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Fluctuations and waves in fluidized bed systems:

The influence of the air-supply system

Srdjan Sasic, Bo Leckner, Filip Johnsson*

Department of Energy Conversion, Chalmers University of Technology, S-412 96 Goteborg, Sweden

Received 30 June 2003; received in revised form 29 October 2004

Available online 10 May 2005

Abstract

A general model of the response of a fluidized bed to disturbances is formulated, and the information provided by the model with respect

to the dynamics of the bed, the bed plus the air-plenum and the bed plus the entire air-supply system, is investigated. Expressions given in

literature on the fundamental frequency of the bed-plenum system are analyzed, and it is shown that they are a special case of the general

model. In order to simulate various types of interaction between the bed and the rest of the system, experiments were performed in a cold

fluidized bed unit operated under both non-circulating and circulating conditions. At low velocity, three regimes were identified: the multiple

bubble regime with almost no interaction between bed and air-plenum, the single bubble regime with the interaction between bed and air-

plenum only, and a regime with numerous irregular bubbles, where the bed interacted with the entire air-supply system. At high fluidization

velocity, the exploding bubble regime was identified, with the same dominant frequency as that of the single bubble regime (the interaction

with the air-supply system remains at that frequency). The models investigated correctly reproduce the dynamics when the bed is independent

of the other parts of the system, or when the bed interacts only with the air-plenum. However, the models are only partially applicable when

the bed interacts with the entire air-supply system. The reasons for this are investigated. In the case of system interaction, pressure waves,

generated in the bed, interact with pressure pulsations from the air-supply system. This results in a coupled system, which is not covered by

the models. Pressure waves resulting from events in the bed, are recognized as the coherent part of the cross power spectra of pressurefluctuations measured in the bed and the air-plenum.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Gas– solid fluidized beds; Dynamics; Fluctuations; Waves; Interaction; Air-supply system

1. Introduction

The most commonly studied quantities in dynamical

analyses of gas–solid fluidized beds are pressure fluctua-

tions e.g. [1–5]. They are used to characterize the fluid-

ization [4] or to model the response of a dynamical system(the fluidized bed) to any input or disturbance [2,6].

Although measurements of pressure fluctuations are easy

to perform, the interpretation of such signals is not

straightforward. It is generally accepted that the fluctuations

come from bubble motion within the bed, but the origin of

the fluctuations is usually not clear. A pressure disturbance

is propagated as a wave through the system [7]. Such waves

in fluidized beds have been the subject of a few publications

[7–9]. The studies originate from two view-points that

created separate paths in the research: to explain the nature

of pressure fluctuations in a bed e.g. [1] and to study the

conditions of fluidization e.g. [10]. The latter usually

concerns the onset of bubbles and their effect on the(in)stability of uniform fluidization, whereas the former

deals with the overall hydrodynamic behavior of the bed and

fluidization regimes.

A previous study [11] identified three principal cases

with respect to the boundary limiting the physical phenom-

ena that may be relevant for the dynamics of the system:

a) Only the fluidized bed is concerned. This is the case

when the pressure drop across the air-distributor is in the

order of magnitude or higher than the pressure drop of

0032-5910/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.powtec.2005.03.012

* Corresponding author. Tel.: +46 31 772 1449.

E-mail address: [email protected] (F. Johnsson).

Powder Technology 153 (2005) 176 – 195

www.elsevier.com/locate/powtec



the bed, yielding no significant propagation of pressure

and flow waves upstream of the air-distributor.

b) The bed interacts with the air-plenum. This is typically

the case in such laboratory units that are fed from

pressurized air systems and have air-distributors with

reasonably low pressure drop.

c) The bed interacts with the entire air-supply system. Thismay take place in industrial systems (with low pressure

drop air-distributors), normally consisting of fluidized

bed, air-plenum, air-supply lines, and fan (or fans).

Pressure and flow waves propagate through the entire

system and an interaction is expected between these

parts. This may make it necessary to include the entire

system in simulations.

The dashed lines in Fig. 1(a–c) indicate the boundaries

of the system in the three cases.

Flow waves in the air-supply parts of the system and

their influence on the fluidized bed dynamics (Case c) havenot attracted much attention so far. Most studies have been

carried out in laboratory units corresponding to Case b or

Case a. Such devices usually prevent the pressure and flow

waves from propagating throughout the system, whereas in

industrial units with long supply lines and low pressure drop

devices, the opposite is to be expected. The influence of the

flow waves on the bed dynamics is far from clear. Although

this issue has been treated by Moritomi et al. [6] and

Johnsson et al. [11], no model of the interaction between

different parts of the system has been presented.

The aim of this work is to analyze literature models on

the dynamics of fluidized beds and to compare them with

the three experimental configurations corresponding to the

abovementioned Cases (a, b , and c ). The models, assumed

to be general enough to comprise the cases treated, express

the response of a fluidized bed system to disturbances (both

external and internal ones with respect to the bed ). Theanalysis of the experiments and models is used to identify

mechanisms responsible for the interaction between the bed

and the air-supply system.

2. Theory

2.1. Modeling of the response of the dynamical system

The idea behind studies of the response of a dynamical

system is that the fluctuation of a given quantity is affected

by disturbances imposed on t he system. In this manner,Borodulya and Zavyalov [12] studied the behavior of the

instantaneous bed height, Moritomi et al. [6] derived a

model for fluctuations of pressure in the air-plenum,

depending on the input (i.e. disturbances) from the air-

supply system, while Fan et al. [2] characterized the

dynamics of a fluidized bed by the fluctuation of the bubble

residence time in terms of both internal and external

disturbances (t he term ‘‘noise’’ was used in the original

work). Table 1 summarizes the main features of the models

given by these authors.

fluidized bed

air plenum

fludizing air

fluidized bed

air plenum

fludizing air

fan

fluidized bed

air plenum

fludizing air

a) b) c)

Fig. 1. a: Case a . The bed is separated form the air-supply system by a high pressure drop air-distributor. b: Case b . Interaction between the bed and the air-

plenum may take place. c: Case c. The bed, the air-plenum, supply lines and the fan form a coupled system.

S. Sasic et al. / Powder Technology 153 (2005) 176–195 177



The models summarized in Table 1 provide fundamental

information about the system (usually a principal frequency

of pressure fluctuations) and can be used for stability

analysis with respect to disturbances originating from

different physical and operating parameters. In this way,

Borodulya and Zavyalov [12] investigated the effects of bed

height, as well as of some other features of the system, such

as hydraulic resistances of the air-supply system and of theair-distributor, on the stability of fluidization. They con-

cluded that increase of the pressure drop over the distributor

had the tendency to stabilize the process, while the system

remained stable irrespective of the change of the bed height.

