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

Statistical Analysis of Axial Flotsam Particle Mixing in Bubbling Fluidized Beds C. Stuart Daw*,† and Jack Halow‡ †

Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States Separation Design Group, 931 Rolling Meadows Road, Waynesburg, Pennsylvania 15370, United States

‡

ABSTRACT: We present statistical analyses for highly resolved, short and long-time-scale experimental measurements of magnetic flotsam tracer particle motion in bubbling airfluidized beds of Geldart Group B particles. The observed axial tracer trajectories appear to exhibit Weibull distributions over both short and long time scales and suggest possible analogies with other complex dynamic processes. Comparisons with other recently published experimental and simulated flotsam mixing patterns reveal patterns consistent with our measurements, implying that the observed patterns may be general features of bubbling fluidized beds.

above incipient fluidization. To do this, we employed magnetic tracking, which is inherently limited by existing technology to smaller size beds but which also provides levels of measurement detail that are unobtainable (at least so far) with other methods.13−15 Our interest in this approach is to investigate the possibility that there might be dynamic mixing and segregation patterns present that can be used to better understand the fundamental physics behind particle motion in fluidized beds, to provide more accurate metrics for describing behavior and making comparisons with computational model predictions (which are typically constrained to smaller fluidized beds), and to stimulate the development of useful analogies that build on knowledge emerging from the study of other complex physical phenomena. In earlier published results for magnetic particle tracking in bubbling beds, we noted that the time-average probabilities of flotsam tracer particle axial locations in our experiments appeared to follow a Weibull distribution.14−16 We were also motivated to investigate this observation further by the following observations: • Short and long-time-scale Weibull distributions are explicitly generated by correlated random walk models of the type that we previously proposed to explain observed tracer particle motion.16 • In preliminary analyses of the experiments reported here, we observed that particle migration times (e.g., residence times, escape times) through different zones in the fluidized bed appeared to exhibit similarities to material

1. INTRODUCTION AND BACKGROUND Solids mixing and segregation in fluidized beds have been studied intensively for many years.1−8 Current efforts to develop advanced pyrolysis and gasification processes for converting biomass solids and municipal solid waste to fuel and chemical feedstocks are highlighting an ongoing need to improve solids mixing models in fluidized bed reactors. This is particularly true for pyrolysis and gasification of low density biomass and municipal solid waste in bubbling fluidized beds, which are typically composed primarily of denser solids such as sand and catalysts. The lower density carbonaceous solids characteristically tend to segregate from the bed solids, potentially affecting product yields and composition.9,10 In bubbling fluidized beds, the lower density particles tend to segregate upward toward the bed surface and are referred to in the literature as “flotsam”. Conversely, the downward moving bed particles are referred to as “jetsam”.1 Other recent studies of fuel particle segregation have addressed the causes and effects of segregation in industrialsize bubbling beds.11,12 For example, Bruni et al.11 used X-ray imaging to track the axial motion of volatilizing particles in incipiently fluidized beds, where the volatiles released by the particles contributed to their motion. Sette et al.,12 on the other hand, used optical tracking to measure the lateral motion of fuel particles. While both of these studies clearly addressed issues important for the design and operation of industrialscale combustion and gasifier units, the measurement methods and operating conditions employed were quite different from the focus of the present study. Specifically, we are concerned here with the fundamentals of both short and long-time-scale axial motion of flotsam particles designed to emulate nonvolatilizing char-like particles in lab-scale beds of heavy solids (sand) at a wide range of fluidization conditions well © XXXX American Chemical Society

