. This shows a highly dilute gas-solid suspension flow with slight fluctuations of the less dilute to dilute solid concentrations. In the wall region especially for r/R ≥ 0.8 shapes of PDF in the entrance section, developing flow and the transition sections (Z ≤ 5.90 m) differs from that in the fully developed section. In the entrance and developing flow sections, profiles are either tall at the left with long tail towards higher solid concentration, i.e. positively skewed distribution; or single peaked with tails both sides; or double peaked profiles. But

in the fully developed section Z > 5.90 m PDF are tall in the left with long tail in the right towards higher solid concentration. These observations show that in the entrance and developing flow sections there are high flow non-uniformities as compared to the fully developed section. Profiles in the centre region along the axial direction have no significant differences in shapes showing the increase in velocity have no significant influence on solid distribution in the core region. In the wall region, profiles show differences by having single and double peaks which shows coexistence of both dilute and dense phases.

When the gas velocity, U_{g} = 5.5 m/s and solid flux, G_{s} = 300 kg/m^{2}s, PDF profiles in the entrance, developing flow and the transition sections are dominated by fluctuations in the wall region with higher solid concentration and right skewed profiles. Towards the wall (0.87 ≤ r/R ≤ 0.98), in the levels from 3.96 m ≤ Z ≤ 6.43 m, PDF profiles are double peaked with tails towards the right. Such observation suggests coexistence of a dilute and dense phase. In the fully developed section, Z ≥ 6.43 m, PDF profiles are tall on the left with longer and flat tail towards the right. This indicates that in the fully developed section is dominated with relatively dilute uniform flow.

When the operating condition is U_{g} = 8.0 m/s and G_{s} = 300 kg/m^{2}s, PDF profiles in the wall region shows high non-uniformities in the entrance and developing flow sections. At Z ≤ 2.73 m the wall region is dominated by single and double peaks PDF with long tail to the right, in the higher concentration direction. This indicates high solid fluctuations with a highly segregated flow structures with coexistence of dilute and dense phase flow. In the transition section PDF are tall in the left and skewed to the right as seen at 3.96 m < Z ≤ 6.34 m for r/R = 0.87 and 0.98. This indicates that the suspension flow is dominated with low solid concentration fluctuating with less dense phases. In the fully developed section the flow is relatively uniform and dominated with very dilute suspension.

For U_{g} = 10 m/s and G_{s} = 300 kg/m^{2}s the PDF profiles in the centre region are tall in the left with long tail towards high solid concentration, that is, positively skewed profiles. This show the centre region is dominated by the dilute gas-solid suspension flow. PDF profiles in the wall region of the entrance section are positively skewed towards high concentration which shows fluctuations between dilute and dense suspensions. Such profiles are found at 1.52 ≤ Z ≤ 6.34 m in the wall region for 0.74 ≤ r/R ≤ 0.98. However, from the transition section towards the exits, i.e. at Z ≥ 5.13 m and r/R = 0.74 and 0.87; PDF are positively skewed with left peak and long tails towards the high concentration direction. Such PDF are also found from Z = 8.74 m to Z = 9. 42 m at r/R = 0.92 and 0.98. Likewise the same PDF shapes are found in the centre region. This suggests presence of uniform low solid concentration suspension flow.

Further observation of the shapes of the PDF reveals four different types, i.e. A, B, C and D as summarised in

Description | ||
---|---|---|

Type A | ・ Right tailed PDF ・ Tall and narrow to the left with longer tails to the right ・ Flow is dominated with low concentration, that is dilute suspension flow ・ Relatively very low fluctuations | |

Type B | ・ Single peak to the left and longer tail towards the right ・ Flow is dominated with low solid concentration ・ High fluctuation between dilute and dense suspension | |

Type C | ・ Single peak at the centre, two tailed profile ・ Flow is dominated with single phase, in this case the dense phase around the mean ・ Existence of low variations in solid fluctuations | |

Type D | ・ Double peak, right tailed ・ Segregated flows ・ Coexistence of the dilute and dense phases ・ High fluctuations |

Type A: PDF profiles are tall at the left with longer tail to the right. These PDF are found mostly in the centre and the middle region of the riser from the bottom to the top section. Also they are found in the wall region of the fully developed section at Z ≥ 8.74 m. this shows the gas-solid flow is dominated with a dilute or very low solid concentration.