In summary, the models investigated describe the system

in the frequency domain, either by a first-order transfer

function (relating any two of several properties of the

system, for example the amplitudes of pressure fluctuations

in the plenum and in the bed)

G ixð Þ ¼ 1

1 þ i

x

x0

;

where x0 is the principal frequency of the system, or, by a

second-order transfer function

G ixð Þ ¼ 1

1 x2

x20

þ i2x

n

x0

;

n being the damping constant.

The first-order transfer function results in the following

ratio of the amplitudes of air-plenum and in-bed pressure

fluctuations P ap

P bed

¼ 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ x2

x20

s :

It is obvious that the interaction between the bed and the

air-plenum (Case b in Fig. 1 b) decreases as the principal

frequency decreases (and it decreases as the pressure drop

over the air-distributor increases). These conclusions fully

agree with the experimental work of Svensson et al. [13],

who also reported that the interaction between the bed and

the air-supply system decreases with increase in the pressure

drop of the air-distributor.

A general formulation of the model employed by the

abovementioned authors can be derived. The fluctuating

pressure in the plenum is chosen to characterize the

dynamics of the system, and the applicability of the

model to Cases a to c is discussed. The aim is to treat

both the properties of the internal bed structure (bubble

eruptions at the surface of the bed) and external

disturbances (effects from the air-supply system and theconfiguration of the apparatus). The system to be treated

is shown in Fig. 2. The model is based on the following

assumptions:

& The pressure fluctuations in the bed and in the air-plenum

are the result of external disturbances from the air-supply

system (flow fluctuations) and from the action of

bubbles.

& The entire bed mass moves vertically or moves and

expands at the same time, while the mass of the bed

remains constant.

&

There is a periodic eruption of bubbles (or of a single bubble) at the surface of the bed.

The derivation of the model comprises a mass balance

over the air-plenum, a momentum balance over the bed, an

expression for the pressure drop over the distributor and an

equation that links the flows on both sides of the distributor.

The details of derivation are presented in Appendix A. After

fluidized bed

air plenumexternaldisturbance:fluidizing air

internal disturbance:bubble activities

Fig. 2. The dynamical system of interest for the study of the response of a

fluidized bed to disturbances.

Table 1

Summary of models expressing the response of a fluidized bed to disturbances

Author(s) Modelled property of the

system

Disturbances taken into account Type of equation Model outputs

[12] Instantaneous bed height External (from the air-supply system) Non-linear differential Stability analysis, Well-posedness of the

mathematical description

[2] Bubble residence time Internal (bubble activity) and external(from the air-supply system) Second-order linear differential Principal frequency of pressure fluctuations,Stability analysis

[6] Fluctuating pressure in the

air-plenum

Internal (bubble eruption) and external

(from the air-supply system)

Second-order linear

differential

Principal frequency of pressure fluctuations,

Pressure responses in the plenum depending

on various disturbances

S. Sasic et al. / Powder Technology 153 (2005) 176–195178



some algebra, a linear differential equation for the fluctuat-

ing pressure in the plenum is obtained

d2 pap

dt 2 þ C 2

C 1

d pap

dt þ 1

C 1 pap ¼ C 3

C 1

dqex

dt þ C 4

C 1qex þ 1

C 1 P in

ð1

Þwhich can be compared with the general equation describing

a harmonic oscillation of a mechanical system, where both

external excitation ( F (t )) and damping are present

d2 x t ð Þdt 2

þ 2nx0d x t ð Þ

dt þ x2

0 x t ð Þ ¼ F t ð Þ ð2Þ

yielding x0 ¼ 1= ffiffiffiffiffiffi

C 1p

as the natural frequency of the

nondamped oscillations and n ¼ C 2=2 ffiffiffiffiffiffi

C 1p

as the damping

factor. After introducing the equivalent height of the plenum

( H eq= V ap / Aap), the coefficients become

1

C 1 ¼ qfb

qap ! g

H eq

b P ap

D P fb

A bed

Aap

;

C 2

C 1¼ qfb

qap

! D P dist

D P fb

mg

U eq

A bed

Aap

; ð3Þ

C 3

C 1¼ b P ap

Aap H eq

; C 4

C 1¼ qfb

qap

! D P dist

D P fb

mg

U eq

b P ap

Aap H eq

A bed

Aap

ð4ÞThe application of the analogy between the fluidized bed

and the mechanical system (Eq. (2)) to simulate the bed

dynamics requires that the terms on the right hand side of

Eq. (1) be imposed on the system as indicated in Fig. 2, and

that these terms be independent of events taking place

within the boundary. This means that the pressure fluctua-

tions in the bed and the flow fluctuations in the air-supply

system have to be defined independently and then used as

input to the simulation. Consequently, in the present

application, the disturbance originating from bubble activ-

ities in the bed (pressure pulse P in in Eq. (1)), which is an

internal disturbance, formally has to be treated as if it were

an external disturbance term ( F (t ) in Eq. (2)). Eq. (1) was

implemented in a dynamical solver (SIMULINK) for

simulating the system. The simulations were compared with

experiments corresponding to the three Cases (a, b , and c ).All measured signals were characterized in the frequency

domain by power spectra, obtained as an average from a

number of sub-spectra (the number being chosen to obtain a

good trade-off between frequency resolution and statistical

significance). Each data set contained 512 points and the

total duration of each run was 240 s.

There are investigations in literature [3,14,15] where the

fluidized bed dynamics are expressed by a single frequency

(called ‘‘natural’’, ‘‘dominant’’, or ‘‘principal’’). The princi-

pal frequency of the system is assumed to be a result of the

oscillations of the bed mass due to bubble passages and can

be identified from a peak in the power spectrum of the static

in-bed pressure fluctuation. These expressions are special

cases of the general model (Eq. (1)), where damping and

disturbances are not accounted for. They give the same

principal frequency of nondamped oscillations as the

general model, but no additional information (e.g. stability

analysis) is obtained. The frequencies calculated will be

compared with the ones obtained in the experiments, and it is therefore necessary to analyze the basic assumptions of

the models.