Received: Revised: Accepted: Published: A

August 27, 2017 December 5, 2017 December 13, 2017 December 13, 2017 DOI: 10.1021/acs.iecr.7b03547 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research component ‘lifetimes’. Since the latter are known to be well represented by Weibull statistics, we decided that a more in-depth investigation of possible Weibull properties for tracer particle motion would be justified. • As is widely known, particle circulation in bubbling beds is driven largely by bubbles. A recent study by others17 (discussed below) has indicated that bubble sizes in fluidized beds follow a Tsallis distribution, which is closely related to the Weibull. In its simplest form, the Weibull distribution is usually represented by p(x) = (k /λ)(x /λ)k − 1exp[ − (x /λ)k ]

and granular systems,30,31 it seemed reasonable to us to consider this possibility. In the following sections, we describe our latest magnetic tracer experiments, which we designed specifically to resolve both short and long-time-scale details about axial flotsam particle motion. In addition to analyzing our own measurements, we also evaluated recent experimental and computationally simulated results published by others for char flotsam circulation. Based on these results, we suggest how the physics of flotsam mixing in fluidized beds may be fundamentally related to processes in other particulate systems.

2. EXPERIMENTAL MATERIALS AND METHODS The experimental measurements described here were obtained by tracking a single magnetically tagged flotsam particle as it circulated in a 55-mm-diameter air-fluidized bubbling bed of Geldart Group B glass beads.28 As summarized in Table 1, the

(1)

where p is the probability density for observing a quantity x ≥ 0 and p(x) = 0 for x < 0. The parameter k is a characteristic exponent (shape) factor, and λ is a characteristic scale factor (consistent with the units of x).18,19 For additional details on the impact of these two parameters on the Weibull probability density, the reader is advised to consult the cited statistical references.18,19 The cumulative (integrated) form of the Weibull distribution is given by C(x′) = 1 − exp( − (x′/λ)k )

Table 1. Experimental Parameters of the Laboratory Fluidized Bed and Tracer Particle bed 55 mm ID, glass cylinder dimensions distributor porous polyethylene fluidizing gas air at ambient temperature and pressure bed solids glass beads 142 μm (weight mean diameter) 2500 kg/m3 density

(2)

where C(x′) is the fraction of values for x ≤ x′.18,19 An important characteristic of eq 2 is that it scales as a power law with λ. That is, at any chosen value for λ (which can be thought of as a kind of magnification factor), the cumulative probability C(x*) for a fixed value of x*= x′/λ remains the same for constant k. Interestingly, this is also a defining property of fractal systems; that is, key statistical properties of such systems are preserved for all scales of magnification. In cases where power law relationships are found to hold, the statistics associated with features of interest are said to be selfsimilar; that is, one observes the same type of patterns repeatedly over a range of scales. As we discuss below, the presence of Weibull distributed patterns in the space and time characteristics of flotsam tracer trajectories appears to imply that these trajectories might be the result of self-similar processes. This could have fundamental importance because the Weibull distribution has been applied to a huge range of physical phenomena including particle comminution and fragmentation, multiphase percolation, weather forecasting, electrical networks, communication systems, disease spread, and hydrology networks.18−29 One reason for its broad applicability appears to be its relationship to power law processes, which appear to be almost universal features of nonlinear systems and networks. For example, Brown demonstrated that Weibull distributions arise naturally in particle grinding as a result of the fractal branching of intraparticle cracks, which can be described by a power-law mapping between larger particles and their smaller fractured products.27,28 Similar arguments have been made to explain the presence of Weibull and power law features in percolation networks in particulate media, where the presence of random transport pathways is associated with the onset of “critical” transitions.29 If Weibull statistics and power law scaling features can be confirmed as a major characteristic for particle motion in fluidized beds, physical models developed for other complex systems might have some relevance to fluidized bed analysis and modeling. Since power law and fractal scaling features have also been reported for complex mixing in fluids

tracer particle

5 mm basswood sphere with imbedded 1.6 mm diameter × 1.6 mm long cylindrical neodymium magnet

