Article pubs.acs.org/IECR
Random Walk Model for Biomass 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
‡
S Supporting Information *
ABSTRACT: We propose a physically motivated random walk model to describe the spatial and temporal mixing of a single biomass particle in a bubbling fluidized bed. The model parameters are estimated from measurements of a magnetically tagged simulated biomass particle in a laboratory fluidized bed. We demonstrate that Monte Carlo simulations using the model match key statistical features of the observed behavior reasonably well. These results suggest that a model of this type can simulate the effects of biomass particle mixing in biomass conversion reactors. We suggest possible improvements to the random walk model and propose how it might be used in conjunction with computational fluid dynamics simulations.
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INTRODUCTION AND BACKGROUND Bubbling bed fluidized reactors are widely used to pyrolyze and burn biomass particles for the production of biofuels and heat.1,2 Although it is now possible to computationally simulate many details of the key mixing, heat transfer, and reaction processes in these reactors,3,4 the computational overhead and time required make it extremely challenging to use such simulations routinely for rapid evaluations of reactor parametric trends and scaling effects. Rapid parametric and scaling evaluations are critical in the assessment of the potential economics and life-cycle impacts of competing technology options. Thus, we consider lower order modeling approaches that run very rapidly and complement more computationally intensive simulations. A traditional approach for quantifying particle mixing rates in fluidized beds has been to utilize particle dispersion coefficients.5,6 This approach has been successful in contexts where longer time scale mixing rates are important and/or where the particles of concern have properties similar to those in which they are dispersing (e.g., when recirculated bed particles are mixed back into the reactor). In the case of some biomass reactors, however, such as in those used for fast pyrolysis, the particles of interest are hydrodynamically very different from the majority of bed particles and have a strong tendency to segregate. This segregation tendency is difficult to accurately represent with dispersion coefficients alone. Also, the short time scale mixing, heatup, and reaction of individual biomass particles passing through the reactor can have a major impact on the chemistry and composition of the products. These effects are also difficult to represent with time-averaged dispersion models. Thus, it becomes important to have more explicit descriptions of particle motion that take these factors into account. Our objective in this study is to investigate a physically motivated approach for modeling the temporal and spatial statistics of the biomass particle mixing in bubbling bed reactors. We are specifically interested in reactors for which the relative number of biomass particles compared to primary bed © 2014 American Chemical Society
particles (e.g., sand or limestone) is small. We are also motivated to keep our model simple to make it computationally inexpensive to use for rapid process simulations. We validate our proposed modeling approach by comparing the predictions with previously reported experimental measurements of simulated biomass particle mixing in a laboratory fluidized bed.7 As demonstrated below, our low-order model appears to match the observed mixing behavior closely with very little computational overhead, and it suggests a path forward for developing other similar models that could be used in rapid process simulations.
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EXPERIMENTAL SETUP AND OBSERVATIONS Details of the experimental apparatus and analysis methods used to generate the simulated biomass particle mixing measurements discussed here have been published previously by the authors.7,8 Briefly, the measurements were made using a novel magnetic particle tracking system to follow the highspeed motion of simulated biomass particles in a 55 mm diameter laboratory bubbling bed of Geldart Group B glass particles operating under ambient temperature and pressure for a range of fluidizing gas flows. De Souza-Santos1 gives an explanation of the Geldart Groups. Individual simulated biomass particles of different sizes and densities were constructed by embedding tiny neodymium magnets in balsa or basswood particles. The main design and operating parameters of the bubbling bed experiment and simulated biomass particles are summarized in Table 1. In the following text, each of these particles is referred to as 1, 2, or 3, respectively. The magnetic particle tracking system had externally mounted sensors that made it possible to monitor the 3D spatial coordinates (horizontal and axial) of single simulated Received: Revised: Accepted: Published: 15836
April 1, 2014 September 6, 2014 September 22, 2014 September 22, 2014 dx.doi.org/10.1021/ie501343q | Ind. Eng. Chem. Res. 2014, 53, 15836−15844
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Table 1. Experimental Parameters of the Laboratory Fluidized Bed and Simulated Biomass Particles Used To Generate Particle Mixing Measurements bed dimensions distributor fluidizing gas bed solids
simulated biomass particles
55 mm i.d., glass cylinder porous polyethylene air at ambient temperature and pressure glass beads 207 μm weight mean diameter 2500 kg/m3 density (1) 4.5 mm D × 4.5 mm L cylinder (2) 4.8 mm × 4.8 mm × 6.2 mm slab (3) 5.5 mm sphere
85 mm static height 50 μm pore size 1.5−5.0 Umf Umf = 0.063m/s
760 kg/m3 density 620 kg/m3 density 410 kg/m3 density
biomass particles as they migrated through the bubbling bed under different fluidization conditions. Because the particle motion was sampled at very high rates (100 Hz) and continuously for long periods (5 min), it was possible to obtain very detailed statistics about the temporal and spatial mixing. This provided a rigorous basis for making quantitative comparisons with our model. The deconvoluted magnetic probe signals resolved individual particle trajectories as multidimensional time series, X(t), Y(t), and Z(t). Examples of the axial and horizontal time series are depicted in Figure 1 for particle 1 at 3Umf. We see here that the Figure 2. Short time scale autocorrelations for the time series in Figure 1.