For a system equipped with a low-resistance air-distrib-

utor and operating at low velocities (Case b), a model

proposed by Davidson [14] could be applied. The bed is

assumed to behave like a mass that oscillates on top of a gas

volume with a frequency (to be detected both in the air-

plenum and in the bed) given by the expression

f 0 ¼ 1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij P ap A bed

q bed H bedV ap

s ; ð5Þ

where P ap is the absolute pressure in the air-plenum,

q bed= qs(1 q)+qg q the bulk density of the bed, A bed is

the bed area, V ap is the volume of the plenum, and H bed is

the height of the bed. Baskakov et al. [3] derived their

expression for the fundamental frequency under conditions

that were similar to those of Davidson [14] using the

analogy with oscillations of an ideal fluid in a U-tube. This

results in

f 0 ¼ 1

t n¼ 1

p

ffiffiffiffiffiffiffiffiffi g

H mf

r ; ð6Þ

where H mf is the bed height at incipient (minimum)

fluidization. The frequency is inversely proportional to ffiffiffiffiffiffiffiffi H mf

p and independent of voidage. The principal frequency

of the bed was derived by Roy et al. [15] for incipient

fluidization in an isothermal system. The oscillations were

assumed to be analogous to those in an organ pipe closed at

one end and open at the other. The relative motion between

particles and gas is neglected. The frequency of the

oscillation is f 0= uc/ k (k = 4 H being the wavelength of the

oscillations, H the characteristic dimension along the main

flow direction and uc the propagation velocity of pressure

waves), which, after introducing the expression for the wave

velocity, finally becomes

f 0 ¼ 4 H

uc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffije qs 1 eð Þ þ qge

qg

s !1

: ð7Þ

To calculate the dominant frequency by Eq. (7), the

propagation velocity of pressure waves has to be

evaluated. Assuming that a gas– solid mixture is a

continuous compressible medium, the wave velocity is

calculated in this work by using a separated flow theory

[8,15]. The theory takes into account the relative motion

between the phases and assumes the gas phase to be




compressible and the particulate one to be incompressible.

This results in

uc ¼ c0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqf

qse 1 eð Þ1 þ x2s2

P B

1 þ x2s2P B

2

s ;

where, B ¼ qge

qs 1eð Þþqge, sP is a particle relaxation timescale, in which the drag is assumed to follow Stokes’ law,

and x is the angular wave frequency (rad/s). The

separated flow model was derived for small perturbations

in a dispersed gas– solid mixture, and therefore its

application for bubbling and especially circulating fluid-

ized beds remains questionable. Also, it is necessary to

know the angular frequency prior to the use of the model.

The analysis of Roy et al. [15] assumes that the pressure

waves are totally reflected at the air-distributor (the node is

assumed to be at the air-distributor, while the antinode is

located at the bed surface), and therefore Eq. (7) is limited to

Case a. Still, Cases b and c may be treated if the effect fromthe interaction of the system is small (such as when the air-

plenum is relatively small). Also, it is worth noticing that

this is the only expression in literature where the frequency

is inversely proportional to H ffiffiffi

qp

.

As indicated in Table 1, the principal frequency may also

be derived from the models that express the response of a

fluidized bed to disturbances. Fan et al. [2] and Moritomi et

al. [6] provide identical expressions to the one of Davidson,

Eq. (5), for nondamped oscillations (assuming isothermal

conditions)

f 0 ¼ 1

2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ap A bed

qmf H mf V aps

: ð8ÞFrom Eq. (1) one obtains the fundamental frequency for

nondamped oscillations as

f 0 ¼ 1

2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqfb

qap

! g

H eq

b P ap

D P fb

A bed

Aap

v uut ; ð9Þ

but there is also an influence on the frequency from the

damping that can be written as

f

¼ f 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1

4

qfb

qap ! D P 2dist

D P fb

m2 g

U 2eq

H eq

b P ap

Aap

A bed v uut ;

ð10

Þwhere f 0 is the frequency for nondamped oscillations (Eq.

(9)). It can be shown with some transformations (see

Appendix A) that Eq. (9) is similar to Eq. (8). Eq. (10),

however, shows for which parameters oscillations are

present. When the expression under the square root is less

than zero, the system is completely damped.

2.2. Identification of pressure waves in fluidized beds

In order to investigate the coupling between the fluidized

bed and the rest of the system, it is necessary to identify the

mechanisms through which this interaction is obtained. Van

der Schaaf et al. [16] proposed a method to recognize

pressure waves from pressure time series recorded in the

system. The power ( P yy) of the time series measured at

position y in the bed, is related to the time series simulta-

neously measured in the air-plenum ( x) and expressed as the

sum of the coherent (COP xy) and incoherent (IOP xy) part as

P yy f ð Þ ¼ COP xy f ð Þ þ IOP xy f ð Þ; ð11Þwith

COP xy f ð Þ ¼ C xy P yy; ð12Þ

IOP xy f ð Þ ¼ 1 C xy

P yy: ð13Þ

The coherence between measurement positions (C xy) was

given

C xy f

ð Þ ¼ P xy f ð Þ P 4 xy f ð Þ P xx f ð Þ P yy f ð Þ

; 0

C xy f

ð Þ 1:

ð14

ÞThe power spectrum of the coherent part identifies

pressure waves, originating in the bed and possibly

propagating through the entire system, whereas the incoher-

ent part expresses the effect of local events in the bed (gas

voids and turbulence).

3. Experiments

In order to illustrate the ability of Eq. (1) to reproduce

the dynamics of a fluidized bed system and the ability of

the method proposed (Eqs. (12) and (13)) to recognize pressure waves from pressure signals in such a system,

experiments are designed to correspond to the Cases a, b,

and c. The experiments comprise measurements of in-bed

and air-plenum pressure together with flow fluctuations in

the air-supply part of a cold circulating fluidized bed

(CFB) unit. The system is outlined in Fig. 3 with the

operating conditions summarized in Table 2. A large

pressure drop over the air-distributor (a perforated plate

with 2 mm holes and 0.4 % open area) in Case a makes

the configuration of the air-supply system irrelevant for the

fluidization situation in the bed. For Cases b and c, the

flow can be controlled by a valve, which is located either

near the air-plenum (a gate valve—Valve 1, Fig. 3) or

upstream of the fan (a nozzle valve—Valve 2, Fig. 3),

air plenum

fluidizedbed

Pitottube

Valve 1fanValve 2

Fig. 3. Simplified sketch of the air-supply system of the fluidization unit.




yielding one case with the air-supply system consisting of

the air-plenum only ( L ¨0) and one case with a long air-

supply tube ( L =30 m). The air was supplied by a radial,

high-pressure fan operated at constant speed. Its character-

istic curve for Cases b and c and the characteristics of

Valve 2 are given in Fig. 4a, with the stars indicating the

time-averaged operation points in the cases studied. The

characteristics of Valve 1 are given in Fig. 4 b. Time

averaged pressure along the air-supply system (from the

fan to the bottom of the bed) is shown in Fig. 4c for Cases

b and c. The air-distributor was a perforated plate with 2

mm holes and 6.2% open area.

