Mutual Information Functions of Differential Pressure Fluctuations in

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Ind. Eng. Chem. Res. 2004, 43, 5754-5762

Mutual Information Functions of Differential Pressure Fluctuations in Spouted Beds Jian Xu, Xiaojun Bao,* and Weisheng Wei The Key Laboratory of Catalysis, China National Petroleum Co., University of Petroleum, Beijing 102249, China

Hsiaotao Bi, John R. Grace, and Jim C. Lim Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, V6T 1Z4 Canada

The objective of this paper is to determine whether the flow regimes in spouted beds can be recognized by the mutual information function of pressure fluctuation signals. Differential pressure fluctuation time series were obtained in a gas spouted bed of 0.12-m i.d. at different axial and radial positions, with glass beads and silica gel particles. It was found that the packed bed has a low level and fast decay rate of mutual information, suggesting a quite random signal. Unstable spouting exhibits a high level of mutual information and strong periodic motion. While for a stable spouting regime, the level of mutual information and the decay rate around time zero are between those for the packed bed and unstable spouting regimes. The mutual information function for signals measured on the wall has higher predictabilities compared to those measured inside the bed. The determination of optimum time delay by the aid of the first minimum of the mutual information function was discussed. It has been found that the first minimum of the mutual information function can be used as the optimum time delay if it does exist. In the case that no first minimum exists in the mutual information function, the suggestion of Daw and Halow (Daw, C. S.; Halow, J. S. AIChE Symp. Ser. 1993, 89, 103-122) that 1050% of the maximum value of the mutual information resulted in too small a time delay. An appropriate time delay can be determined by comparing the similarity between phase space trajectories and pseudo phase space trajectories. Introduction Spouted bed technology has been widely used for a variety of applications, such as drying, coating, granulation, and chemical reactors.1 To optimize the design and operation of spouted beds, the different flow regimes and their transitions should be well characterized and defined. Previous recognition of spouted bed flow regimes reported in the literature had usually been obtained from visual observations, which inevitably introduced some arbitrariness and subjectivity. Moreover, most industrial vessels are not transparent and visual observations are even impossible. Thus new nonvisual approaches are required. Although various methods based on analysis of pressure fluctuation signals have been extensively utilized in fluidized beds, their applications to spouted beds are scarce. As demonstrated by the research group of this contribution in previous research work, pressure fluctuation analysis appears to be a promising technique for spouted bed regime identification.2-4 The mutual information function, first suggested by Shannon and Weaver,5 is similar in some aspects to the autocorrelation function. It provides information about the time evolution of the process such as how well one can predict the future behavior of the process or how periodic or how chaotic the signal recorded is. The mutual information function does not have an underlying linear relationship as does the autocorrelation * To whom correspondence should be addressed. Tel: (8610)89734836. Fax: (8610)89734979. E-mail: [email protected].

function and thus suits itself for the analysis of nonlinear processes as an important property of dynamical systems.6 The concept of entropy is central to mutual information theory, which has been used mostly in the area of data communication and is now extended to the analysis of experimental data. The mutual information function can be calculated directly from experimental time series data. The procedure was described by Daw and Halow7 and Karamavruç et al.8 Consider an arbitrary discrete data set obtained by experimental observation so that x(t) ) {x(t1), x(t2), x(t3), ..., x(tN)}. Values of x may be divided into bins, each with a range, and denoted by values x1, x2, ..., xNbins. For any data set, the probability of any value of x falling into a specific bin is P(xi). Hence, a set of probabilities P(x1), P(x2), P(x3), ..., P(xNbins) can be created from the original data set. The average entropy corresponding to this data set is defined as Nbins

H(X) ) -

P(xi) log 2[P(xi)] ∑ i)1

(1)

where X denotes the whole system that consists of all measured data and Nbins the number of bins. The logarithm is usually taken to the base two so that the entropy has units of bits. Entropy is a good measure of the uncertainty of the outcome of a probabilistic experiment. If one is absolutely certain about the outcome of an event before it happens, the probability will be one and consequently the entropy will be zero. However, if all the probabilities are equally distributed, then P(xi)

10.1021/ie049751q CCC: $27.50 © 2004 American Chemical Society Published on Web 07/15/2004

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5755

Figure 1. Mutual information functions for sinusoidal and random time series. Table 1. Properties of Bed Materials material

dp (mm)

Fb (kg/m3)

Fp (kg/m3)

glass beads silica gel

1.600 1.660

1530 430

2560 720

Figure 2. Schematic of the experimental setup.

