Experimental study of the interaction of a dense gas–solid fluidized bed with its air-plenum

Powder Technology 202 (2010) 118–129

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Experimental study of the interaction of a dense gas–solid fluidized bed with its air-plenum F. Bonniol, C. Sierra ⁎, R. Occelli, L. Tadrist Laboratoire de l'IUSTI (UMR CNRS 6595) - Université de Provence, Technopôle de Château Gombert, 5, rue Enrico Fermi, 13453 Marseille Cedex 13, France

a r t i c l e

i n f o

Article history: Received 10 July 2009 Received in revised form 19 February 2010 Accepted 20 April 2010 Available online 20 May 2010 Keywords: Gas–solid fluidized bed Plenum Air-supply system Hilbert–Huang Transform Pressure fluctuations

a b s t r a c t The influence of the air-plenum on the bed dynamics that can occur for low pressure drop distributors is studied from an experimental point of view and compared with different models from the literature. These experiments are done for a wide range of conditions: A, B and D Geldart particles, different inlet flow velocities and plenum volumes. The effects of the plenum on the bed dynamics are investigated by means of pressure measurements analyzed with an original and accurate time–frequency method, the Hilbert–Huang Transform. The present work clarifies former observations of the literature and specifies the action of the sole plenum, independently of the rest of the air-supply system. © 2010 Elsevier B.V. All rights reserved.

1. Introduction In gas–solid fluidized beds design, the air distributor is known to have a major influence on the bed two-phase dynamics. Its main role is to ensure a uniform injection of the gas at the bottom of the bed in order to achieve a homogeneous fluidization (see e.g. [1,2]). In order to fulfill this function correctly, Kunii et al. [3] propose from empirical considerations that the pressure drop ratio between the distributor and the bed should be between 0.2 and 0.4. Industrial applications usually respect this criterion as a minimum rule in order to limit purchase and consumption fan costs. However, in this range, Svensson et al. [4] observed a dramatic change of the bed dynamics due to the interaction between the bed and the plenum. Indeed, for rather small pressure drop distributors, pressure waves can propagate into the plenum, the free volume under the bed, and induce a feedback interaction that may alter the intrinsic bed dynamics. Svensson et al. explained in their study that for high pressure drop distributors (with Geldart B bed particles), the bed shows a “multiple bubble regime” which is nothing but the classical behavior of a free bubbling bed. However, for sufficiently low pressure drop distributors, they observed a significant correlation between pressure signals at the bottom of the bed and in the air-plenum. This latter observation was already reported by Lirag and Littman [5]. In this case, the pressure signals show large amplitudes and regular oscillations that characterize the so called “single bubble regime” where a unique bubble erupts at the bed free surface in a very regular way. The authors

⁎ Corresponding author. E-mail address: [email protected] (C. Sierra). 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.04.026

attributed this behavior change to an interaction between the bed and the plenum. From the experimental point of view there are few and rather old works that focused on the plenum influence. The first who studied the plenum contribution were Baird and Klein [6] and Kage et al. [7]. They looked at the bed frequency and compared it with Davidson's [8] spring-mass model for different plenum volumes. They observed that the major bed oscillation frequencies decrease with the plenum volume and, in the same time, the bed pressure drop signal shows more regular and high amplitude fluctuations. Vakhshouri and Grace [9] recently studied the formation of bubbles from a single orifice submerged in a gas fluidized bed taking into account the influence of the plenum. They found that large plena promote the formation of large bubbles and lower the detachment frequency. These results are similar to those reported for gas–liquid systems and are consistent with previously cited works on gas fluidized beds. However for free bubbling beds, to our knowledge, the most recent experimental work about the effect of bed pressure fluctuations that may propagate to the entire air-feed system is from Johnsson et al. [10]. They concluded that transient effects on the air flow arising from the plenum and the pipings strongly influence the bubbling behavior of the bed so that the whole air-feed system should be taken into account for future numerical simulations. However in their work, the influence of sole plenum is not studied systematically and few quantitative information are available on its specific role on the bed dynamics. Consequently, the present article focuses on its distinctive effect with an original experimental approach. This work encompasses and prolongs the former cited studies as it aims at isolating the action of the plenum separately from other elements of the air-feed system (pipes, valves, fan type…). The experimental analysis proposed

F. Bonniol et al. / Powder Technology 202 (2010) 118–129

hereunder relies on an original time–frequency method, the Hilbert– Huang Transform (HHT), first used by Briongos et al. [11] that studied the different spatial scales involved in the dynamics of bubbling fluidized beds. The impact of the plenum is investigated through the variation of its volume and the consequent bed dynamics is analyzed by means of the HHT applied on recorded pressure signals. The study is conducted for a wide range of particles (Geldart class A, B and D) and also for various static bed heights and mean fluid inlet velocities. Experimental measurements are also compared, in terms of characteristic frequencies and level of interaction between the bed and the plenum, with theoretical predictions from the model of Bonniol et al. [12]. 2. Theory Bonniol et al. [12] carried out a theoretical analysis1 about similitude parameters for a dense gas–solid fluidized bed in potential interaction with the air-supply system (limited to the distributor and the plenum). In addition to the classical set of dimensionless numbers controlling the sole bed dynamics (e.g. [13–15]), the influence of the distributor and the plenum respectively leads to the definition of two new numbers θp and θd. The extended set writes, Re; Ar; De; ψ; θp ; θd with θd =