The pressure fluctuations were measured (Honeywell

pressure transducers, type 143PC03D) at the wall at 0.2 m

above the air-distributor and in the air-plenum. The in-bed

measurement position was within the bottom bed, but high

enough to avoid the influence from the entrance effects from

the air-distributor. The sampling frequency was 20 Hz

(enough to capture the main frequency content in fluidized

beds). The inflow of the air to the air-plenum was deducedfrom the velocity as measured by a Pitot tube (Furnace

Control, FC014). Following the procedure suggested by van

Ommen et al. [17], the experimental setup was tested for

distortion of the amplitude and phase of fluctuations and no

significant influence was found for the range of frequencies

concerned in the study.

4. Experimental results

Fig. 5 presents power spectra of pressure fluctuations in

the bed and in the air-plenum when the unit is operated

with the high pressure drop air-distributor (Case a).

Compared to the other two cases, the power spectrum is

broad with an irregular peak region. The fluidization

regime was previously termed multiple bubble regime

[13] and is characterized by many bubbles, evenly

distributed over the cross-section of the bed. There are

no significant flow fluctuations recorded in the air-supplyduct in this case (pressure and flow waves are assumed to

be almost fully reflected from the air-distributor), since

the fluidized bed and the air-supply system are independ-

ent with respect to pressure fluctuations. Still, there is a

low amplitude peak of approximately 0.9 Hz in the

pressure spectrum of the plenum. The reason for this peak

is not known, but a possible explanation is that, in this

case, a standing wave was established in the air-supply

system due to operation of the fan against a nearly closed

valve [13].

Figs. 6–9 present power spectra of pressure fluctuations

in the bed and in the air-plenum, as well as the flowfluctuations in the air-supply duct for Cases b and c. At

U =0.8 m/ s and L¨0 (Case b, the flow is controlled by

Valve 2, Fig. 2), one identifies a regime termed single

bubble regime [13]. A piston-like movement of the bed as

a whole and a strong periodicity of formation and eruption

of bubbles are observed. The bubble rises through the

centre of the bed. Fig. 6a or c reveals that the dominant

frequencies (the centre points of the spectra) of the

pressure signals in the bed and in the air-plenum are

identical. The flow fluctuations in the air-supply system

show a dominant peak at the same frequency as the

pressure fluctuations in the bed and in the air-plenum (1

Hz, Fig. 6 b or d). At the same velocity and L =30 m (Case

c, flow control by Valve 2, while Valve 1 is fully open,

Fig. 2), it is not straightforward to recognize the fluid-

ization regime (Fig. 7a). Note that the vertical scale differs

from that in Fig. 6a. The dominant frequency in the bed

decreases to approximately 0.5 Hz, whereas the pressure

fluctuations in the air-plenum have two peaks, one that

coincides with the dominant frequency in the bed (0.5 Hz)

and one around 3 Hz that is not readily noticed in the bed

if linear scales are used on both axes. In logarithmic scales

one observes that the peak at 3 Hz is, indeed, present in

the bed, but with far lower energy than the 0.5 Hz peak

(Fig. 7c). The flow fluctuations show peaks whichcoincide with the dominant frequencies identified above

(0.5 Hz and 3 Hz), Fig 7d. Since the natural frequency of

the duct is estimated to be around 3 Hz, it is likely that the

corresponding frequency content in the air-plenum and to

some extent in the bed represents effects from the air-

supply system. In this regime, visual observation reveals

that the bubbles are not as large as in the single bubble

regime, and also that they are less uniform, despite the

same fluidization velocity in both cases (b and c).

At U =2.2 m/s and L ¨0 (Case b), bubbles of an

exploding character, extended vertically all the way from the

air-distributor to the surface of the bed, dominate the flow

Table 2

Experimental data and operating conditions

Temperature Ambient

Cross-section of the bed (m m) 0.120.7

Cross-section of the air-plenum

(mm)

0.80.8

Height of the unit (m) 8.5Bed material Silica sand

Particle diameter (mm) 0.32

Minimum fluidization velocity (m/s) 0.085

Terminal velocity of a single

particle (m/s)

2.0

Air-supply system Case a Case b: flow control

by Valve 1; Case c:

flow control by Valve 2

Fluidization velocity (m/s) 0.6 0.8; 2.20

Air-distributor pressure drop,

D P dist (Pa)

4200 1000 (at 0.80 m/s);

3200 (at 2.20 m/s)

Bed pressure drop, D P fb (Pa) 5050 5300 (Cases b and c )

Bottom bed height (m) 0.38 0.40 (at 0.80 m/s);

0.47 (at 2.20 m/s)

Volume, air-plenum (m

3

) – 0.45Volume, air-supply duct (m3)

(320 mm ID)

– 2.43

Flux of solids (kg/m2 s) Non circulating conditions: 0,

Circulating conditions: ¨ 1




field. A large portion of gas is in the form of a high

throughflow, exceeding the gas flow in the particulate phase

and in the visible bubble flow [18]. The dominant frequency

in the pressure fluctuations in the bed and the air-plenum

(Fig. 8a) coincides with the one observed in the single

bubble regime (1 Hz). The same can be said regarding the

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4x 105

frequency (Hz) frequency (Hz)

P S D ( P a

2 / H z )

P S D ( P a

2 / H z )

air-plenum

in-bed

10-2 10-1 100 101

100

105

air-plenum

in-beda) b)

Fig. 5. Case a, U =0.60 m/s. Power spectra of a) pressure fluctuations, b) pressure fluctuations (logarithmic scales).

a) b)

c)

Case b

20

16

12

P r e s s u r e [ k P a ]

P r e s s u r e [ k P a

]

1612

8

4

00 10 20

Length [m]

Length [m] Length [m]

Length [m]30

0 10 20 30

0

Flow control upstream of fan

Non-circulating conditions, u =0.8 m/s

Flow control near air-plenum

Non-circulating conditions, u =0.8 m/s

Flow control upstream of fan

Circulating conditions, u =2.2 m/s

Flow control near air-plenum

Circulating conditions, u =2.2 m/s

10 20 30

0 10 20 30

16

12

8

4

0

1612

8

4

0

16

12

8

4

0 P r e s s u r e [ k P a ]

P r e s s u r e [ k P a

]