) 1/Nbins and the entropy, H(X) ) log2(Nbins), will be maximized. Consider now a long time series as x(t) ) {x(t1), x(t2), x(t3), ..., x(tN)} and x*(t) ) {x(t1 + τ), x(t2 + τ), x(t3 + τ), ..., x(tN + τ)}, where x*(t) is the τ delayed version of x(t), with the set of probabilities P(x*1), P(x*2), P(x*3), ..., P(x*Nbins). The mutual information function can be described as

I(X,X + τ) ) H(X) + H(X + τ) - H(X,X + τ) (2) where H(X) is given by eq 1, and H(X + τ) and H(X,X + τ) are expressed as follows Nbins

H(X + τ) ) -

P(x*j) log2[P(x*j)] ∑ j)1

(3)

NbinsNbins

H(X,X + τ) ) -

∑ ∑ P(xi,x*j) log 2[P(xi,x*j)] i)1 j)1

(4)

The quantity P(xi,x*j) is the joint probability that measurement x(t) falls into bin xi and its delay version x(t + τ) falls into bin x*j. In words, the mutual information function I(X,X + τ) represents the average number of bits that can be predicted correctly for a subsequent measurement τ time steps in the future based on knowledge of measurement X. The exact nature of dependence between the two measurements is unspecified (as compared to the autocorrelation function), and thus, the mutual information function I(X,X + τ) is completely generic. For truly random signals, there is no deterministic relationship between successive measurements and knowing past history provides no advantage in predicting future behavior. As suggested by Figure 1, random signals show a complete loss of mutual information between measurements if the signal is delayed by one or more points. For truly periodic deterministic signals, on the other hand, there is a direct and fully predictable relationship between successive measurements. In such

cases, it is thus possible to predict future behavior over an unlimited time span. The mutual information function for a computer-generated sinusoidal signal with a frequency of 10 Hz is shown in Figure 1. It displays a high level of mutual information shared between measurements of the signal regardless of the time delay. It also implies that the average information transmitted between x(t) and x(t + τ) reaches maximum at the period and half-period of the signal (10 and 5 s, respectively, in this case). Deterministic chaotic signals fall between these two limiting types of behavior. They are deterministic and therefore fully predictable in principle, but because of exponential sensitivity to initial conditions, only short-term predictions are practically possible. The mutual information of signals obtained in fluidized beds has been applied to determine the appropriate time delay τ for reconstruction of the phase space by time delay embedding9 and to identify the periodicity and the predictability of the measured signals. Daw and Halow7 analyzed the pressure fluctuation signals from a bubbling fluidized bed and found that the measured data exhibited signature characteristics of deterministic chaos. Mutual information functions for slugging, bubbling, and turbulent regimes have been found to fall into bands lying between purely periodic and purely random patterns. Impetuous changes from one band to another often accompany transitions in flow regimes. They found that the behavior of the mutual information function conforms to the visual complexity of the various fluidization regimes; for example, more complex bubble and/or turbulence patterns have pressure drop signals with more rapid decay of mutual information. Using mutual information theory, Karamavruç et al.8 analyzed temperature and differential pressure data recorded from a horizontal heat-transfer tube in a cold bubbling bed and established the relative predictability of the signals recorded around the heat transfer tube, illustrating that local differential pressure signals are comparatively more unpredictable than local temperature signals. Karamavruç and Clark10 used mutual information function for preliminary analysis of the predictability

5756 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004

from data measured at different gas velocities. The information content decreased sharply in the first few seconds and then reached a plateau value around which the mutual information function revolved. The loss of information was more dramatic at lower gas velocities, suggesting a more chaotic solids flow pattern for those conditions. In the present work, differential pressure fluctuations, recorded at different operating and geometric conditions in spouted beds, are analyzed using the mutual information function. Experiment Details and Calculation Procedure

Figure 3. Schematic of pressure ports.