θp =

2ρg hξd ihud i2 hΔpd i =4 hΔpb i hΔpb i ′

2γ〈pp 〉Σhud i g hcp hΔpb iΩ

ð1aÞ

−1

the plenum–distributor association does not need to induce large fluid velocity fluctuations to accommodate the pressure changes imposed by the bed. Asymptotically, it's shown that when θp → ∞, fluid velocity fluctuations amplitude at the bed entrance tend to zero. On the contrary, when τb ≪ τpd i.e. for small θp values (obtained in our experiments for large plenum volumes) it means that the plenum cannot adapt rapidly to pressure change imposed by the bed. In this case, it takes time for the plenum to fill-up (resp. to empty) when its pressure is increased (resp. decreased) faster than τpd allows it. This forces the plenum to generate large amplitude of fluid velocity fluctuations at the bed entrance. For intermediate values of θp, the amplitude of fluid velocity fluctuations and their phase lag with the bed pressure oscillations continuously vary with its value. By means of a linear stability analysis of the differential equations governing the bed, the distributor and the plenum dynamics, Bonniol et al. investigated the evolution of the amplification rate of perturbations (say A⁎) and the dominant frequency of the bed f with θp and θd. Gray-level maps in Fig. 1 illustrate the existence of a zone where A⁎ N 0 (with a maximum for a given θp = θ+ p ∼ 5 in this example, see Fig. 1c) signature of a coupled dynamics accompanied with a frequency modification (Fig. 1b). Outside this coupled zone A⁎ → 0 so that the bed is not coupled with the distributor and the plenum and undergoes its own intrinsic dynamics at a constant oscillation frequency. The uncoupling for large (resp. low) θp values roughly occurs for θp ∼ 100 (resp. θp ∼ 0.01). This transition also appears for 0.1 b θd b 10. 3. Experiments 3.1. Experimental set-up

!−1 = 2 ð1bÞ

As pointed out in the introduction, the ratio θd is empirically known to affect the spatial air distribution uniformity at the bottom of the bed. Additionally, Bonniol et al. showed that the latter is also involved in the control the temporal flow fluctuations at the bed entrance. This ratio reflects the ability of the pressure waves be transmitted from the bed to the plenum through the distributor and vice versa. The other new dimensionless parameter θp relates the temporal derivative of pressure fluctuations generated by the bed to the fluctuations of inlet fluid velocity at its bottom [12]. This number can also be written as, θp = θd τb τpd

119

ð2Þ

with two characteristic times τpd = 2Ω〈Δpd〉/γ〈pp〉Σ〈ud〉 and τb = (hcp/ g′)1/2. It has been shown by Sierra et al. [16] that the dynamics of the pressure–distributor association is governed by a first order ordinary differential equation so that any sudden change of the pressure in the plenum relaxes exponentially (by adjusting the downstream fluid velocity to the bed since the velocity at the plenum entrance is fixed) 1 represents the characteristic with this characteristic time τpd. τ− b oscillation frequency of a fluidized bed of static height hcp. Thus, for a given θd, this expression shows that θp compares these two time scales. When τb ≫ τpd i.e. for large θp values (obtained in our experiments for small plenum volumes) the plenum adapts very quickly to the pressure fluctuations imposed by the bed. Since the fluid flow rate is prescribed constant at the plenum entrance, the only way for the plenum to accommodate pressure changes imposed by the bed is to vary its outlet flow rate i.e. the bed entrance one. As long as its characteristic time τpd is small compared to τb, we can say that 1 This section is a summary of this theoretical work. Details can be found in the cited reference.

The experimental apparatus (Fig. 2) is made up of a 1-m high transparent square glazed riser (cross-section Σ = 0.1 × 0.1 m2). The air distributor at the bottom is a perforated steel grid with 1 mm holes (40% open area) that is welded to a very low pressure drop metallic fabric with 40 μm holes that prevents particles from falling into the plenum. Such design of the air distributor is purposely chosen to allow a complete communication between the bed and the plenum: for all the experiments done 0.06 b θd b 6. The plenum is made up of a large barrel connected to the bed upstream the distributor. Its volume Ω can be adjusted by changing the amount of water inside so that 0 b Ω b 220 l. The bed is fluidized by means of a centrifugal pump (MPR Industries CL 80/01). An isolation grid between the plenum and the rest of the air-supply system (pipes and fan) is used to create a large pressure drop that ensures a complete isolation with these elements and guarantees that only the plenum can interact with the bed. Three differential pressure transducers (Sensym SCX05, 0–5 PSI range) are disposed along the system: one at the bottom of the bed, one in the plenum and one upstream the isolation grid; all connected to a PC data acquisition system. In order to isolate the pressure probes from high frequency electro-magnetic noise, the signal is low-pass filtered at 20 Hz. A hot-wire anemometer measures the air velocity and temperature downstream of the isolation grid. 3.2. Experimental procedure As the aim of this study is to analyze the influence of the sole plenum on the bed dynamics expressed through θp, the chosen variable parameters are: the plenum volume Ω, the mean gas inlet velocity 〈ud〉 and the static bed height hcp. Additionally, the influence of the Geldart type of the particles is also explored (see particle properties and experimental configurations in Table 2). For each particle system, the experimental minimum fluidization velocity umf is measured with the classical procedure, on the 〈Δpb〉 = fcn(〈ud〉) curve (with 〈ud〉 = 〈ua〉Σa/Σ). From then it is possible to adjust at will, with the hot-wire anemometer, the velocity ratio

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Fig. 1. (a) Map of the amplification rate A⁎ for 〈ud⁎〉 = 2. The gray levels from dark to light indicate the transition from uncoupled to coupled bed dynamics. (b) Map of the oscillation frequency f for 〈ud⁎〉 = 2. The gray levels indicate the bed oscillation frequencies, high in the uncoupled zone (dark gray), they tend to decrease for stronger coupling (light gray). (c) Amplification rate vs. θp for θd = 0.66. (d) Frequency vs. θp for θd = 0.66. These maps and graphs are drawn for zirconia particles B1 in Table 2.