P r e s s u r e [ k P a ]

Flow Q [m3 /s] Flow Q [m3 /s]

8

4

20

16

12

∆ p v

[ k P a ]

8

4

0 0.05 0.1 0.15 0.2 0 0.05 0.1 0.15 0.2

Non-circulating conditions

Non-circulating conditionsNon-circulatingconditions

Circulatingconditions

Case b

Case b

Case c

Case c

Circulating conditions

Circulating conditions∆p v

∆p v

∆p v∆p fan

∆p fan

Fig. 4. a: The characteristics of the fluidization fan and of Valve 2. The stars indicate time averaged operation points in the cases studied. b: The characteristics

of Valve 1. The stars indicate time averaged operation points in the cases studied. c: Time average pressure along the air-supply. The pressure drops at Valve 1

(left-hand diagrams) and Valve 2 (right-hand diagrams) and finally at the air-distributor.




flow fluctuations in the air-supply system (Fig. 8 b). Finally,

at U =2.2 m/s and L = 3 0 m (Case c), the dominant

frequency of the pressure fluctuations in the bed and in

the air-plenum (Fig. 9a) remains at 1 Hz (i.e., it is not

changed with a change of the configuration of the air-supply

system as in the low velocity case). Again, as was observed

at the lower velocity for the same configuration, the

fluctuations in the air-plenum show the presence of another

peak at around 3 Hz (Fig. 9a), not seen in the bed, even in

logarithmic scales (Fig. 9c). Figs. 8d and 9d illustrate that

the flow fluctuations have the same frequency distribution

as the pressure fluctuations recorded in the air-plenum.

5. Comparison with models

Table 3 summarizes the results of the fundamental

frequencies as obtained from the different literature expres-

sions and from the general model (Eq. (10)). In the

expression of Roy et al. [15], the velocity of the pressure

wave, us, was evaluated by the separated flow theory

discussed above.

All models yield a principal frequency lower than 3

Hz. In Case a there is no clear dominant bed frequency,

since there are many bubbles passing simultaneously

through the bed, all influencing the pressure measure-

ments. The maximum amplitude in the spectrum is around

3 Hz. The properties of this regime differ from the

assumptions of the models (with the dominant frequency

calculated by Eq. (5)) listed in Table 1 and of the model

(Eq. (6)) of Baskakov et al. [3], which is the reason why

the model results are not included in Table 3. In Case b

and at U =0.8 m/s, fair agreement is achieved, especially

with the model of Davidson. This is expected, since in

this case the model assumptions correspond to the

experimental configuration (insignificant damping and a bed resting on an isolated air-plenum). The model of

Baskakov et al. [3] somewhat over-predicts the measured

frequency, although the derivation of the model was

similar to that of Davidson. The fundamental frequency

calculated by the model of Roy et al. [15] under-predicts

the measured one, the reasons probable being the

difficulty to correctly evaluate the propagation velocity

of the pressure waves and the wavelength of oscillations

(k). The general model (Eq. (1) or (10)) successfully

predicts the absence of periodicity in Case a (the system

is overdamped and there are no oscillations), and a good

agreement with measurements is achieved in Case b.

0 2 4 6 8 100

2

4

6

8

10

12

14x 107

frequency (Hz)

P S D ( P a

2 / H z )

air-plenum

in-bed

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3x 10-3

frequency (Hz)

P S D

( Q 2 / H z )

flow into plenum

10-2

10-1

100

101

102

104

106

108

1010

frequency (Hz)

P S D ( P a

2 / H z )

air-plenum

in-bed

10-2

10-1

100

101

10-6

10-5

10-4

10-3

10-2

frequency (Hz)

P S D ( Q 2 / H z )

flow into plenum

a) c)

b) d)

Fig. 6. Case b , U =0.80 m/s. Power spectra of a) pressure fluctuations, b) flow fluctuations, c) pressure fluctuations (logarithmic scales), d) flow fluctuations

(logarithmic scales).




There is even a displacement of the principal frequency

because of the damping as seen in Eq. (10). In Case b

and at U =2.2 m/s, the model of Davidson (Eq. (5)) and

the present model (Eq. (10)) seem to yield the correct

dominant frequency of the in-bed pressure fluctuations,

although the fluidization regime and the pressure drop of

the air-distributor deviate from the assumptions of the

model (the model of Baskakov et al., Eq. (6), provides the

same result as in the low velocity case, since, according to

their model, the actual expanded bed height is irrelevant

for the calculation of the dominant frequency). In this

case, the exploding bubbles simply occur at approximatelythe same frequency as the single bubbles. The properties

of the exploding bubbles are not well understood at

present and their position (frequency) is not accurately

predicted. In Case c and at U =0.8 m/s, only Eqs. (5) and

(10) obtain satisfactory results for the lower peak of the

measured in-bed frequency (0.5 Hz, Table 3, Fig. 7a or c),

but only when the effective gas volume, represented by

H eq, comprises the volume of the entire air-supply system.

However, none of the models is able to reproduce the

higher frequency peak (around 3Hz, Table 3, Fig. 7c),

which was shown to originate from the air-supply system.

Finally, in Case c and at U =2.2 m/s, the assumptions in

the models differ from the fluidization situation in the bed

(the dominance of short-cuts of gas), and therefore, an

agreement between the measured and calculated frequen-

cies may be regarded as a coincidence.

As a further step in the analysis, the general model (Eq.

(1)), expressing the response of a fluidized bed to

disturbances, is tested. Power spectra of measured and

simulated pressure fluctuations in the air-plenum (which in

most cases agree with the in-bed pressure fluctuations) are

compared (Figs. 10 and 11) for the two cases when there is

an interaction of the bed with other parts of the system.

There are two possibilities to directly introduce flowfluctuations into Eq. (1), and both are investigated. First,

measured flow fluctuations are used as an input to

simulations (Fig. 10), although they cannot strictly be

regarded as independent signals, as requested by Eq. (1).