of the local differential pressure signals measured in a slugging bed and found a clear decrease in the level of the mutual information function with the increase in air flow rate. Pence et al.11 examined instantaneous heat-transfer signals acquired in the bubbling flow regime in a gas fluidized bed for evidence of deterministic chaotic behavior. They used the mutual information function to determine the most appropriate time delay for embedding the time series and providing qualitative information about the degree of chaos in the recorded signals. They found that the values of mutual information tend to be dependent on fluidization velocity. For the 715-µm glass bead particles investigated in their work, higher fluidization velocities resulted in higher mutual information. By comparing the work of Skrzycke et al.12 and Daw and Halow,7 they found that trends in mutual information of pressure drop signals might be a function of particle size and possibly also a function of particle density. Skrzycke et al.12 performed mutual information function analyses on pressure drop signals acquired by fluidizing 175-µm stainless steel particles with air and reported a trend opposite in nature to that of Daw and Halow,7 who fluidized 3175-µm nylon spheres with air. In their analysis of three-dimensional trajectory time series from a radioactive solid tracer freely moving in a conical-based spouted bed reactor, Cassanello et al.13 compared two mutual information functions obtained

The spouted bed in the present study was a plexiglass cylindrical column with a conical steel base. The cylindrical section is 0.12 m in inner diameter and 1.7 m in height. The conical base has an internal angle of 60°. Two sizes of gas orifices with diameters of 6 and 10 mm were used. The schematic diagram of the experimental setup is shown in Figure 2. Air, as spouting gas, was provided by an air compressor, controlled by a pressure regulator and measured by rotameters. Narrow sizedistributed glass beads and silica gel particles of different mean diameters were used as bed materials; their properties are listed in Table 1. Differential pressures were measured using rapid-response pressure transducers (Omega, PX164-010D5V). To prevent blockage by fine particles, each port was covered by a porous screen and purged frequently with air. Pressure signals were logged into a computer via an A/D converter with 12bit resolution. Three different sampling frequencies were used for each measurement: 200, 400, and 600 Hz; 65 536 data points were collected in each case. The measurements were taken at different locations, the arrangement of the pressure taps is shown in Figure 3. Total pressure drops of the spouted beds were taken from the wall (i.e., r ) R, where r is the radial distance between the tip of the pressure tube and the axis of the column and R is the radius of the column) of the section between points 1 and 4; differential pressures of the upper portions of the columns were measured between points 3 and 4, both from the wall; three measurements were taken for the lower portions of the cylindrical columns, all for the same height interval (between points 2 and 3) but with different radial positions: one with the tips of the pressure tubes inserted to the axis of the column (r ) 0), one inserted to r ) R/2, and the

Figure 4. Mutual information function by varying the number of bins.


Figure 5. Mutual information functions derived from total pressure drop signals at different sampling frequencies (Dc ) 120 mm, Di ) 10 mm, 1.6-mm glass beads, Hb ) 345 mm, Ug ) 1.01 m/s).

Figure 7. Mutual information functions for different flow regimes (a) and their power spectral densities (b) (1.66-mm silica gel, Dc ) 120 mm, Di ) 10 mm, Hb ) 345 mm).

Figure 6. Mutual information functions for different flow regimes (a) and their power spectral densities (b) (1.6-mm glass beads, Dc ) 120 mm, Di ) 10 mm, Hb ) 345 mm).

last was from the wall; differential pressures of the conical section were measured between points 1 and 2, from the wall. The mutual information function depends on the number of bins used to evaluate the probabilities. However, it is not obvious how to choose the right number of bins. Fraser and Swinney9 suggested a small

number of bins including more points in each so that an estimation of the average probability is more accurate. However, this causes an inaccurate estimate of the average joint probability, resulting in underestimation of mutual information. On the other hand, using a higher number of bins has the advantage that the joint probabilities P(xi,x*j) follow each other very closely. Due to the limited sample size, however, this interprets small fluctuations as small-scale structures in P(xi,x*j), causing overestimation of mutual information. Based on their experience, Daw and Halow7 suggested Nbin < (0.1N)0.5, where Nbins is the number of bins and N is the total number of data points. Although this rule of thumb was claimed to have been followed by Karamavruç et al.8,10,14 and Kang et al.,15 a much larger number of bins was used by them. The criterion “the number of bins should not be more than 10% of the total number of data” was followed instead. The mutual information functions for the pressure fluctuation signals obtained in the present study with different numbers of bins were given in Figure 4. With an increasing number of bins, although the level of mutual information increases overall, no shift of the maximum and minimum of the mutual information was found. The number of bins in the present work was determined based on Nbin < (0.1N)0.5. With 65 536 data point, Nbin is fixed at 80 for all the calculations. The procedure used to calculate mutual information function is from Hegger et al.16


Figure 8. Mutual information functions at different measurement locations for unstable spouting (a) and their power spectral densities (b) (1.66-mm silica gel, Dc ) 120 mm, Di ) 10 mm, Hb ) 345 mm).