85 mm static height 50 μm pore size 1.7−3.0 Umf Umf = 0.029 m/s expanded bed height = 85−125 mm 1120 kg/m3 density

bed was operated at ambient conditions over a range of flows, and the properties of the tracer particle were adjusted such that it would behave similarly to dense biomass (e.g., pelletized wood). Bed expansion varied up to 50% for the highest velocity. In all cases in this study, we observed that the tracer particle acted as flotsam and thus tended to migrate toward the surface. Even though the minimum fluidizing and terminal velocities of the tracer particle are much larger than the bed material in this case, the tracer flotsam behavior is consistent with many other segregation studies in the past (see for example Rowe and Nienow2) in which large lower density particles (e.g., coal and char) have been observed to typically segregate toward the surface in beds of smaller, high density solids such as sand. As suggested by Rowe and Nienow, this phenomenon appears to involve fluid-like effects similar to buoyancy in bubbling beds, where the bulk density of the bed is greater than the particle density of the flotsam. We previously reported the magnetic particle tracking method used in these experiments in detail elsewhere.14−16 Some improvements have been made to the experimental set up to increase the spatial resolution available, including: addition of more rigid supports; increased shielding of the signal cables (to reduce noise), addition of a vertical pair of Helmholtz coils (to control the vertical ambient magnetic field),32 and further refinement of the algorithms used to map magnetic field variations to spatial coordinates. Improvements in the original algorithms involved compensating for geometric biases by adding a correction based on measured positions versus magnetic field strength and positioning the probes to minimize the effects. Figure 1 illustrates the principal components of the set up. B

DOI: 10.1021/acs.iecr.7b03547 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 1. Magnetic particle tracking apparatus.

We initiated each experimental run by dropping the magnetically tagged tracer into the already fluidized bed. Four externally mounted sensors were used to track the particle trajectory by variations in the magnet field. The magnetic signals were sampled at 200 Hz for 5-min periods and digitally recorded for post facto conversion into spatial coordinates vs time. This calculation is reasonably fast requiring about a minute computation time on a up to date standard desk or laptop. This resolved the tracer trajectory over 5 min into a sequence of 60 000 spatial positions at 0.005 s time steps. For purposes of this study we focused specifically on axial position in the bed, which involved an approximate maximum measurement error of ±0.25 mm with the current set up. This estimate of error is calculated from the observed fluctuations or noise in the magnetic field measurements revealed by repeated position measurements for a stationary tracer. For incremental position changes of the tracer position (determined by differencing sequential axial positions), this suggests a maximum error of twice the above or ±0.5 mm. As the sampling time was fixed electronically to be very precise, we estimate that our maximum error for measured tracer velocities was about 0.1 m/s. The actual error is typically much less than the maximum. As an example in Figure 3 below, the error is typically about half or this or less.

Figure 2. Axial position of the tracer (as measured relative to the distributor) in the fluidized bed at U = 1.7 Umf.

would rise quickly, sometimes going all the way to the bed surface. This is more clearly illustrated by examining a typical cycle shown in Figure 3 for the time period 38−43 s. The upper plot shows the vertical position versus time while the lower the velocity versus time. During the period between 38 and 41 s, the tracer in the present example moved slowly downward. Then between 41

3. RESULTS AND DISCUSSION 3.1. Axial Position Time Series. An example of the measured axial tracer position versus time is depicted in Figure 2 for the bed fluidized at 1.7 Umf. As can be seen, the tracer particle followed a looping, zigzag trajectory that brought it down into the bed to a finite depth above the distributor and then back toward the upper levels of the bed. The tendency of the tracer particle used in these experiments to spend most of the time in the upper regions of the bed is indicative of its flotsam properties. As depicted in the example time series, we deleted the first 20 s of measurements for each experiment to remove possible transients associated with introducing the tracer particle into the bed. In all our experiments, the tracer exhibited temporally correlated axial motion that included repeating sequences of relatively large orbits which were occasionally interrupted by shorter complex circulations. When following a large orbit, the tracer started at or near the bed surface (around 12 cm) and descended relatively deeply into the bed, at which point it

Figure 3. Details of tracer axial position and velocity over one transit cycle. C