Similar autocorrelation decay times were observed for the other particles and fluidizing conditions as well. These finite but short decay times imply that there is an inherent short-time memory in the particle migration before information about the past is erased by turbulence. Because this is similar to time scales over which significant changes can occur in biomass particles undergoing conversion (e.g., heatup and reaction times of biomass particles in fast pyrolysis), accurate mixing models for such processes need to replicate this feature. To globally resolve the overall spatial trajectories of the particles, the measured coordinates can also be plotted as in Figure 3. These plots map the spatial locations visited by particle 1 over a 5 min period at three different fluidization states. Although there were slight differences in some details among the different particles in Table 1, all three exhibited similar behaviors. Over long times, the instantaneous particle locations formed a cloud of points resembling an inverted cone, which was narrower near the bottom and spread outward near the top of the bed. Visual observations during our experiments confirmed that the outward spreading seen in the spatial trajectories appeared to be related to the bubble splash zone, where both biomass and bed particles were thrown outward toward the walls by erupting bubbles. The bubble eruptions also created a characteristic depression in the distribution cloud in the centerline at the bed surface, resulting in a kind of global cardioid shape. We targeted these global patterns as important features that we wanted to replicate with our model. It is also possible to quantify the global mixing rates of our simulated biomass particles in terms of dispersion coefficients using the mean-squared displacements per unit time in the axial and horizontal directions. Our estimates for the experimental conditions investigated here indicate global axial dispersion coefficients for the experimental simulated biomass particles between 1.2 × 10−4 m2/s near minimum fluidization (e.g., 1.5 times Umf) and 1.2 × 10−3 m2/s for well-fluidized conditions
Figure 1. Example measured time series of the axial and horizontal positions of simulated biomass particle 1 at 3.0Umf.
simulated biomass particle did not mix completely in the axial direction, but instead it hovered about a mean location above the bed midpoint (that is, above the point halfway between the bottom and top of the particle trajectory). This same type of behavior was exhibited by all of the particles over the entire range of fluidizing air flows investigated. This tendency to act like flotsam in bubbling beds has been widely observed for biomass and coal particles.5,9,10 We determined previously that the time-average axial distribution, which is upwardly skewed by segregation, can be approximated by a Weibull distribution.7 Rinne11 provides a detailed discussion of this distribution and of its many applications. On the other hand, horizontal mixing of our simulated biomass tended to be symmetric, with the time-average particle location on the centerline. Autocorrelations in the time series also reveal time scales for the particle mixing. Figure 2 illustrates that the autocorrelation functions of both the axial and horizontal cylindrical particle positions decayed rapidly over a time scale of approximately 1 s. 15837
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Figure 3. Example spatial distributions of simulated biomass particle 1 at three different fluidization states over a 5 min period. The top row is a top view (X−Y plane), and the bottom row is a side view (X−Z plane).
(e.g., 3−5 times Umf). For the radial direction, we estimate lower particle dispersion coefficients of (1.1−2.0) × 10−4 m2/s. These values are significantly lower than the dispersion coefficients reported by others for smaller particles under similar degrees of fluidization. Mostoufi and Chaouki report axial dispersion coefficients between 3.3 × 10−3 and 5.6 × 10−2 m2/s and radial coefficients between 2.6 × 10−4 and 1.5 × 10−3 m2/s based on radioactive tracking measurements of bed particles in group B bubbling beds.12 This implies that at least for larger biomass particles similar to those investigated here, the relative rates of dispersion are likely to be significantly less than for the smaller bed particles, which is perhaps not surprising given their much larger mass. Whereas relative dispersion rates can be a useful global indicator, they clearly cannot explain the above nonuniform features in the spatial particle distribution described above.