Effect of Sampling Rate To resolve the detailed decay of mutual information at the beginning of the curve, Daw and Halow7 recommended (based on their experience) that the sampling rate be sufficient to observe a minimum of five points in the upper portion of the exponential decay region. Also, in their case, they found that the sampling rate had to be above 125 Hz to meet the resolution requirement. Mutual information functions for differential pressure signals recorded at different sampling frequencies of 200, 400, and 600 Hz in this study for the same operating condition are compared in Figure 5. It can be seen that, for all the sampling frequencies, more than five points were obtained in the upper portion of the exponential decay regions, suggesting that the sampling frequencies employed in this study are high enough to meet the resolution requirement. Comparison shows that the mutual information functions for 200 and 400 Hz are almost the same. The level of mutual information for 600 Hz is a little higher than for 200 and 400 Hz. However, the maximum and minimum of the mutual information functions do not shift. The higher level of mutual information at the higher frequency is reasonable, since from eqs 2 and 4, the mutual information depends on joint probabilities, and smaller time intervals cause a close link between the joint probabilities, so that higher mutual information is obtained. All data presented in the following discussions were measured at 600 Hz. Results and Discussion For a given combination of gas, solids, and vessel configuration, spouting occurs over a limited range of

Figure 9. Mutual information functions at different measurement locations for stable spouting (1.66-mm silica gel, Dc ) 120 mm, Di ) 10 mm, Hb ) 345 mm).

gas velocities. With the increase in inlet gas velocity, three flow patterns were observed in the present system: packed bed, stable spouting, and unstable spouting. The packed bed and stable spouting are distinguished by the formation of the stable spout/ fountain. Unstable spouting is characterized by swirling and pulsation of the spout with time. The mutual information functions of total pressure drops measured in the packed bed, stable spouting, and unstable spouting regimes are illustrated in Figures 6a and 7a, respectively. Figure 6a is for 1.6-mm glass beads, and Figure 7a is for 1.66-mm silica gel. The static


Figure 10. Mutual information functions for different regimes for a shallow bed (a) and their power spectral densities (b) (Dc ) 120 mm, Di ) 10 mm, 1.6-mm glass beads, Hb ) 175 mm).

bed heights for both are 345 mm. It can be seen that the mutual information functions of the various flow regimes appear to fall into bands between purely periodic and purely random patterns, demonstrating that chaotic (nonlinear) characteristics exists for all three flow regimes. In general, as the superficial gas velocity decreases, the level of mutual information becomes lower, and the decay speed of the mutual information around time zero becomes faster, indicating more chaotic (nonlinear) flow patterns at lower flow rates. This is consistent with the findings of Cassanello et al.,13 who analyzed three-dimensional trajectory time series from a radioactive solid tracer moving freely in a conical-based spouted bed reactor. For the packed bed flow regime at low superficial gas velocity, it can be seen that the mutual information functions decay rapidly to a value close to zero, which is quite similar with the case for the random signal shown in Figure 1. This suggests that the fluctuations in the packed bed regime are quite random. This is in agreement with the results of frequency analysis, which showed that power spectral densities lacked a specific dominant frequency (see Figures 6b and 7b; detailed discussion about the power spectral density in spouted bed was presented in ref 3). The data recorded under unstable spouting conditions exhibit a high level of mutual information and strong periodic motion. The