DOI: 10.1021/acs.iecr.7b03547 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research and 42 s, it accelerated rapidly upward for a short interval, reaching about 25 cm/s before decelerating rapidly near the surface. This general pattern of relatively long, slow descents interrupted by short, upward accelerations appeared to be typical for all the conditions we studied, with the upward events becoming more numerous, rapid, and complex as the fluidization intensity increased. The above particle motions appear to be consistent with the physical circulation mechanisms described by Geldart,33 who suggested that: • While moving downward, particles in bubbling beds tend to move slowly and steadily in response to the mean solids circulation that accompanies the upward rising bubbles. • Particles move upward by either (a) becoming entrained in bubble wakes and rising rapidly with them or (b) rising more slowly in response to “drift” effects produced by the presence of transient higher voidage regions (trails) left behind nearby passing bubbles. When particles rise in close contact with bubble wakes, their velocities should approach that of bubbles. In the present case, bubbles were estimated from slow motion video of the bed surface to be from about 1 to 2 cm in diameter. Bubble velocities for these diameter (using the standard Davidson and Harrison relationship33) are approximately 20−30 cm/s. The 25 cm/s velocity peak in Figure 3 is in this range. Likewise, we interpret the periods of lower rise velocity (e.g., as the tracer approaches the bed surface) as corresponding to moments when drift forces are dominant. 3.2. Longer Term Statistical Patterns. To evaluate longer term (time-average) statistical trends, we analyzed the axial distributions of tracer position during each entire 5-min experiment. Because the upward and downward motions are known to have different physical mechanisms, we consider these two motions separately. Figure 4a−c illustrates the probability density distributions for the incremental downward flotsam tracer particle velocities during each time interval (i.e., Δzd/Δt) at three different fluidization air flows. The observed probability densities (indicated by crosses in the plot) are consistent with a Weibull distribution (indicated by the solid line) and might imply that the tracer motion might have similarities to particle fragmentation as proposed by Brown and Wohletz.27,28 Specifically, the argument is made that the Weibull distribution can arise naturally in physical processes where there is a repeated power law mapping of an original length and/or time scale into shorter length and/or time scales. When this mapping involves a nonintegral power (such as in particle comminution where crack propagation occurs), the resulting distribution of particle fragments is distributed according to a power law, which may have fractal properties. This suggests to us that there might be a similar conceptual model for repeated disturbances of the tracer particle trajectory under the influence of the surrounding bed particles such that the tracer particle trajectory is ‘diverted’ repeatedly to produce modified trajectory segments and correspondingly modified residence times in each vertical region of the bed. In simple terms, we suggest that the accumulated downward steps taken by the tracer reflect the combined effects of bulk particle circulation and perturbations due to passing bubbles. In subsequent equations, the term Δzd1 refers to a characteristic incremental axial length scale used in defining the power law mapping.

Figure 4. Probability density distributions of incremental downward velocities for the flotsam tracer particle at a fluidizing air flow corresponding to (a) U/Umf = 1.7, (b) U/Umf = 2.1, and (c) U/Umf= 2.5. Crosses indicate experimental measurements, and the solid line is a least-squared best fit for a Weibull distribution (eq 3c) with the indicated parameter values. Maximum estimated error limits in values for vd1 and a are ±16% and ±7%, respectively.

Given the above analogy, we consider the possibility that incremental downward tracer movements could diverge (due to the action of passing bubbles) into a population of perturbed movements whose axial component is distributed according to ∞

p(Δzd) =

∫Δz

p(Δz˜ d)f (Δz′d → Δzd) dΔz′d

(3a)

where p(Δzd) is the probability of observing an incremental tracer movement between Δzd and Δzd + dΔzd). The function f(Δz′d → Δzd) = (Δzd/Δzd1)a represents a probability density mapping (with units of 1/m such that the units of the product of the two right-hand terms, 1/m multiplied by m, has units of differential probability). The resulting integral units are thus equivalent to the units of the left-hand side. Over long times, a multitude of downward steps of size Δzd (with units of m) are associated with diversions of unperturbed downward steps of size Δz′d. The integrated result is a set of modified trajectories with characteristics similar to that of a branched tree or dendritic network, where the observed sequences of incremental downward steps are the axial components of the repeated diversions (or branchings) of the trajectory. The reader is referred to Brown and Wohletz27,28 for more details about the derivation of the analogous equation for D