Fig. 2. Scheme of the experimental set-up.

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〈ud⁎〉 = 〈ud〉/umf. Once the fluid velocity and the bed height are chosen, the plenum volume is varied over prescribed values (see Table 1). For each plenum volume, the pressure data is collected at a sampling rate of 200 Hz with more than 32 768 samples recorded (N 2 min 45 s) for the three sensors. For all experiments, it was verified that there is no hysteresis effect and that the behavior of the bed is robust by an incremental and decremental scan of plenum volumes. All data extracted from these experiments are averages of these ‘up’ and ‘down’ measurements.

determination of the instantaneous frequency fi(t) associated to each ci by using the Hilbert transform on each component,

3.3. Hilbert–Huang Transform

with

Pressure signals are common measurements used to study fluidized beds because they provide low-cost, easy to set-up and global nonintrusive information about the bed dynamics. These signals are routinely analyzed by means of linear procedures such as statistical time domain analysis (average, standard deviation, kurtosis, skewness, and correlations) or frequency domain analysis with the Fourier Transform to obtain spectral distributions. These analyses are easy to use, and their associated theory is well understood. More recent methods like the short time Fourier transform or the wavelet transform [17,18] are suitable for the analysis of non-stationary signals, commonly observed in fluidized beds. However, all these methods remain linear in essence and inevitably introduce distortions when applied to non-linear signals. To overcome this issue, a method called the Hilbert–Huang Transform [19] was developed originally for oceanographic and meteorological signal treatments. HHT is an empirical method that decomposes the original signal into finite number of zero-mean and narrow band components called Intrinsic Mode Functions (IMF). This decomposition, called Empirical Mode Decomposition (EMD), is adaptive and based on the local time scale of the data. On each IMF, the Hilbert transform can be applied to yield the instantaneous frequency content of the component. This method is particularly interesting since it prevents harmonic distortions like in the Fourier analysis and decomposes the original signal into orthogonal components from which it is easier to identify physical phenomena and associated time scales. To our knowledge, Briongos et al. [11] were the first to apply this method to study fluidized bed dynamics as it is completely adapted to non-linear or non-stationary data. These authors proposed time–frequency signal representation of pressure signals. In their work each IMF component is associated to a particular oscillation mode (with a characteristic frequency) that reflects the different spatial scales involved in the global bed dynamics: particle, local bubble and bulk scales. In HHT method, the original signal is decomposed by means of the EMD (see algorithm details in Huang et al. [19], with Szero method for the interpolation procedure (Peel et al. [20]) and the Rilling criteria is used as the sifting termination condition [21]) as,

ai ðt Þ =

n

X ðt Þ = ∑ ci ðt Þ + rn ðt Þ

ð3Þ

i=1

where ci are orthogonal IMF and rn(t) corresponds to the mean tendency of the original signal X(t). The following step is the

  ′ 1 ∞ ci t ′ dt ; H ½ci ðt Þ = ∫ π −∞ t−t ′

ð4Þ

from which an analytic signal is defined as, zi ðt Þ = ci ðt Þ + jH½ci ðt Þ = ai ðt Þe

jϕi ðt Þ

ð5Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2i ðt Þ + H 2 ½ci ðt Þ

ð6Þ

and ϕi ðt Þ = arctan

H½ci ðt Þ : ci ðt Þ

ð7Þ

The instantaneous frequency is then given by, fi ðt Þ =

1 dϕi ðt Þ : 2π dt

ð8Þ

When the two above steps are completed the original signal finally writes as, n

X ðt Þ = ∑ ai ðt Þe

jϕi ðt Þ

ð9Þ

i=1

This decomposition enables to get a representation of the frequency fi(t) and of the amplitude ai(t) associated to each IMF as functions of time, which is designated as the Hilbert amplitude spectrum, H(ω, t) (with ω = 2πf), or simply Hilbert spectrum. We can also define the marginal spectrum h(ω) as, T

hðωÞ = ∫ H ðω; t Þdt

ð10Þ

0

which allows to get, in a way that recalls the Fourier Transform, a global amplitude–frequency representation of the total signal that is convenient to detect a dominant frequency. It gives a measure of the total amplitude (or energy) contribution from each frequency value. It represents the cumulated amplitude over the entire signal in a probabilistic sense. However as pointed out by Huang et al. [19], the frequency in either H(ω, t) or h(ω) has a different meaning from the Fourier spectral analysis. In the Fourier representation, the existence of certain level of energy at a given pulsation ω means that the original signal can be represented as a combination of constant amplitude sine or cosine functions throughout the time span of the data. In the marginal Hilbert spectrum, the existence of energy at a pulsation ω indicates a higher likelihood to have an oscillation with such a frequency in a statistical sense. In the following work the Hilbert spectrum is used to characterize the frequency content of the original signal as a function of time and the marginal Hilbert spectrum is used to identify the dominant frequency and the associated spectrum width.