mutual information functions for unstable spouting show a gain in information for a fixed time lag as time evolves. The peak-to-peak distances approximate well the dominant frequency. For example, the time interval between two nearby peaks in the mutual information function diagram for unstable spouting in Figure 6a is ∼0.08 s, corresponding to a frequency of ∼6.3 Hz. This value is consistent with the power spectral density in Figure 6b. For stable spouting, the level of the mutual information functions is between those for a packed bed and for unstable spouting. For the particles studied in this work, i.e., glass beads and silica gel particles, it can be concluded that as the superficial gas velocity increases, the level of the mutual information, and thus the predictability, increase. Simultaneously recorded differential pressure signals at different locations have also been analyzed using the mutual information function. The results for the unstable spouting regime were illustrated in Figure 8a. These data were collected in a 120-mm-i.d. spouted bed with 1.66-mm silica gel particles at a superficial gas velocity of 0.485 m/s. In general, it can be seen that, for unstable spouting, mutual information functions of pressure signals measured at the wall have higher levels of mutual information than those measured in the interior of the bed. This indicates that pressure fluctuations measured at the column wall are qualitatively more predictable than inside the beds. It can also be seen that the mutual information for pressure signals obtained from the wall of the conical section has a lower level and a faster decay rate than at the cylinder wall (both upper and lower sections), suggesting that the gas-solid dynamics in the conical section is subject to greater uncertainty than near the column wall. The mutual information function for signals recorded in the spout region has the lowest information level and fastest decay rate. This seems to be reasonable since the gassolid dynamics in the spout region are expected to be more complex than in the annular region. The mutual information functions of pressure fluctuations at the upper and lower sections along the cylinder wall have approximately the same level of mutual information. This implies that the locations along the cylinder wall experience similar gas-solid dynamics. The peak-topeak distances for various measurement locations remain unchanged as shown is Figure 8a, suggesting that the same dominant frequencies can be obtained at all locations. This is consistent with the frequency analysis result (Figure 8b). A comparison of mutual information functions at various measurement locations for glass beads leads to similar results. Figure 9 plots mutual information functions obtained from pressure signals recorded simultaneously at various locations in the spouted bed operating in the stable spouting regime. The data were collected in the 120mm-i.d. column with 1.66-mm silica gel at a superficial gas velocity of 0.415 m/s. Trends similar to those in the unstable spouting regime discussed above were found, with the mutual information for signals measured on the wall generally greater and a lower decay rate than inside the spouted bed (in the spout region or inside the annular region). This indicates faster information loss inside the bed, suggesting more complex gas-solid behavior. Note that the mutual information functions for pressure signals measured in the spout region and inside the annular region decay rapidly after time zero and then level off around a fixed value, with no


Figure 11. Phase space construction of pressure signals (a), and pseudo-phase space construction of the pressure signal using τ ) 10 (b), τ ) 20 (c), and τ ) 40 (d).

Figure 12. Phase space construction of pressure signals (a) and pseudo-phase space construction of the pressure signal using τ ) 10 (b), τ ) 20 (c), and τ ) 40 (d).

information gain being found as time evolves. It can also be seen that no first minimum points exist in the mutual

information functions for signals measured in the spouted region and inside the annular region.


Figure 13. Phase space construction of pressure signals (a) and pseudo-phase space construction of the pressure signal using τ ) 1 (b), τ ) 9 (c), and τ ) 15 (d).

Figure 10a compares the mutual information functions of pressure signals measured in the spouting flow regime and in the packed bed regime for a shallow bed (Hb/Dc ) 1.45). In a shallow spouted bed, by increasing the gas velocity, two flow regimes, i.e., packed bed and stable spouting, can be achieved. The mutual information function for the packed bed regime has a lower level and a faster decay rate than that in the spouting regime, indicating more random motion in a packed bed. Comparing with the mutual information function obtained in a deeper spouted bed (Figure 6a), we see that the shallow bed shows no gain in information as time evolves. This is consistent with the frequency analysis result, in which a wide-band frequency distribution with no dominant frequency was found for shallow spouted beds (Figure 10b). In the chaotic analysis of experimentally measured time series, it is important to choose a proper time delay to reconstruct the phase space. If the time delay is too small, the trajectories lie near the diagonal of the embedding space. With the increase in time delay, the trajectories will expand from the diagonal so that they fill the complete m-dimensional phase space. Therefore, in the presence of experimental noise, the measured signal will be indistinguishable if a small time delay is used. On the other hand, a large time delay causes an uncorrelated time series in embedding space.8 Fraser and Swinney9 suggested that the first minimum of the mutual information function will provide the most qualitative time delay. However, it is difficult to determine the first minimum of the mutual information. There are some other cases where the first minimum of the mutual information function does not exist.7,11,12