DOI: 10.1021/acs.iecr.7b03547 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research comminution, which eq 3a mirrors in terms of incremental tracer steps instead of particle fragments. In the present case, we consider the mapping function to represent the probability density of any particular axial component of a trajectory branch rather than the probability density of particle fragments from grinding. To complete the picture, the parameter Δzd1 represents a characteristic length scale for downward (axial) motion (corresponding to a characteristic downward velocity vd1 associated with the global solids circulation), and a represents a characteristic scaling exponent. We apply similar reasoning when considering downward tracer velocities (vd) instead of steps in position. Thus, in terms of incremental downward velocity (Δzd/Δt) during the measurement time interval, eq 3a becomes ∞

p(vd) =

∫Δz

p(v′d )f (v′d → vd) dv′d

(3b)

Following an argument similar to Brown, the probability density for such a process becomes p(vd) = (a /vd1)(vd /vd1)a − 1exp[ − (vd /vd1)a ]

(3c)

Similar plots for upward incremental tracer motion are depicted in Figure 5a−c. Note that the characteristic shapes are somewhat different, but the probabilities for upward tracer movements are also described by a Weibull distribution with different parameter values. In an analogous matter, the upward motion can be described by p(Δz u) = (b/Δz u1)(Δz u/Δz u1)b − 1exp [−(Δz u/Δz u1)b ] (4a)

where p(Δzu) is the probability of an incremental upward tracer movement between Δzu and Δzu + dΔzu. Similar to eq 3a, the variable Δzu is the upward moving analog of the downward incremental step Δzd, and Δzu1 represents a characteristic length scale for upward motion (corresponding to a characteristic upward velocity vu1). The exponent b reflects a scaling process for the upward motion. In terms of incremental upward axial velocity (Δzu/Δt) evaluated during the measurement time interval, eq 4a becomes p(vu) = (b/vu1)(vu /vu1)b − 1exp [−(vu /vu1)b ]

Figure 5. Probability density distributions of incremental upward velocities for the flotsam tracer particle at a fluidizing air flow corresponding to (a) U/Umf = 1.7, (b) U/Umf = 2.1, and (c) U/Umf = 2.5. Crosses indicate experimental measurements, and the solid line is a least-squared best fit for a Weibull distribution (eq 4b) with the indicated parameter values. Maximum estimated error limits in values for vu1 and b are ±13% and ±10%, respectively.

(4b)

where p(vu) is the probability of observing an incremental velocity between vu and vu + dvu. Establishing precise confidence limits for the fitted parameter estimates is problematic in this context because of the highly nonlinear, non-Gaussian character of the distributions and the temporal correlations between the measurements (which make it impossible to precisely determine the appropriate number of degrees of freedom involved). Nevertheless, we were able to estimate uncertainties in the parameter values by comparing variations in the least-squares estimates over time. Briefly, we did this by making an initial least-squares estimate of the distribution parameters using all 5 min from each experimental time series. We then separately made leastsquares parameter estimates for the front and back halves of each data set (each consisting of 2.5 min of data). The differences among the least-squares fitted parameter values resulting from these three different fits provided the uncertainty estimates listed in each figure caption. Nonintegral values for the exponents a and b in the above examples imply that the observed tracer particle motions may have exhibited fractal-like behavior; that is, the axial move-

ments appeared to exhibit some degree of clustering rather than smoothly filling the entire range. Table 2 summarizes additional time-average statistics for the incremental tracer motions corresponding to Figures 4 and 5 that reveal more about differences between up and down motion. Specifically, the frequency and mean step sizes for upward and downward motion are distinct, which is consistent with previous studies of flotsam mixing.1−4 Another unique characteristic of experiTable 2. Additional Statistics for Incremental Axial Tracer Motion Corresponding to the Experimental Conditions in Figures 4 and 5a U/Umf

Δz u (m)

Pu

Δzd (m)

Pd

P0

1.7 2.1 2.5

0.016 0.031 0.032

0.405 0.399 0.400

−0.011 −0.021 −0.022

0.595 0.592 0.598

0 0.009 0.002

a

Δz u and Δzd are the mean axial increments for upward and downward motion, respectively. Pu, Pd, and P0 are the probabilities of the tracer moving up, moving down, or remaining fixed, respectively. E