Table 1 Explored plenum volumes. Exp # Ω [m3]

1 3.08 × 10− 4

2 6.54 × 10− 4

3 1.06 × 10− 3

4 2.80 × 10− 3

5 1.38 × 10− 2

6 1.90 × 10− 2

Exp # Ω [m3]

7 2.42 × 10− 2

8 3.19 × 10− 2

9 4.48 × 10− 2

10 5.77 × 10− 2

11 7.06 × 10− 2

12 8.35 × 10− 2

Exp # Ω [m3]

13 9.63 × 10− 2

14 1.22 × 10− 1

15 1.48 × 10− 1

16 1.77 × 10− 1

17 1.99 × 10− 1

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4. Results 4.1. The example of Geldart B particles In order to describe and analyze the effects of the plenum on the bed dynamics, zirconia particles (referenced B1 in Table 2 for 〈ud⁎〉 ≃ 2 and hcp = 10 cm) are chosen as a paradigm of the general tendencies that can be observed among all particle systems, although each of the latter have their own peculiarities that are discussed later. This analysis relies on two sets of graphs exposed in Figs. 3 and 4. Fig. 3a shows the evolution of the standard deviation σ of the bed instantaneous pressure drop fluctuations with θp. Starting from high values of θp (small plenum volumes) σ increases until it reaches a maximum at a given θ+ p and decreases for lower values of θp (larger plenum volumes). This evolution indicates that the bed progressively increases its interaction with the plenum generating pressure fluctuations of larger amplitude until an optimal plenum volume for which the behavior of the bed is qualified to be in its most coupled state. At this point, the interactive dynamics between the bed and the plenum is the most effective. The bed oscillations are such that they induce pressure fluctuations in the plenum that modulates the inlet fluid velocity in a constructive manner that promotes and maintains a high amplitude bed motion. This evolution is to be qualitatively compared with success with the one of A⁎ in Fig. 1 given by the model of Bonniol et al. [12]. Indeed, these authors showed that an increase of A⁎ for decreasing θp and the presence of a maximum is the indicator of an increase of the coupling efficiency between the bed and the plenum until it reaches a maximum that finally degrades at lower values of θp. They found that this phenomenon is driven by two antagonist effects, the evolution of phase lag and the amplitude of the air flow fluctuations at the bottom of the bed, that simultaneously and respectively tend to promote and reduce the coupling efficiency from low to high values of θp (see Fig. 5 in reference [12] for details). This progressive change of the coupling efficiency is accompanied with a frequency shift of the dominant bed oscillation frequency (Fig. 3b). From high to low θp values, the frequency diminishes in a very similar way as predicted by the model although quantitative agreement is not found. For descriptive purposes, the θp range is divided in two zones + (zone 1: θp N θ+ p and zone 2: θp b θp , see Fig. 3) in which the bed undergoes a peculiar dynamics which can be further analyzed by means of the HHT as illustrated in Fig. 4. For high values of θp (rightmost of zone 1) the dynamics between the bed and the plenum is said to be uncoupled. This means that the bed undergoes its own intrinsic bubbling dynamics without interaction with the plenum. Fig. 4b1 shows the Hilbert spectrum for the highest θp value (rightmost white point in Fig. 3a). It is characterized by a continuous mix of high and low frequencies that randomly appear in the pressure signal. The marginal large-band Hilbert spectrum confirms this observation (Fig. 4c1). The dominant frequency around 4 Hz must be seen as the most probable frequency at which the bed tries to oscillate. This value fairly matches the classical Baskakov model frequency [22] f = (g/hcp)1/2/ π = 3.15 Hz which is no surprise since Müller et al. [23] showed that this expression is usually in good agreement with experiments for bubbling beds of type B/D particles (as the present B1 zirconia particles). As pointed out by Briongos et al. [11], this frequency is also

Fig. 3. (a) Standard deviation of the bed pressure drop fluctuations as a function of θp. The maximum of coupling occurs at a given θ+ p . (b) Dominant oscillation frequency as a function of θp. Comparison of the experiments with various models from the literature. Gray lines are guides to the eye. The particle and bed configurations are the same as in Fig. 1.

associated to a characteristic spatial scale that is the average bubble size present in the bed. Such bubbling beds also produce larger and smaller bubbles because of merging and splitting processes. The flatness and the width of this spectrum reflect these phenomena. A further decrease of θp is characterized by a progressive modification of the bed dynamics that gets coupled with the plenum. As observed by former authors [6,7], the principal frequency decreases continuously as plenum volumes grow larger. This is coherent with Davidson's spring-mass model developed to evaluate the oscillation frequency of piston-like bed motions. It's is noteworthy that as θp decreases in zone 1, the experimental frequencies get closer to this model indicating that the bed tends to adopt this particular kind of high amplitude piston-like motion. This observation is confirmed on Fig. 4b2 and c2. The Hilbert spectrum and the marginal spectrum taken at the maximum of coupling θ+ p (mid white point in

Table 2 Particles properties, associated characteristics and range of operating conditions. Set

Material

d (μm)

ρs (kg/m3)

Ar

Geldart classification

umf (m/s)

Remf

〈ud⁎〉

hcp (m)

A B1 B2 D1 D2

Glass Zirconia Glass Glass Polystyrene

69 670 475 1500 2640

2500 3800 2500 2500 650

29.8 4.15 × 104 9.73 × 103 3.06 × 105 4.27 × 105

A B B D D

0.0188 0.616 0.28 1 0.69

0.0867 27.5 8.95 100.5 121.44

3–6–9 1.5–2–3 2–4–6 1.5–2 1.5–2–2.5

0.1–0.2 0.1–0.15 0.1–0.2 0.1 0.1–0.2

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Fig. 4. (a) Original pressure signals recorded at the bottom of the bed. (b) Hilbert spectra over a 30 s time period. Frequency scale is from 0 to 10 Hz. The gray levels intensity indicates the amplitude. (c) Marginal spectra obtained from the time integration of the Hilbert spectra. (1) The uncoupled case that corresponds to the rightmost white symbol in Fig. 3. (2) The maximum coupling case that corresponds to the white symbol at the peak in Fig. 3a. (3) Beyond the maximum coupling that corresponds to the leftmost white symbol in Fig. 3. The fluid inlet velocity and distributor properties are the same as in Fig. 1.