To overcome this difficulty, Daw and Halow7 suggested that 10-50% of the maximum value of the mutual information be used to set the time delay. Packard et al.17 showed that the τ shifted version of measured signal x(t), i.e., x(t + τ), is related to dx(t)/dt. Therefore, both x(t + τ) and dx(t)/dt should have similar properties in phase plane. So the pseudophase space, two-dimensional phase portrait, constructed by plotting x(t + τ) against x(t), must be geometrically similar to two-dimensional phase space, the plots dx(t)/dt versus x(t). By comparing the geometric similarity between the phase space trajectories and the pseudophase space trajectories, an appropriate time delay τ can be determined. The present work found that the first minimum of the mutual information function, if it exists, can be used as the appropriate time delay. For instance, the mutual information function for the total pressure drop signal in Figure 8 reaches its first minimum at τ ) 20. Figure 11a illustrates the phase space trajectories by plotting signals versus its first time derivative, Panels b-d in Figure 11 show the pseudophase space using τ ) 10, 20, and 40, respectively. The trajectories of Figure 11c (τ ) 20) show more similarities to the phase space trajectories (Figure 11a) than both trajectories in Figure 11b and d. A smaller time delay (as τ ) 10 in Figure 11b) compresses the trajectories toward the diagonal of the embedding space. While larger time delay (as τ ) 40 in Figure 11b) compresses the trajectories toward the opposite diagonal. Figure 12 demonstrates the phase space trajectories and pseudophase space constructed by using τ ) 10, 20, and 40, respectively, for total pressure drop signals whose mutual information func-


tion was shown in Figure 9, where a first minimum at about τ ) 20 is found. Again, the pseudophase space trajectories constructed by the time delay determined from the first minimum of mutual information (Figure 12c) shows the most similarities to the phase space trajectories (Figure 12a). It is therefore concluded that if the first minimum of the mutual information function exists, it can give the appropriate time delay for the embedding. In the case that no first minimum exists in the mutual information function, this work found that 10-50% of the maximum value of the mutual information as suggested by Daw and Halow7 seems to be inapplicable to setting the time delay. For example, for the mutual information function in spouted bed in Figure 10, 1050% of the maximum value of the mutual information are corresponding to a time delay of τ ) 1-9. The pseudophase space trajectories constructed using τ ) 1 and 9 were illustrated in Figure 13b and c, respectively. It can be seen that the trajectories were compressed along the diagonal, suggesting the time delay used are too small. Pseudophase space trajectories using a larger time delay of τ ) 15 (Figure 13d) shows better similarity to the phase space (Figure 13a). Conclusion The results discussed above showed that different characteristics have been found for the mutual information functions obtained in the different flow regimes in the spouted beds. The packed bed flow regime has a low level and fast decay rate of mutual information, suggesting a quite random signal. The unstable spouting flow regime exhibits a high level of mutual information and strong periodic motion. While for the spouting regime, the level of mutual information and the decay rate around time zero are between those for the packed bed and unstable spouting regimes. The mutual information function for signals measured on the wall has higher predictabilities compared to those measured inside the bed. A comparison with the frequency analysis suggests that the mutual information function depends strongly on the frequency contents, as mentioned by Karamavruç et al.8 The mutual information functions of the flow regimes in the present study appeared to fall into bands between purely periodic and purely random patterns, indicating chaotic (nonlinear) characteristics for all three flow regimes. The first minimum of the mutual information function can be used as the optimum time delay if it does exist. In the case that no first minimum exists in the mutual information function, we follow the suggestion of Daw and Halow7 that 10-50% of the maximum value of the mutual information resulted in a too small time delay. An appropriate time delay can be determined by comparing the similarity between phase space trajectories and pseudophase space trajectories. Acknowledgment The authors are grateful to acknowledge the Ministry of Education of China for financial support through the Doctorate Discipline Foundation (Grant 2000042503). Nomenclature Dc ) column diameter, m Di ) orifice diameter, mm dp ) particle diameter, mm H(X) ) entropy associated with x(t), bits H(X*) ) entropy associated with x*(t), bits