DOI: 10.1021/acs.iecr.7b03547 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research ments such as these is that the net axial motion of the tracer particle over long intervals of time should approach zero, since the particle is constrained in these experiments to never leave the bed. This is reflected in Table 2, where we see that upward particle motions are characteristically larger but less frequent than downward particle motions. There is also a finite probability for no motion at all, at least within our ability to detect it. When both upward and downward average motions are considered along with their different probabilities, we find that the net translation over the experimental time interval for each case is effectively zero, confirming that the time-average axial tracer behavior is indeed stationary. Weibull patterns in the incremental particle motion suggested to us that the integral tracer trajectories might exhibit similar properties, since each axial position represents the accumulated effects of many previous incremental steps. To investigate this we evaluated the cumulative distributions of the time-average axial positions of the tracer particle as well as the time intervals spanned by the tracer orbits to see if Weibull features might be present. Since the downward incremental motion of the tracer reflects processes that counter the natural tendency for flotsam to segregate toward the bed surface (and thus are a kind of mixing/segregation metric), we found it useful to quantify axial position in terms of “penetration depth”, that is, the relative distance downward that the flotsam tracer had moved below its maximum height. Figure 6 illustrates cumulative distributions for instantaneous penetration depth for the same three conditions corresponding to Figures 4 and 5 and Table 2. Not surprisingly, the cumulative distributions of the tracer penetration appear to be consistent with cumulative Weibull distributions, subject to shifts in the scaling and shape parameters as the fluidization intensity (fluidization air flow) is changed. In this case, the scaling parameter reflects a characteristic penetration depth, and the exponent reflects the tendency for the tracer to visit lower regions of the bed. The main deviation between the observed and cumulative Weibull distribution profiles appears to be at large penetration depths (near the distributor), where distributor boundary effects are present. Since deep penetration events are rare, the observed frequencies involve higher statistical uncertainty, which leads to larger errors in time-average estimates. To evaluate the distribution of circulation time scales for the tracer particle, we also computed cumulative distributions for the time intervals between crossings of the mean axial position visited by the tracer in that experiment (also referred to as mean return times for each tracer axial position time series) as depicted in Figure 7. As with the integral penetration locations, these return times, which define a type of characteristic time scale for axial tracer orbits, appear to be Weibull distributed for each 5-min experimental run. The time intervals for both up and down crossings (that is, the time intervals for successive upward transits of the mean axial location vs time intervals for successive downward transits of the mean axial location) were tracked separately and are depicted with different symbols in the plot. As before, precise determination of confidence limits for the parameter estimates is challenging, so we used the differences in the estimated Weibull parameters for each type of event (up vs down crossing) to estimate the maximum uncertainties in the Weibull parameter values listed in the figure caption. 3.3. Comparisons with Other Studies. Considering the above patterns in axial tracer velocity, it appears that others

Figure 6. Cumulative distributions of instantaneous axial tracer position (measured as penetration below the bed surface) at fluidizing air flows corresponding to (a) U/Umf = 1.7, (b) U/Umf = 2.1, and (c) U/Umf = 2.5. The solid red line indicates experimental measurements, and the dashed blue solid line is a Weibull distribution fit for the indicated parameter values. Maximum estimated error limits in values for λp and kp are ±10% and ±6%, respectively.

have seen similar trends in a different laboratory-scale experimental fluidized bed, implying that the general mixing patterns described above may not be highly dependent on a specific experimental setup.13 To further investigate the possibility that the flotsam profile patterns we observed were not specific to just our experiments, we also investigated timeaverage axial flotsam mixing profiles reported by two other groups in the recent literature.7,8 The former study by Park et al.7 was an experimental investigation of flotsam char mixing in a bubbling bed of Group B sand particles, which we expect to have some reasonable similarity with our experiments. In contrast, the latter study by Zhao et al.8 was an attempt to simulate the Park et al.7 experiment with a two-fluid computational fluid dynamic (CFD) model. This provided a way to assess whether the model physics replicate the flotsam profile patterns in both our experiments and those of Park et al.7 F

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Figure 8. Cumulative distribution of the experimental time-average penetration depth for char flotsam in a sand fluidized bed reported by Park et al. (2013) (U/Umf = 1.35). Experimental measurements are indicated by points and the least-squared cumulative Weibull fit by solid lines. Listed parameter values yield the minimum least-squared deviation from a cumulative Weibull distribution. For the fitted results, R2 > 0.99.