Fig. 3a) show that frequencies are much more collapsed around a single frequency than in the previous case. The bed oscillates in a very regular way and almost no ‘parasitic’ frequencies appear during time. This is the signature of quasi-single mode of oscillation that fairly resembles a piston-like motion of the bed with the regular eruption of a single bubble. The dominant frequency is lower with a larger peak amplitude (roughly 15 times larger than the most uncoupled case). In zone 2, where θp b θ+ p , the frequency slightly continues its decrease (less quickly than before) and fairly follows Davidson's model. However, σ now diminishes, indicating that the coupling weakens. Fig. 4b3 and c3 (corresponding to the leftmost white point in Fig. 3a) indicate that the bed, during time, mainly oscillates at a frequency around 1 Hz but that there exist sporadically lower and higher frequencies gusts that indicate an alteration of the almost ideal piston-like motion of the bed at the end of zone 1. Indeed, careful visual observations of the bed dynamics revealed, between two

sequential eruptions of a bunch of bubbles at the bottom of the bed, the existence of transient and small swirly bubble-like voids that produce those ‘parasitic’ frequencies when the bed is in this pseudofluidized state. In this zone, the bed dynamics is more and more controlled by the charge and discharge of the plenum volume. Larger plenum volumes (not explored in these experiments) would produce relaxation oscillations where the bed does not interact dynamically anymore with the plenum but follows the dynamics of the latter (see [12] for details). 4.2. Influence of the particle Geldart type The paradigmatic example of zirconia particles B1 helped to analyze the overall influence of the plenum on the bed dynamics. Although in broad lines, all tested particles show a similar behavior, they have their own peculiarities that are discussed here.

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4.2.1. Standard deviation of pressure fluctuations Fig. 5 shows the evolution of the standard deviations σ with θp. All systems were explored for the same plenum volumes, but their θp range differs since their particle characteristics are different. These curves all show a bell-shaped appearance with a peak at a given θ+ p . This confirms a universal coupling/uncoupling mechanism common to all Geldart particle types as proposed by Bonniol et al. [12]. However, the first difference that can be noticed concerns the amplitude of pressure fluctuations in the most uncoupled state (rightmost point of each curve) where all particle systems show a different level of standard deviations. From a heuristic argument based on the energy balance of a fluidized bed, Punčochař and Drahoš [24] found that for a free bubbling bed, the standard deviation of pressure fluctuations “is proportional to the increase of normalised rate of dissipated energy caused by the excess of fluidizing air above the minimum bubbling conditions” i.e. σ ∝ Δe = (1 − εmf)(ρs − ρg)g (〈ud〉 − umf). Except for particles D1, all standard deviations follow this linear relationship as illustrated in Fig. 6 (for uncoupled conditions, see black-filled symbols). The success of this scaling argument shows that, as supposed by these authors, all beds are in their intrinsic bubbling mode of oscillation, totally uncoupled from the plenum : the amount of excess energy brought by the fluid above incipient fluidization conditions is dissipated by the motion of particles (kinetic energy) induced by the multiple bubble dynamics. In the most coupled state, the measured standard deviations (hollow symbols) still show a correct agreement with the proposed scaling. This suggests that, although the overall bed dynamics may be different with bigger and more regular bubbles, the dissipation process is merely identical to the free bubbling conditions. However, the larger slope indicates that the coupling with the plenum actually acts as an amplifier of the bed dynamics and of the joint dissipation. The plenum plays the role of a storage tank of the air compressive potential energy. This energy storage and release during each oscillation cycle manifests by the regular modulation of the air inlet velocity at the bed entrance. When this modulation is tuned with the bed dynamics in terms of phase lag and amplitude, the bed is in its most coupled state with a regular and high amplitude motion [12]. Another important difference concerns the relative variation of the standard deviation δ = σmax/σmin − 1 between the most uncoupled and coupled dynamics. Fig. 7a presents the evolution of δ for all particle systems as a function of their respective position in the Geldart classification. The variation of the pressure standard deviation between the uncoupled and the most coupled states diminishes from 169% for Geldart A to 59% for Geldart D particles. From the three glass particle systems (A, B2 and D1) it can be seen that this diminution

Fig. 5. (1) Standard deviation of the pressure fluctuations a function of θp for all particle systems in Table 2 except B1. For particle A, 〈ud⁎〉 = 3, θd = 0.066; B2, 〈ud⁎〉 = 2, θd = 0.37; D1, 〈ud⁎〉 = 2, θd = 1.61; D2, 〈ud⁎〉 = 2, θd = 4.37. The bed height for all particle systems is hcp = 10 cm. Gray lines are guides to the eye.

Fig. 6. Standard deviation of all particle systems as a function of the increase of energy Δe proposed by Punčochař and Drahoš [24]. Black and white symbols respectively refer to the most uncoupled and coupled regimes. The black continuous and the dashed lines correspond to a linear fit of the points (D1 excluded). The gray lines delimit a ±15% zone around the fits.