H(X,X*) ) entropy associated with joint of x(t) and x*(t), bits Hb ) bed height, mm I(X,X*) ) average mutual information, bits N ) number of samples, [-] Nbins ) number of bins, [-] P(xi) ) probability associated with xi P(xi,x*j) ) probability associated with xi and x*j r ) radial distance between the tip of the pressure tube and the axis of the column, mm R ) radius of the column, mm Ug ) superficial gas velocity, m/s x(t) ) time series t ) delay time in integer number of sampling time intervals

Literature Cited (1) Epstein, N.; Grace, J. R. Spouting of particulate solids. In Handbook of Powder Science and Technology, 2nd ed; Fayed, M. E., Otten, L., Eds; Chapman & Hall: New York, 1997; Chapter 10, pp 532-567. (2) Wang, M.; Xu, J.; Wei, W.; Bao, X.; Bi, H. Pressure fluctuations in gas-solids spouted beds. In Fluidization X; Kwauk, M., Li, J., Yang, W. C., Eds.; Engineering Foundation: New York, 2001; pp 149-156. (3) Xu, J.; Bao, X.; Wei, W.; Bi, H.; Grace, J. R.; Lim, J. C. Statistical and frequency analysis of pressure fluctuations in spouted beds. Powder Technol. 2004, 141, 142-155. (4) Xu, J.; Bao, X.; Wei, W.; Bi, H.; Grace, J. R.; Lim, J. C. Characterization of gas spouted beds using the rescaled range analysis. Can. J. Chem. Eng. 2004, 82, 37-47. (5) Shannon, C. E.; Weaver, W. The Mathematical Theory of Communication; University of Illinois Press: Urbana, IL, 1949. (6) Williams, G. P. Chaos Theory Tamed; Joseph Henry Press: Washington, DC, 1997. (7) Daw, C. S.; Halow, J. S. Evaluation and control of fluidization quality through chaotic time series analysis of pressure-drop measurements. AIChE Symp. Ser. 1993, 89 (296), 103-122. (8) Karamavruç , A. I.; Clark, N. N.; Halow, J. S. Application of mutual information theory to fluid bed temperature and differential pressure signal analysis. Powder Technol. 1995, 84, 247-257. (9) Fraser, A. M.; Swinney, H. L. Independent coordinates for strange attractors from mutual information. Phys. Rev. A 1986, 33, 1134-1140. (10) Karamavruç , A. I.; Clark, N. N. Local differential pressure analysis in a slugging fluidized bed using deterministic chaos theory. Chem. Eng. Sci. 1997, 52, 357-370. (11) Pence, D. V.; Beasley, D. E.; Riester, J. B. Deterministic chaotic behavior of heat transfer in gas fluidized beds. J. Heat Transfer 1995, 117, 465-472. (12) Skrzycke, D. P.; Nguyen, K.; Daw, C. S. Characterization of the fluidization behavior of different solid types based on chaotic time series analysis of pressure signals, Proceedings of the 12th International Conference on Fluidized Bed Combustion; San Diego, 1993; pp 155-166. (13) Cassanello, M.; Larachi, F.; Legros, R.; Chaouki, J. Solids dynamics from experimental trajectory time-series of a single particle motion in gas-spouted beds. Chem. Eng. Sci. 1999, 54, 2545-2554. (14) Karamavruç , A. I.; Clark, N. N. A fractal approach for interpretation of local instantaneous temperature signals around horizontal heat transfer tube in a bubbling fluidized bed. Powder Technol. 1997, 90, 235-244. (15) Kang, Y.; Cho, Y. J.; Woo, K. J.; Kim, K. I.; Kim, S. D. Bubble properties and pressure fluctuations in pressurized bubble columns. Chem. Eng. Sci. 2000, 55, 411-419. (16) Hegger, R.; Kantz H.; Schreiber, T. Practical implementation of nonlinear time series methods: The TISEAN package. Chaos 1999, 9, 413-435. (17) Packard H. H.; Grutchfield, J. P.; Farmer, J. D.; Shaw R. S. Geometry of a time series. Phys. Rev. Lett. 1980, 45, 712-715.

Received for review March 30, 2004 Revised manuscript received May 30, 2004 Accepted June 9, 2004 IE049751Q

Mutual Information Functions of Differential Pressure Fluctuations in

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