As far as frequency is concerned, the similarity between

Eq. (1) and the frequency models (Eqs. (5) –(7)) reflects

itself in a very good agreement with measurements in Case

b (Fig. 10a and c). In Case c, the model either reproduces

the lower frequency peak only (U =0.8 m/s, Fig. 10 b) or

does not reproduce the dynamics of the system at all

(U =2.2 m/s, Fig. 10d). Fig. 11 compares measurements

with simulations in Case b (single bubble regime) when

0 2 4 6 8 100

0.5

1

1.5

2x 10

6

frequency (Hz)

P S D ( P

a 2 / H z )

air-plenum

in-bed

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5x 10

-3

frequency (Hz)

P S D (

Q 2 / H z )

flow into plenum

10-2

10-1

100

101

102

103

104

105

106

107

frequency (Hz)

P S D ( P a

2 / H z )

air-plenum

in-bed

10-2

10-1

100

101

10-6

10-5

10-4

10-3

10-2

frequency (Hz)

P S D ( Q

2 / H z )

flow into plenum

a)

b)

c)

d)

Fig. 7. Case c, U =0.80 m/s. Power spectra of a) pressure fluctuations, b) flow fluctuations, c) pressure fluctuations (logarithmic scales), d) flow fluctuations





imposed signals are input to the simulations. Constant

amplitude of the signal and five frequencies from 0.3 to 4

Hz were arbitrarily chosen, as shown in the figure. The

fluctuations become significant only when the frequency of

the disturbances (1.1 Hz in this case) is close to the natural

frequency x0. This is a well known property of the

mechanical system described by Eq. (2). Similar results

were obtained for other cases.

Finally, a different approach from the ones used above is

tried. The general model (Eq. (1)) is understood as a linear

system with multiple inputs and a single output. Its responsecan be investigated in the form of a transfer function. A

block diagram of the system is presented in Fig. 12a, with

the corresponding input–output form

y ¼ G 1; G 2½ u1

u2

; ð15Þ

with y denoting the output of the system ( pap), u i two inputs

(external and internal disturbances) and G i transfer func-

tions of the subsystems (e.g. G 1 represents the relation

between the first input and the output). The response of such

systems is usually illustrated by Bode plots, which show the

amplitude change (from the input to the output) as a

function of the frequency of the input. The corresponding

transfer functions from Eq. (1) read:

G 1 sð Þ ¼ P ap sð ÞQex sð Þ ¼

C 3

C 1 s þ C 4

C 1

s2 þ C 2

C 1 s þ 1

C 1

ð16aÞ

G 2 sð Þ ¼ P ap sð Þ P in s

ð Þ ¼

1

C 1

s2 þ C 2

C 1 s þ 1

C 1

ð16bÞ

By using the data presented in Table 2, transfer functions

for the interaction cases (Cases b and c , for both velocities

used in this work) have been calculated. The corresponding

Bode plots are given in Fig. 12 b (for the fluidization

velocity of 0.8 m/s) and 12c (for U = 2.2 m/s) with the

contribution of each of the inputs presented separately. If we

compare Fig. 12 b and c with a Bode plot of an idealized

second-order mechanical system (available in any textbook

dealing with system modeling and control, e.g. [19]), we

may realize the following: an intersection of low-frequency

and high-frequency asymptotes determines the natural

0 2 4 6 8 100

0.5

1

1.5

2

2.5x 10

7

frequency (Hz)

P S D ( P

a 2 / H z )

air-plenum

in-bed

0 2 4 6 8 100

0.002

0.004

0.006

0.008

0.01

frequency (Hz)

P S D (

Q 2 / H z

)

flow into the plenum

10-2

10-1

100

101

102

104

106

108

frequency (Hz)

P S D ( P

a 2 / H z )

air-plenum

in-bed

10-2

10-1

100

101

10-6

10-5

10-4

10-3

10-2

frequency (Hz)

P S D (

Q 2 / H z )


a)

b)

c)

d)

Fig. 8. Case b , U =2.20 m/s. Power spectra of a) pressure fluctuations, b) flow fluctuations, c) pressure fluctuations (logarithmic scales), d) flow fluctuations





frequency (in system control, this frequency is usually

referred to as cornering frequency), which corresponds to

the values obtained from Eq. (10) presented in Table 3.

Furthermore, from the shape of the curves, it is seen that in

all the cases studied the systems are dearly underdamped

(nb1) and finally, all curves experience a high-frequency

roll-off of 40 dB/decade (which is a general property of

second-order linear systems). The analysis presented is not

able to point out the existence of the higher-frequency peak

observed in the bed and the air-plenum in Case c (U =0.8

m/s) or in the air-plenum only (Case c, U =2.2 m/s).

In the end, the coherence analysis, Eqs. (11)–(13), is

applied to cases where the interaction between bed and air-

supply system is expected. Fig. 13a and b show that in the

single bubble regime (Case b, U = 0.8 m/s), pressure

fluctuations in the bed and in the air-supply system are

strongly coherent. At the same fluidization velocity, but

with a longer air-supply duct, Fig. 13c illustrates that the in-

bed and air-plenum pressure signals are predominantly

coherent at 0.5 Hz (there is also some incoherence at that

frequency too, Fig. 13d), but also that there is some

coherence at 3 Hz (effects from the air-supply system). At

0 2 4 6 8 100

0.5

1

1.5

2 x 10

6

frequency (Hz)

P S D (

P a

2 / H z )

air-plenum

in-bed

0 2 4 6 8 100

0.005

0.01

0.015

0.02

0.025

0.03

frequency (Hz)

P S D (

Q 2 / H z

)


10-2

10-1

100

101

102

103

104

105

106

107

frequency (Hz)

P S D ( P

a 2 / H z )

air-plenum

in-bed

10-2

10-1

100

101

10-6

10-5

10-4

10-3

10-2

10-1

frequency (Hz)

P S D (

Q 2 / H z )


a)

b)

c)

d)

Fig. 9. Case c, U =2.20 m/s. Power spectra of a) pressure fluctuations, b) flow fluctuations, c) pressure fluctuations (logarithmic scales), d) flow fluctuations


Table 3

Measured and modeled dominant frequencies for the cases investigated

Case Measured frequency (Figs. 5–9) Modeled frequency (Hz)

Fan et al.; Moritomi et al.;

Davidson; Eq. (5)

Baskakov et al.;

Eq. (6)

Roy et al.;

Eq. (7)

Eq. (10)

Case a No clear peak. Centre of spectrum at

3 Hz in the bed

/ / 0.63 No oscillations

Case b (0.8 m/s) 1 (bed and plenum) 1.10 1.57 0.60 1.06

Case b (2.20 m/s) ¨ 1 (bed and plenum) 1.10 1.57 0.60 1.04

Case c (0.8 m/s) 0.5 (bed and plenum) 3.5 (bed and plenum) 1.10a (0.44 b) 1.57 0.51 1.06a (0.33 b)

Case c (2.20 m/s) ¨ 1 (bed and plenum) 3.5 (air-plenum only) 1.10a (0.44 b) 1.57 0.51 0.67a (0.27 b)

a Volume of the air-plenum. b Volume of the entire air-supply system.