Type B: PDF profiles are single peaked at the left with a longer tail to the right. These profiles are found in the entrance, developing flow and transition sections when U_{g} = 5.5, 8.0 and 10 m/s and G_{s} = 300 kg/m^{2}s as shown in

Type C: PDF profiles are single peaked at the centre with tails on both sides. These type of profiles are found in the wall region at r/R = 0.87 and Z = 2.73 when the operating condition is U_{g} = 8 m/s and G_{s} = 300 kg/m^{2}s. But also they are found at r/R = 0.81, 0.87 and 0.92 at Z = 2.73 m when U_{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s as shown in

Type D: PDF profiles are double peaked. Such profiles are found in the wall region at r/R = 0.87 and Z = 1.52 m as shown in _{g} = 8.0 m/s and G_{s} = 300 kg/m^{2}s. They are also found in the wall region from Z = 3.96 to 6.43 m when the operating condition is U_{g} = 5.5 m/s and G_{s} = 300 kg/m^{2}s as shown in

The summary of the spatial locations of various types of the PDF obtained in the analysis are shown in

From this study the following conclusions can be made:

・ The gas-solid suspension flow in the riser is dominated by low solid concentration at the centre and high solid concentration in the wall region forming a core-annulus flow structure.

・ The mean solid concentration in the wall region decreases with riser height from the dense bottom section to less dense in the fully developed flow section.

・ The gas-solid suspension flow in the centre region is dominated with dilute uniform flow while the wall region is dominated with high fluctuations in solid concentration.

・ The bottom and developing flow sections of the riser exhibit high flow non-uniformities than the fully developed flow section of the riser.

Operating Condition | Location | |||
---|---|---|---|---|

Type A | Type B | Type C | Type D | |

U_{g} = 5.5 m/s G_{s} = 300 kg/m^{2}s | ・ At the centre region (r/R = 0.0) ・ At r/R = 0.74 from Z = 5.90 to 9.42 m ・ At r/R = 0.81 from Z = 6.43 to 9.42 m ・ At r/R = 0.92, from Z = 8.74 to 9.42 m ・ At r/R = 0.98, at Z = 9.42 m | ・ At r/R = 0.74 from Z = 1.52 to 3.96 m ・ At r/R = 0.74 from Z = 1.52 to 3.96 m ・ At r/R = 0.81 Z = 1.52 m and Z = 3.96 m | ・ At Z = 2.73 from r/R = 0.81 to 0.92 | ・ At r/R = 0.87, Z = 3.96 m ・ At r/R = 0.92 from Z = 3.96 to 6.43 m ・ At r/R = 0.98, from Z = 5.90 to 6.43 m |

U_{g} = 8.0 m/s G_{s} = 300 kg/m^{2}s | ・ At r/R = 0.0 from Z = 1.52 to 9.42 m ・ At r/R = 0.74 and r/R = 0.81 from Z = 3.96 to 9.42 m ・ At r/R = 0.87 from Z = 5.90 to Z = 9.48 m ・ At r/R = 0.92 and r/R = 0.98 from Z = 8.72 to 9.42 m | ・ At r/R = 0.74 from Z = 1.52 and 2.73 m ・ r/R = 0.81 from Z = 1.52 to Z = 513 m ・ At r/R = 0.87, Z = 3.96 and 5.13 m ・ r/R = 0.92, from Z = 2.73 to 6.34 m ・ At r/R = 0.98, Z = 1.52, 3.96 to 6.34 m | ・ At r/R = 0.87, Z = 2.73 m ・ At r/R = 0.92, Z = 2.73 | ・ At r/R = 0.87, Z = 0.87 m |

U_{g} = 10 m/s G_{s} = 300 kg/m^{2}s | ・ At r/R = 0.0 form Z = 1.52 to 9.42 m ・ r/R = 0.74 from Z = 3.96 to 9.42 m ・ r/R = 0.81 from Z = 5.13 to 9.42 m ・ r/R = 0.87, Z = 5.13 to 9.42 m ・ found from r/R = 0.92 to 0.98 at Z = 8.74 to 9.42 m | ・ At Z = 1.52 m, r/R = 0.74 to 0.87 m ・ At Z = 2.73, r/R = 0.81 to 0.92 m ・ r/R = 0.81 to 0.92 from Z = 1.52 to 3.96 m ・ r/R = 0.92 to 0.98 at Z = 5.13 to Z = 6.34 m | - | - |