Figure 9. Cumulative distribution of the computationally simulated time-average penetration depths for char flotsam in the Park et al sand fluidized bed reported by Zhao et al. (2017) (U/Umf = 1.35). Predicted time-average measurements are indicated by points and the least-squared cumulative Weibull fit by solid lines. The listed parameter values yield the minimum least-squared deviation from a cumulative Weibull distribution. For the fitted results, R2 > 0.99.

Figure 7. Cumulative distribution of tracer return times at fluidizing air flows corresponding to (a) U/Umf = 1.7, (b) U/Umf = 2.1, and (c) U/Umf = 2.5. The points are experimental measurements, and the blue solid line is a Weibull distribution fit for the indicated parameter values. Maximum estimated error limits in the values for λT and kT are ±6% and ±18%, respectively.

3.4. Diverted Trajectory Hypothesis. The statistical patterns in both the incremental and integral axial motion of flotsam particles in this study appear to support the idea that particle motion in bubbling fluidized beds is controlled by power law or fractal mixing processes that naturally lead to Weibull distributions in space and time. Bubbles are likely candidates for the source of such mappings in bubbling fluidized beds, since they tend to be the major driver of particle motion. The connections to Weibull statistics would also seem to be supported by the experimental findings of Gheorghiu et al.,18 which imply that bubble size distributions may follow Tsallis distributions, which are closely related to Weibull distributions.34,35 Based on the above arguments, we propose the conceptual mechanism for flotsam particle mixing depicted in Figure 10, which we refer to as the diverted trajectory hypothesis. We begin by noting that the global solids circulation associated with rising bubbles tends to draw all particles (flotsam and jetsam included) toward the bottom of the bed until they are carried upward again by wake or drift events created by rising bubbles.33 Thus in the absence of any random variations in

As illustrated in Figures 8 and 9, cumulative distributions of the axial penetration for flotsam char in both these prior studies appear to have Weibull distributions. Unfortunately, we are constrained by the resolution of the measurements reported by these investigators, but as with our experiments, the goodness of fit levels for both the experimental and simulated measurements are >0.99 with only two adjustable parameters. It is also not possible to develop a reliable estimate of parameter uncertainties in these studies from the available information. Although the characteristic parameter values for the experiment and simulation are somewhat different, the general patterns appear to be similar. While not conclusive, these results suggest that the patterns we observed for flotsam mixing are not just unique to our experiments but have more general significance. It also appears that the computational simulations of Zhao et al.8 effectively include the physics necessary to produce the Weibull patterns seen here. G

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4. CONCLUSIONS AND RECOMMENDATIONS Based on the experimental measurements of the axial motion of a magnetically tagged flotsam particle in a lab-scale bubbling fluidized bed with air flows well above incipient fluidization and comparisons with recent literature that are summarized above, we conclude the following: • Both incremental and integral flotsam particle axial motion appear to be consistent with two-parameter Weibull statistics. • The observed Weibull parameters reflect characteristic space or time scales and exponents that quantify the diversity and complexity in particle trajectories. • The good agreement with Weibull statistics suggests that the accumulated set of circulation trajectories followed by individual particles may have an underlying fractal structure, which may have important similarities to other complex processes such as fragmentation. Considering the above, we recommend the following future studies: • The connection between correlated random walks in bubbling fluidized beds and the tracer behavior reported here. • Identification of characteristic scaling factors and exponents based on the statistics reported here for verifying and validating physical models and computational simulations of bubbling fluidized beds. • Measurement of the impact of flotsam population size (i.e., the impact of flotsam particle concentration in the bed) on the statistics of flotsam particle motion. • Investigation of the axial mixing of flotsam particles in larger-scale fluidized beds operating well above incipient fluidization where large numbers of bubbles are present. • Investigation of the statistical distribution of horizontal and 3-dimensional fluidized bed tracer particle trajectories.