Axial Motion. Like Langevin, we assume that the particle motion of interest (in this case motion of biomass particles) can be described by an instantaneous force balance over each particle. We include in our balance a friction term, a steady force that accounts for segregation tendency, and a transient force due to turbulent interactions with bed particles and gas. In the axial direction this gives mp
dvpz dt
= −kzvpz + fsz + ftz
(1a)
Here mp is the particle mass, vpz is the axial particle velocity, kz is an axial coefficient of friction, fsz is the mean segregation force acting in the axial direction, and f tz is a fluctuating axial force due to fluid turbulence. The segregation force, fsz, arises because of the different size and density of the biomass particles relative to the bed particles, and it promotes axial migration of the biomass particle relative to the rest of the bed. The turbulent force results from transient interactions between the biomass particles and the surrounding gas and solids. These include both short-range collisions with “thermalized” bed particles and longer range interactions with rising bubbles. Thus, we are attempting to account for both long and short time scale stochastic processes that cause the particle to move relative to the bed. If we consider the particle motion over small but finite time steps, Δt, we assume that we can approximate the mean particle velocity during the next time interval as vpz ≅ (z(t + Δt) − z(t)/Δt). We also assume that the mean frictional dissipation in axial particle momentum during this interval would be −kz vpz Δt. Likewise, we can approximate the velocities at times t and t + Δt as vp(t) ≅ (z(t) − z(t − Δt)/Δt) and vp(t + Δt) ≅
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MODELING APPROACH We restrict ourselves here to steady-state fluidization conditions in which the biomass particles are relatively low in number compared to the bulk of the bed particles. Thus, the fluidization hydrodynamics are dominated by the properties of the bed particles and do not significantly change as the biomass particles move. We also assume that the biomass particles are hydrodynamically different from most of the bed particles and can freely move both vertically and horizontally relative to the mean position of the bed. Instead of adopting the traditional approach of dispersion coefficients for describing particle mixing,5 we begin by hypothesizing a modified version of the more fundamental particle dynamics model first proposed by Langevin for Brownian particle motion.13 15838
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(z(t + Δt − z(t) − z(t))/Δt), respectively. Considering eq 1a in momentum form, the mean impulses generated by the segregation and turbulence forces in this interval can be approximated by fszΔt and f̅ tz Δt, respectively. We designate the turbulent force here with an overbar to indicate that it is a short time scale average value resulting from potentially multiple small time scale fluctuations that are smaller than the observation time, Δt. We do not use an overbar for the segregation force because we assume that this force is effectively steady over the observation time as indicated above. Using the above assumptions, we can approximate the particle velocity change over the time interval in discretized form as
on the simulated particle versus the bed particles. In the model, the segregation parameter, c, has units of length and represents the expected incremental axial movement due to segregation. Its minimum absolute value is 0, but its sign can be positive or negative, depending on the direction in which segregation acts. In general, for biomass particles in dense particle beds, we expect c to be positive. The balance between dissipative and segregating forces also leads to establishment of an equilibrium axial location, Ze = c(1 − a − b), toward which the particle will tend to move in the absence of stochastic perturbations. Without stochastic inputs, repeated iterations of eq 2 always converge toward an equilibrium point, which must be below the maximum bed height as long as the particle remains in the bed. Once the expanded bed height, He, is known from measurements or correlations, it should be possible to establish an upper limit for c, because Ze < He and thus c < He(1 − a − b). We assume the turbulence term,sz(t), can be approximated as a stochastic process that follows some specific probability distribution. In this study, we assume that sz(t) is Gaussian distributed with a specified standard deviation (or amplitude). We explain below how we determined the strength of the Gaussian fluctuations below for our experiments, but physically one expects that this should be closely related to the intensity of bubbling in the bed. All of the above approximations are based on choosing a specific incremental observation time (Δt) for tracking the particle’s motion. We emphasize that it is important to choose appropriate time increments that are shorter than the autocorrelation time scales described above. Otherwise, important dynamical features are not resolved. For our experiments, this implies that Δt should be 2.5), the estimated parameter values changed little with air flow. Likewise, the different particles behaved similarly at high fluidization rates, with the slab-shaped particle (2) exhibiting the biggest differences. Combined Axial and Horizontal Motion. Because the convective term in our proposed horizontal random walk model (eq 4) includes a dependence on axial position, it is necessary to iterate both eqs 2 and 4 together to simulate horizontal particle motion. This complicates the horizontal parameter estimation process because it is not possible to independently determine horizontal motion apart from axial motion. An additional complication is that we did not know the convective stochastic function sr(t)g(Z) a priori. As a starting point, we conjectured that sr(t) is a uniformly distributed random number, and g(Z) = ∝Zβ. As for the axial direction, we assumed that sx(t) is Gaussian distributed with a constant specific standard deviation. To estimate the five horizontal parameters in eq 4 (e, f, α, β, and σx) for our experimental cases, it was necessary to resort to nonlinear regression, using the previously determined axial parameter values determined by linear least-squares as 15840