Fig. 7. (a) Evolution of the relative variation of the standard deviation δ between the most uncoupled and coupled dynamics as a function of the particle Geldart type. (b) δ as a function of the apparent average bed permeability 〈kapp〉 defined in Eq. (12).

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ð12Þ

fluidized bed in interaction with its plenum – of this particular damping mechanism. The absolute standard deviations of pressure fluctuations are correctly ranked, either in the uncoupled or in the most coupled regime, by the scaling of Punčochař and Drahoš [24]: the larger the particles density and/or the larger their diameter, the larger the amplitude of pressure fluctuations. On the contrary, the relative increase of pressure fluctuations between both regimes does not obey this rule: δ is high for small particles (Geldart A) with low permeabilities and decreases for coarser particles with higher permeabilities (Geldart D). Small particles show a lowest propensity to let the gas accumulated in the plenum percolate through the bed, their relative coupling efficiency is then much stronger than for coarse particles: the bed interaction with the plenum is then relatively more constructive. By opposition, the ability of coarse particles to compress the plenum is weaker and induced fluctuations of the inlet fluid velocity that do not succeed to generate an intense coupling with a high amplitude motion of the bed. Consequently, the relative difference of pressure fluctuations amplitude between both regimes is also weaker.

For viscous flows, Ar ∝ Remf so that 〈kapp〉 ∝ d2 and for inertial flows, Ar ∝ Re2mf so that 〈kapp〉 ∝ d: in all cases, the apparent permeability of the bed increases with d. Fig. 7b shows the evolution of δ as a function of 〈kapp〉 for all particle systems. The decrease of δ with the permeability is, to our knowledge, the first experimental evidence – for a freely oscillating

4.2.2. Dominant frequency and spectrum content Fig. 8 shows the evolution of the dominant oscillation frequency for each particle system as a function of θp. These frequencies are determined from smoothed marginal Hilbert spectra using the Savitzky–Golay adjacent averaging procedure. Vertical error bars indicate the range of frequencies around the identified peak for which the amplitude is between 95% and 100% of the maximum. This gives a

does depend neither on their chemical composition nor on their density but only on their diameter. This observation can be related to the work of Wong and Baird [25] that proposed a modified version the Davidson's spring-mass model by including the permeability of the bed. From this refinement, they suggested that the motion of coarse particles should be subjected to a higher damping compared to small particles because of their greater permeability of the bed. In order to test this idea, it is interesting to define, from the Darcy–Forchheimer pressure drop relationship, the apparent average permeability 〈kapp〉 of a particle bed at a given fluid velocity 〈ud〉 as, −

μg μg hΔP b i 2 E hud i hu i + βρg hud i = D = hki d hhi kapp

ð11Þ

Under equilibrium conditions, 〈Δpb〉 = (ρs − ρg)(1−〈ε〉)g so that, D E T D E ud μ g hud i Remf 2  d kapp =  = Ar ð1−hε iÞ ρs −ρg ð1−hεiÞg

Fig. 8. Dominant oscillation frequencies as a function of θp. Comparison of the experiments with various models from the literature. Fluidization conditions for each particle system are the same as those in Fig. 5. Gray lines are guides to the eye.

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qualitative measure of the spectrum spread around the dominant frequency. Horizontal bars give the estimated error on θp due to the error on each parameter in Eq. (2). For all particles system, the measured frequencies qualitatively follow the aspect of the theoretical curve of Bonniol's model. The latter does not predict accurately these frequencies which is no great surprise since it reduces the bed to one-dimensional homogeneous oscillator which is rather far from a real bed dynamics. However, contrary to other models, it gives a good and continuous estimation of the relative frequency shift over the explored θp range. This is an indication that the leading order of the bed dynamics is nevertheless captured by this model. For the uncoupled points, the model of Baskakov appears as a good estimation of the dominant bubbling frequency, except for particle D1 and, to a less extent, for particles D2. For particles D1, the difference is particularly obvious and results from an inhomogeneous fluidization of the bed observed during these experiments. This observation is corroborated by the amplitude of pressure standard deviation exposed in Fig. 6. The mass of particles involved in the motion is lower than the whole bed mass so that the absolute amplitude of pressure fluctuations are lower than expected and frequencies higher. As this situation perdures from the uncoupled to the coupled regime, this artifact does not modify the relative amplitude increase δ. When in the most coupled state, dominant frequencies of all systems are fairly well predicted by Davidson's model as for B1 particles. This again indicates that the simple piston-like dynamics is a

good approximation of the bed motion for which, more or less, a single bubble erupts during each oscillation cycle. This last observation can be refined by the analysis of the marginal Hilbert spectra of all particle systems exposed in Fig. 9 for the uncoupled and the most coupled regimes. For particles A and B2 (similarly to B1) the regime change is clearly identified by a modification of the spectra: large for the uncoupled case and narrow for the coupled dynamics. However for particles D1 and D2, except the frequency lowering, the spectra in both cases are very similar. These observations make a coherent picture with all results presented above. Small particles have a great mobility that makes them produces an uncoupled dynamics with small and numerous bubbles with a rich splitting and merging dynamics. Due to their ability to interact with the plenum (low permeabilities: 〈kapp〉 ≈ 1.5 × 10− 10 m2) they undergo a dramatic change of dynamics when coupled with the latter. As particles get larger, they progressively lose these characteristics and tend to produce, even in the uncoupled regime, large bubbles. For shallow beds like those studied here, it can give, for this kind of particles, a pseudo-slugging dynamics as reported by Brzic et al. [26]. Due to their high permeabilities (〈kapp〉 ≈ 1 × 10− 8 m2), their ability to interact with the plenum is poor so that their dynamics is almost unchanged between both regimes, only the dominant frequency is lowered. 4.2.3. Locus of maximum coupling The comparison between model and experimental θp-loci at the maximum coupling is given in Fig. 10 for all explored bed heights and

Fig. 9. Marginal Hilbert spectra for all particle systems (except B1). Fluidization conditions for each particle system are the same as those in Fig. 5. Continuous and dashed lines are respectively for the most uncoupled and the most coupled regimes.