Figure 10. Conceptual representation of the hypothesized diverted trajectory mechanism for flotsam particle circulation statistics. Particle motion is repeatedly diverted by rising bubbles according to power law mapping probabilities. This action produces a complex subset of actual trajectories with reduced penetration depths and return times. Over extended periods, Weibull spatial and temporal distributions result.

global circulation, all particles should tend to cycle over most of the height of the bed from top to bottom. This tendency is reflected in the persistent axial circulation (orbits) in the flotsam trajectory depicted in Figures 2 and 3 above. The rising bubble population in the bed exhibits complex variations over time and space, so that particle trajectories are randomly perturbed and diverted from their ideal global orbits. Because of their different hydrodynamic properties, flotsam and jetsam particles are perturbed differently, resulting in the tendency for axial particle segregation. We propose that the accumulated effect of repeated trajectory diversions over sufficient intervals produces the observed Weibull statistics in axial flotsam distributions. Because of the similarity in the statistical patterns between flotsam mixing and particle grinding, we also suggest that the perturbations to solids circulation in bubbling beds may be consistent with some type of power law process. If the probabilities for particle trajectory diversions follow power laws with nonintegral exponents, one would expect the subset of spawned trajectories to have similar statistical features to those reported here. In this regard, there may be important similarities to grinding, in which fractal crack propagation controls the creation of small particles from larger particles.27,28 To carry this idea forward, one might think of the flotsam particle motion in fluidized beds as being modified by the action of bubbles to produce networks of complex paths with fractal-like properties. The axial components of such paths would be represented by the time series tracer measurements described here. This scenario suggests a potential connection to the physics of other network-based systems such as percolating granular media and distributed communication networks.29 If these connections can be confirmed, it would also imply that fluidized beds should exhibit other characteristic features of power law networks, such as self-organized critical transitions.36

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AUTHOR INFORMATION

Corresponding Author

*Tel.: 865-946-1341. E-mail: [email protected]. ORCID

C. Stuart Daw: 0000-0002-1656-8720 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was partially supported by the Bioenergy Technologies Office of the U.S. Department of Energy under Contract No. DE-AC0500OR22725 with the U.S. Department of Energy. The United States government retains and the publisher, by accepting the article for publication, acknowledges that the United States government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States government purposes. The U.S. Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doepublic-access-plan). H

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SYMBOLS a,b = characteristic Weibull exponents for downward and upward incremental motion, respectively (−) C(x′) = cumulative probability of observing 0 < x ≤ x′ f(Δz′d → Δzd) = mapping function defining the probability density that any incremental downward motion (Δz′d) will be diverted into a different incremental downward motion (Δzd) k = general Weibull characteristic exponent (−) kp = Weibull characteristic exponent for axial penetration (−) kT = Weibull characteristic exponent for mean crossing time (−) R2 = correlation coefficient (−) U = superficial gas (fluidizing) velocity (m/s) Umf = minimum superficial fluidizing velocity (m/s) Pd = probability of incremental downward tracer motion (−) Pu = probability of incremental upward tracer motion (−) P0 = probability of no incremental axial tracer motion (−) p(x) = probability density function x = independent measurement variable Δt = measurement (incremental) time interval (s) vd1 = characteristic downward incremental velocity (m/s) vu1 = characteristic upward incremental velocity (m/s) Δz = incremental change in axial position (cm) Δzd = average incremental downward axial tracer motion (cm) Δz u = average incremental upward axial tracer motion (cm) λ = general characteristic Weibull scale factor (same units as x) λp = characteristic Weibull scale factor for axial penetration (cm or m) λT = characteristic Weibull scale factor for mean crossing time (s or measurement increments)

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