U =2.2 m/s and for both interaction Cases (b and c), the

signals are coherent at the frequency of the exploding

bubbles (Fig. 14a and c). Fig. 14a also reveals that the

coherence is almost zero at 3 Hz, indicating that the effects

from the air-supply system are not felt in the bed in this

case. It is likely that the air-distributor pressure drop is high

enough to almost eliminate the coupling between the bed

and the rest of the system at that frequency. Fig. 14 b and d

simply show that the signals are not entirely coherent at the

frequency of the exploding bubbles.

6. Discussion

The application of Eqs. (5–7) shows that the dynamical

behavior of the fluidized-bed system is successfully

reproduced when the model assumptions correspond to the

configuration of the case treated. In other cases, the modelsgive only a part of the information about the events taking

place in the system (Case c, U =0.8 m/s), or do not provide

correct information at all. The question is: can the results be

improved? One reason for uncertainty is that the effective

gas volume to be used in calculations in Case c is not

necessarily the same as the one in Case b. It is physically

sound to take the entire air-supply system (air-plenum +

pipe) as an active volume in Case c, but it could be

questioned whether the entire volume is equally active, or if

it is necessary to include energy losses in long supply lines.

As indicated in Table 3, the inclusion of the entire volume

correctly predicts the frequency shift (from 1 Hz to 0.5 Hz)

0 2 4 6 8 100

2

4

6

8

10

12

14 x 10

7

frequency (Hz)

P S D (

P a 2 / H z )

measured

simulated

0 2 4 6 8 100

0.5

1

1.5

2 x 10

6

frequency (Hz)

P S D ( P

a 2 / H z )

measuredsimulated

0 2 4 6 8 100

0.5

1

1.5

2

2.5 x 10

7

frequency (Hz)

P S D (

P a 2 / H z )

measuredsimulated

0 2 4 6 8 100

0.5

1

1.5

2 x 10

6

frequency (Hz)

P S D (

P a 2 / H z )

measured

simulated

a)

c)

b)

d)

Case b Case c

Case b Case c

Fig. 10. Power spectrum of air-plenum pressure fluctuations, a) U = 0.80 m/s, Case b, b) U =0.80 m/s, Case c, c) U = 2.20 m/s, Case b, d) U =2.20 m/s, Case c.

0 2 4 6 8 100

0.5

1

1.5

2x 10

8

frequency (Hz)

P S D

( P a 2 / H z )

solid line: measured

dashed lines: simulated

Case b

Fig. 11. Power spectrum of air-plenum pressure fluctuations, Case b.

Simulations with imposed disturbances.




when the interaction of the bed with the entire air-supply

system is expected (Case c, U =0.8 m/s). A similar type of

uncertainty is encountered when the fundamental frequency

is calculated by the model (Eq. (7)) of [15]. The assumption

that k = 4 H , made in the derivation of the model, requires

that the node and the antinode be located at the air-

y

G1

G2

u1

u2

+

+

10-2 10-1 100 101 102

From: Input 2

Frequency (rad/s)

10-2 10-1 100 101 102

-40

0

40

80

120

From: Input 1

Frequency (rad/s)

M a g n i t u d e ( d

B )

solid line: Case b

dashed line: Case c

solid line: Case b

dashed line: Case c

U = 0.8 m/s

10-2 10-1 100 101 102

From: Input 2

Frequency (rad/s)

10-2 10-1 100 101 102

-40

0

40

80

120

From: Input 1

Frequency (rad/s)

M a g n i t u d e ( d B )

solid line: Case b

dashed line: Case c

solid line: Case b

dashed line: Case c

U = 2.2 m/s

a)

b)

c)

Fig. 12. a: Block diagram representation of the dynamical system of Fig. 2. b: The frequency response of the linear dynamic model (Eq. (1)) in the cases where

interaction between the bed and the air-supply system is expected, b) U =0.8 m/s, c) U =2.2 m/s.




distributor and at the bed surface, respectively. This is

fulfilled in Case a, since the high pressure drop over the air-

distributor gives rise to an almost complete reflection of

waves from the distributor. In Cases b and c, the air-

distributor is not a fixed boundary of the system as in Case

a, and the node is shifted from the air-distributor to a

position somewhere in the air-supply system. Then the

wavelength (k) becomes larger than 4H, and this decreases

the fundamental frequency. Furthermore, the inability of all

models treated to predict the presence of another peak of

measured pressure fluctuations in the bed and in the air- plenum (at 3.5 Hz, Table 3) is not surprising, since the

system is more complex than assumed by the models.

Pressure and flow pulsations from the air-supply system and

their interaction with the waves from the bed, which may be

the cause for the change of the bed dynamics (shown by the

changes in the power spectrum compared to Case b), are not

accounted for in any of the models. Finally, the assumption

that the entire bed area is active introduces an uncertainty.

Perhaps only the part of the bed that is lifted by a bubble

should be considered [20].

The general model (Eq. (1)) takes influencing factors,

such as damping and disturbances, into account, but there is

still a significant disagreement with the measurements in

Case c. Although the model explicitly considers the flow

fluctuations ( qex in Eq. (1)) in the air-supply system, their

effect is not readily observed in the power spectra of

simulated pressure fluctuations in the air-plenum (Fig. 10 b).

Therefore, it seems that the fundamental frequency of the

system (measured both in the bed and in the air-plenum,

Table 3) is successfully reproduced by the model, whereas

the influence of the air-supply system is not. It may be

argued that this is a result of the linearization of the

governing equations (the inertial forces, responsible for theonset of the instabilities in the system, are neglected), and

that thereby only small fluctuations of the relevant quantities

(listed in Table 1) may be studied. However, the principal

drawback with the modeling procedure is that the dis-

turbances, which are input to the model, are affected by the

bed itself. Therefore, it is not mathematically justified to use

the signals measured during operation as an external

disturbance. On the other hand, if fluctuations are imposed

as requested by the model, the information obtained (Fig.