F. Bonniol et al. / Powder Technology 202 (2010) 118–129

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+ + + Fig. 10. (a) Log–log comparison of θ+ p(th) with θp(exp) for all particle systems, bed heights and fluid velocities. (b) Relative error between θp(th) and θp(exp) as a function of the fluid velocity 〈ud⁎〉. Horizontal dashed lines delimit a ±50% zone around the null error.

fluid velocities. The theoretical θp at the maximum of coupling is given in [12] as,

þ

θpðtheoÞ = 4π f

′

g hcp

!−1 = 2

0 B @1 +

1:75ρg ϕcp hud i2 ψρs g ′ d

1 D E3 T h C   3 A hT −ϕcp

ð13Þ

For a given particle set and operating conditions, this expression is numerically evaluated by calculating the average dimensionless bed dilation 〈h⁎〉 = 〈h〉/hcp and f the dominant bed oscillation frequency that depends on the level of coupling with the plenum, respectively from Eqs. (12a) and (13a)–(13b) in [12]. The model predicts, for most particle systems, the order of magnitude of the maximum coupling locus (Fig. 10a). It works slightly better for group B and group D particles. The spread of the points can be attributed to different causes. The first cause comes from the difficulty, for the majority of particle systems, to identify a well defined locus of maximum coupling because there is no sharp peak in the σ vs. θp experimental curves but rather a “smooth hump” that delimits a range of probable θ+ p(exp). Even though there always exist a locus where the coupling is maximum as already seen in Fig. 5, the phenomenon reveals to be, in this zone, poorly sensitive to plenum volumes changes. The maximum of coupling and its associated θ+ p(exp) were practically identified from a

low order polynomial fit of this zone. The vertical error bars (Fig. 10a) are defined from the polynomial as a range of θ+ p(exp) for which the standard deviation is between 99% and 100% of the maximum. The second cause can be identified with Fig. 10a that shows the + + + + relative error Δθ+ p /θp(theo) = (θp(theo) − θp(exp))/θp(theo) as a function of ⁎ the normalized fluid velocity 〈ud 〉. Except for particles A, the model works best at rather low velocities, between 1.5 b 〈ud⁎〉 b 2.5 for Geldart B and D particles. The model predicts almost no influence of the fluid inlet velocity on θ+ p(theo) whereas measurements show an increase. In a similar way as observed at low velocities, the coupling phenomenon still exists at higher velocities: the σ vs. θp experimental curves still show a bell-shaped appearance with an identifiable maximum of coupling2. However, the model that represents the bed as a onedimensional oscillator fails to account for the dynamics modification induced by a higher fluid flow. Although for the largest velocities explored the so called “turbulent fluidization regime” is still far from being attained, all beds already show a more chaotic and turbulent behavior that depart from the simple oscillator picture of the model. This undoubtedly affects the predictive capacity of the latter at high fluid velocities. No influence of the bed height is identifiable over the range studied (10 b hcp b 20 cm).

2 These curves are not shown here not to weight down the article presentation but can be found in [27].

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All these observations have important consequences for the practical design of fluidized beds. Using a low pressure drop distributor opens the possibility of a potential interactive coupling of the bed with its plenum (at least). In most cases this coupling is undesirable because (i) it degrades chemical conversions and heat transfers (large bubbles yield poor mixing and mass transfer) and (ii) high amplitude motion of the bed can be damaging for fluidization devices as it induces a strong mechanical fatigue. In order to avoid this problem, it is of prior importance to choose the volume of the plenum chamber as low as possible (provided that the air-supply system design still ensures the uniformity of the gas injection). Indeed, without particular attention in the design process, the plenum chamber can reveal to be large enough to give rise to an unwanted coupling dynamics if the pressure drop of the distributor is sufficiently low. This is typically what is reported in [4,28]. Very large plenum volumes i.e. θp ≪ θ+ p , although they produce an uncoupled bed dynamics in the sense of low standard deviations of bed pressure fluctuations compared to the most coupled state, remain problematic. Indeed, in this case the bed dynamics tends to show relaxation oscillations totally controlled by the plenum: this kind of dynamics is undesirable for the reasons stated above. Then, in order to avoid the coupling zone, it is necessary to sufficiently diminish plenum volumes to reach large enough θp values. The theoretical model [12] predicts that this can be achieved, for a large range of particle properties, for a + critical θp value typically 100 times larger than θ+ p i.e. θpc ≃ 100θp . Figs. 5 and 8 confirm this criterion. Starting from the most coupled point, the application of this criterion places the uncoupled point in a zone where the standard deviation of bed pressure fluctuations are indeed minimum and the frequencies are on a plateau roughly equal to Baskakov frequencies. Written in terms of plenum volumes, this criterion simply gives the largest plenum volume allowed before triggering an unwanted coupling: Ωc ≃ Ω+/100. As the model [12] is able, at low fluid velocities, to give a fairly correct order of magnitude + and give practical evaluations of of θ+ p it can be used to calculate Ω Ωc. This method can help to reduce the transient interaction of industrial beds with their air-supply systems. It does not suppress the problem of spatial maldistribution and certainly requires to maintain a distributor with a sufficient pressure drop but, without doubts, significantly lower than required when the plenum is too large. This could reduce investments and operating costs of the blowing fans.