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change in mixing. The agreement between the random walk model and the experiment was better for well-fluidized conditions (i.e., for air flows >2.0Umf) for all three simulated biomass particles. The improved match between the model and experiment at higher fluidization rates should not be surprising because near minimum fluidization the bed was less homogeneous, and thus the assumption of uniform random walk parameters would not be as accurate. Overall, the predicted axial probability distributions and autocorrelation functions generated by the Monte Carlo simulations appeared to agree well with the experimental observations (i.e., within the inherent range of variability in the Monte Carlo process plus our estimates of the experimental measurement error; regarding the latter, we typically saw no more than a 5% variation in the average probability of finding the tracked particle at or below the bed midpoint; the time scale for reaching a 50% autocorrelation in particle position typically varied by no more than 0.1 s in repeated experiments). This suggests that models of this general form can indeed effectively replicate the general mixing behavior of particles like those studied in bubbling fluidized beds of the type employed here. It is important to emphasize that the random walk parameters used here were explicitly fitted to the available data. Thus, the present model is not fully predictive in the sense that it can be used to simulate mixing a priori for a given bed from the system and particle properties and operating conditions alone. However, we also repeat that the model formulation does provide important constraints on some of the parameter values. For example, the fitted values for parameters a−c in Table 2 fall within the expected ranges for the hypothetical model discussed above. However, it is also apparent that the relationship between the autoregression parameters a and b is somewhat different from that expected. Specifically, it appears that the first autoregression coefficient value is smaller than expected relative to the second (i.e., a < 1 − b). We conclude from this that some correction is probably needed in our original assumptions. We cannot say at this time what this correction might entail, but possibilities include allowing for spatial variations in the regression parameters or
explained above. This is a nontrivial task, but because of the large number of measurements available from our magnetic tracking experiment and the constraints on some of the parameters implied by our hypothesized physical model, we were able to quickly obtain approximate parameter estimates such as the example cases are listed in Table 3. Although these Table 3. Example Horizontal Regression Parameters for the Three Simulated Biomass Particles example case particle particle particle particle particle
1, 1, 1, 2, 3,
1.5 3.0 5.0 3.0 3.0
Umf Umf Umf Umf Umf
e
f
α
β
σx
1.20 1.00 1.00 1.00 1.00
−0.33 −0.22 −0.22 −0.22 −0.18
4.0e-7 6.0e-5 6.0e-5 8.5e-5 6.0e-5
6.0 4.0 4.0 4.0 4.0
0.12 0.32 0.32 0.28 0.18
estimates may not be fully optimized, they appear to do a reasonably good job of replicating the observed temporal and spatial correlations for the radial component of the particle motion as revealed below. Monte Carlo Simulations. We evaluated the predictive value of our model by carrying out Monte Carlo simulations of a particle’s motion over a 5 min period by iterating eqs 2 and 4 using the above parameter estimates. Typically, each simulation required only about 2 s to complete in Matlab on a desktop computer. Figures 4 and 5 compare the spatial and temporal behavior generated by the random walk model with our experimental observations for particle 1 over a range of air flow. Similar plots for the other particles are provided in the Supporting Information. As expected, each random walk simulation produces a slightly different profile, so we included the upper and lower limits of 30 repeated simulations of the random walk model as dashed curves in each figure to indicate this variability. As expected, axial and radial mixing increased and autocorrelation times in both directions decreased as fluidization intensity (air flow and bubbling level) increased. As air flow was increased to 3Umf or greater, there was little further
Figure 4. Time-average spatial distributions for particle 1 at three different fluidization levels. Dashed lines are upper and lower limits for random walks. Small squares are experimental. 15841
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Figure 5. Autocorrelation in particle 1 axial and radial locations at three fluidization levels. Dashed lines are upper and lower limits of random walks. Solid lines or small squares are experimental.
Figure 6. Comparison of the particle 1 experimental radial and axial locations at 0.1 s intervals versus those resulting from a single Monte Carlo realization of the random walk model at 1.5Umf.
Figure 7. Comparison of the particle 1 experimental radial and axial locations at 0.1 s intervals versus those resulting from a single Monte Carlo realization of the random walk model at 3Umf.
to modifying the effective value of the first autoregression coefficient. Nevertheless, whatever the cause, the general correlated random walk generated by eq 2 appeared to produce
corrections to the friction and/or segregation terms. For example, if the segregation tendency is not constant but varies linearly with location (e.g., c = c0 − c*Z), this could contribute 15842
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Figure 8. Comparison of the particle 1 experimental radial and axial locations at 0.1 s intervals versus those resulting from a single Monte Carlo realization of the random walk model at 5Umf.
ments similar to those utilized here, but with extended ranges of particle size, density, and fluidization hydrodynamics. The ultimate goal should be to confirm the robustness of the basic random walk approach and develop more direct physical correlations between the parameters in the governing equations and the particle properties and fluidization state. We recognize that in many cases there will be significant practical limits to experimental particle tracking measurements. For example, the magnetic tracking method exploited here is not likely to be feasible for very small simulated biomass particle sizes (e.g., particles