U_{g} = 8.0 m/s G_{s} = 400 kg/m^{2}s | ・ At r/R = 0.0, from Z = 1.52 to 9.42 m ・ At r/R = 0.74, from Z = 3.96 to 5.90 m and Z = 8.74 to 9.42 m ・ At r/R = 0.81 to 0.92 from Z = 8.74 to 9.42 m ・ At r/R = 0.98 from Z = 1.52 to 2.73 m and from Z = 6.43 to 9.42 m | ・ At r/R = 0.74 from Z = 1.52 to 2.73 m and Z = 6.42 m ・ At r/R = 0.81 from Z = 1.52 ? 6.43 m ・ At r/R = 0.87, Z = 5.90 m ・ r/R = 0.92, Z = 5.90 m ・ r/R = 0.98 from Z = 5.13 to 5.90 m | ・ At r/R = 0.87, Z = 2.73 m and 6.43 m | ・ At r/R = 0.87, Z = 1.52 m ・ At r/R = 0.92 from Z = 1.52 to 3.96 and Z = 6.43 m ・ At r/R = 0.98, Z = 3.96 m |

U_{g} = 8.0 m/s G_{s} = 550 kg/m^{2}s | ・ At r/R = 0.0 from Z = 1.52 to 9.42 m ・ At r/R = 0.74 Z = 9.42 m ・ At r/R = 0.81 Z = 9.42 m | ・ At r/R = 0.74, Z = 5.13 to 6.34 m ・ At r/R = 0.81, Z = 1.52 to 6.43 m ・ At r/R = 0.87, Z = 5.13, 6.34 and 9.42 m ・ At r/R = 0.92 m, Z = 5.13, 6.34 and 9.42 m ・ At r/R = 0.98, Z = 5.13, 6.34 and 9.42 m | - | - |

Parameter | Significance | Observation | |
---|---|---|---|

Centre/core region | Wall region | ||

Mean | Gives an average distribution of the local solid concentration | Low values with regular flat profiles | Higher values, increase towards the wall |

Standard Deviation | A measure of dispersion or fluctuation of solid concentration | Low values with relatively flat profiles | Higher values towards the wall |

Skewness | Measures the symmetrical properties or degree of asymmetry of PDF around the mean | Higher and relatively flat profiles | Decrease towards the wall with irregular profiles |

Kurtosis | Indicates the tendency of the PDF to be peaked | Higher values and generally flat profiles | Lower, slightly decreases towards the wall |

Intermittent Index | Shows the extent of gas-solid fluctuations or gas-solid suspension flow segregation | Relatively higher/lower values and regular flat profiles | Irregular profiles, Slightly increase towards the wall |

Coefficient of Variation | A measure of relative dispersion of a distribution. It indicates variability or scattering tendency | Relatively higher and regular profiles | Decreases towards the wall |

Average Absolute Deviation | Estimates the width of data around the average values | Constant, higher or lower values with regular profiles | Irregular profiles, relatively increasing/decreasing towards wall |

Probability Density Function | Indicates the distribution property of solid concentration | Tall and narrow left peak with longer tails to the right | Single or double peaks with tails towards the right |

・ The flow non-uniformities in the bottom and developing flow section increase with increase in superficial gas velocity at constant solid flux due to the increase in particle interactions and disorders of the solid particles in the gas-solid suspension.

・ The wall region along the axial direction from the bottom to the top section of the riser has positive skewed, single and double peaked PDF profiles indicating coexistence of both dilute and dense suspension flow.

These conclusions show the strength of statistical analysis as a technique for comprehensive understanding of the gas-solid flow dynamics in the High Flux CFB riser systems using solid concentration signals.

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. (2019) Investigation of Entrance and Wall Dynamics of the High-Flux Gas-Solid Riser Using Statistical Analysis of Solids Concentration Signals. Engineering, 11, 167-187. https://doi.org/10.4236/eng.2019.113013