The present work also highlighted some specificities of each particle type. In particular, the interaction intensity between the bed and the plenum decreases with the bed permeability. When in the most coupled state, the low apparent permeabilities of small particle beds makes them more capable to compress the air in the plenum and though increase the coupling efficiency that manifests by a strong change in the bed dynamics (frequency content and standard deviation amplitude). On the contrary, beds with larger particles show a less pronounced and even no change (except the frequency shift) in the bed dynamics between the coupled and the uncoupled regimes. A final outcome of this study is the practical consequences that can be drawn for the design of plenum volumes. Uncoupling the bed from its air-supply system cannot only be done by increasing the pressure drop of the distributor as it is classically advised in industrial fluidization text-books. The reduction of the plenum volume also leads to the same effect and may limit the required bed to distributor pressure drop ratio. From the model of Bonniol et al. that gives a good order of magnitude of Ω+, the following design criterion Ωc ≃ Ω+/100 should be efficient to reduce or suppress the temporal fluid flow oscillations at the bed entrance that usually induce bad fluidization operating conditions and also mechanical fatigue of the devices. Finally, here are some perspectives to this work. First, it would be interesting to test the coupling effect with the plenum for deeper beds i.e. for larger aspect ratios than studied here. Indeed, these kinds of beds are usually prone to slugging and it is possible that the transition between uncoupled and coupled dynamics would not radically change the slugging dynamics of the bed. Since slugging already manifests itself as a rather regular phenomenon, we can imagine that the coupling with the plenum would mainly change (increase) the amplitude of slugging, certainly decrease the slug formation frequency but not the type of dynamics itself. This is a conjecture that would be worth being tested in future works. Another perspective could be to proceed to a similitude study that would systematically compare the coupled behaviors of dense fluidized beds of different heights, with different particles and different riser, distributor and plenum scales but that match all the dimensionless numbers given in Eqs. (1a) and (1b). Indeed, for all particles tested here, no match of dimensionless numbers was sought (which enabled to highlight differences in their behaviors) but it would be interesting to test if dynamic similarity can be achieved for beds interacting with their upstream flow inlet conditions.

5. Conclusion The influence of the plenum on dense gas fluidized bed dynamics is studied in this paper from an experimental point of view and compared with different models from the literature. For all particle systems tested (A, B and D Geldart type), at all bed heights and fluid velocities, the experiments show the existence of a θp range where the bed dynamics is coupled with the plenum. Within this range there always exist a most coupled state that occurs at a specified θ+ p . These experiments also reveal that the bed dominant frequency varies with the coupling intensity. All these measurements confirm, at least qualitatively, the theoretical predictions of Bonniol et al. [12] model. The present work clarifies former observations of the literature and specifies the plenum effects when a dense gas fluidized bed interacts with it. It is shown that the Davidson's mass-spring model usually used to evaluate the bed dominant frequency in case of interaction with the plenum is only valid beyond the maximum of coupling θp ≤ θ+ p . Baskakov model is confirmed to be correct for an estimation of the dominant frequency when the bed does not interact with the plenum. For the cases in between, none of these models work. The model of Bonniol, without giving a fully quantitative evaluation of the frequency correctly predicts the relative frequency shift and so can be used to evaluate the dominant oscillation frequency is this half-tone coupling region.

Nomenclature amplitude of the analytic signal (−) ai(t) Ar Archimedes number (−) A⁎ amplification rate of perturbations (−) intrinsic mode function (Pa) ci(t) De density ratio (−) d particle diameter (m) f bed dominant frequency (Hz) instantaneous frequency of each ci (Hz) fi (t) g acceleration of gravity (m/s2) g′ g(1 − ρg/ρs) (m/s2) H(ω, t) Hilbert spectrum (−) h(ω) marginal Hilbert spectrum (−) close packing bed height (m) hcp h(t) instantaneous bed height (m) k bed permeability (m2) apparent bed permeability (m2) kapp pressure in the plenum (Pa) pp Re Reynolds number (−) minimum fluidisation Reynolds number (−) Remf mean tendency of the original pressure signal X(t) (Pa) rn(t) fluid inlet velocity through one orifice of the distributor (m/s) ua fluid inlet superficial velocity (m/s) ud

F. Bonniol et al. / Powder Technology 202 (2010) 118–129

umf X(t) zi(t) ε εmf Δpb Δpd γ ϕi(t) μg θd θp ψ ρg ρs Ω Ωc ω ξd Σ Σα σ

minimum fluidization velocity (m/s) original pressure signal (Pa) analytic signal (−) porosity (−) minimum fluidisation porosity (−) bed pressure drop (Pa) distributor pressure drop (Pa) ratio of specific heat (−) instantaneous phase of each ci (−) gas dynamic viscosity (kg/m s) dimensionless number associated to the distributor (−) dimensionless number associated to the plenum (−) particle sphericity (−) gas density (kg/m3) solid density (kg/m3) plenum volume (m3) critical plenum volume (m3) pulsation (rad/s) distributor discharge coefficient (−) distributor cross-section (m2) distributor orifice cross-section (m2) standard deviation of bed pressure fluctuations (Pa)

Subscripts and superscripts * dimensionless variable + variable at the maximum of coupling max maximum value measured min minimum value measured theo theoretical exp experimental 〈〉 time averaged variable

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