11) is limited to what is conventional for this type of

mathematical descriptions (Eq. (2)). For this reason, despite

this deviation from the mathematical correctness, measured

0 2 4 6 8 100

2

4

6

8

10

12

14x 106

frequency (Hz)

C O P x y ( P a

2 / H z )

0 2 4 6 8 100

2

4

6

8

10

12

14x 106

frequency (Hz)

I O P x y ( P a

2 / H z )

0 2 4 6 8 100

1

2

3

4

5x 105

frequency (Hz)

C O P x y ( P a

2 / H z )

0 2 4 6 8 100

1

2

3

4

5x 105

frequency (Hz)

I O P x y ( P a 2 / H

z )

a)

c)

b)

d)

Case b Case b

Case c Case c

Fig. 13. Decomposition of power spectrum of pressure fluctuations in the bed. U =0.80 m/s, a) coherent output (COP xy), Case b, b) incoherent output (IOP xy),

Case b, c) coherent output (COP xy), Case c, d) incoherent output (IOP xy), Case c.




disturbances were applied as input and in one case (Case b)

an excellent agreement is found, whereas this cannot be said

about Case c. In the latter case, the transient effects from the

air-supply system, not accounted for in the model, may have

played an important role. The fluidized bed, the air-plenum,

the air-supply line and the fan, indeed form a coupled system,

as indicated by the dashed system limit in Fig. 3, but, because

of the very nature of the model (one differential equation with

predefined disturbances), it is not possible to extract any

additional information, although the models seem to be

formulated in a general way. Therefore, an understanding of the interaction of the pressure waves generated in a bed with

pressure and flow pulsations from the air-supply system is

needed to solve the problems in Case c. In order to do so, a

method which would recognize the waves from measured

pressure fluctuations is required. For example, in the single

bubble regime, although the correct dominant frequency is

obtained by Eq. (10), the model does not provide a physical

explanation for the mechanisms responsible for the inter-

action between the bed and the remaining part of the system.

The reason for the interaction is probably the creation of a

pressure wave after the collapse of the bed. The wave may

propagate throughout the entire system, since the resistance

of the air-distributor is low. Existence of a sharp peak in the

coherent part of the power spectrum of in-bed pressure

fluctuations (Fig. 13a) indicates that a pressure wave is

indeed created in the bed. The wave is able to propagate both

upwards and downwards in the bed from the point of origin.

The amplitude of the wave does not decrease in the

downward direction [16] and thus, the wave is readily

detected in the air-plenum. Since the peaks in the power

spectrum of pressure fluctuations in Case b (Figs. 6a and 8a)

fully coincide with the peaks in the spectrum of the coherent

part (Figs. 13a and 14a), it is evident that in this case, theinteraction between the bed and the air-supply system is

determined exclusively by pressure waves created in the bed.

At the same fluidization velocity (0.80 m/s) but with a

different configuration of the air-supply system (Case c), the

fluidization situation is different, as shown before. This

implies that, apart from the pressure waves generated in the

bed (determined by the peak at 0.5 Hz in Fig. 7a or c), the

effects from the air-supply system (detected at the natural

frequency of the air-supply duct) significantly influence the

dynamics of the bed. This influence decreases as the

fluidization velocity increases (due to the increase of the

pressure drop over the air-distributor).

0 2 4 6 8 100

0.5

1

1.5

2x 106

frequency (Hz)

C O P x y ( P

a 2 / H z )

0 2 4 6 8 100

0.5

1

1.5

2x 106

frequency (Hz)

I O P x y ( P

a 2 / H z )

0 2 4 6 8 100

2

4

6

8

10x 105

frequency (Hz)

C O P x y ( P a

2 / H

z )

0 2 4 6 8 100

2

4

6

8

10x 105

frequency (Hz)

I O P x y ( P a

2 / H z )

a)

c)

b)

d)

Case b Case b

Case c Case c

Fig. 14. Decomposition of power spectrum of pressure fluctuations in the bed. U =2.20 m/s, a) coherent output (COP xy), Case b, b) incoherent output (IOP xy),

Case b, c) coherent output (COP xy), Case c, d) incoherent output (IOP xy), Case c.




The importance of flow waves on the dynamics of the

system also represents an important step in investigating t he

Case c problems. Moritomi et al. [6] and Fan et al. [2]

introduced flow fluctuations in the air-supply system as an

external noise to the bed and investigated the influence of

flow fluctuations on the dynamical response of the system,

represented by a chosen (single) property (Table 1). Astrong correlation was found between the flow fluctuations

in the air-su pply system and the fluctuations of pressure in

the plenum [6] or of bubble residence time [2]. Johnsson et

al. [11] investigated the dynamics of a complex system,

consisting of a fluidized bed, pipes, valves, a fan and a

plenum. They measured pressure fluctuations in the bed and

in the plenum, as well as the flow fluctuations in the air-

supply system. Pressure and flow fluctuations in the supply

system were also obtained from continuity and momentum

equations for one-dimensional non-stationary fluid flow.

However, all these studies have so far discussed the

influence of flow fluctuations on the fluidized bed’sdynamics. The study of the interaction remains to be done

and would have to comprise also the feedback of the events

in the bed on the pressure and flow pulsations in the air-

supply system.

7. Conclusions

The purpose of this paper is twofold: firstly, to discuss

the present knowledge concerning the dynamical features

of gas– solid fluidized beds expressed by models, and

secondly, to investigate whether these models are sufficient

to reproduce the dynamics of the system in a number of

cases with different configurations of air-supply system. A

general model describing the response of the bed to

disturbances is formulated and its applicability to three

distinct configurations of a chosen fluidization system is

studied. Also, expressions describing the fundamental

frequency of a fluidized-bed system are tested. It is shown

that these expressions are a special case of the general

models, when damping and disturbances are not accounted

for. Comparison of the results from the model with

experiments shows that the model can explain the

dynamics of the system when the bed is decoupled from

the other parts of the system (Case a), and when the bedinteracts with the air-plenum only (Case b). Experiments

show that in Case c, the dynamics of the bed (under non-

circulating conditions) are the result of the following

events: 1) pressure waves generated in the bed and felt

through the entire system, recognized as the coherent part

of the cross power spectra of measured in-bed and air-

plenum pressure fluctuations; 2) local phenomena not to be

felt in other parts of the system, represented by the

incoherent part of the cross power spectra of the signals

treated, and 3) the effects from the air-supply system

recorded at the natural frequency of the air-supply ducts.

The model only partially reproduces the dynamics of Case

c, since it does not reproduce the effects from the air-

supply system, although the flow fluctuations in the air-

supply ducts are explicitly taken into account. Also, it is

shown that no further information can be obtained from

the given models to cover such a case. Accordingly, there

is a need for further information. At high velocity (close to

the terminal velocity of a single particle), the beddynamics are dominated by structures (termed exploding

bubbles) which constitute a gas by-pass from the air-

distributor to the splash zone. The position of the peak that

represents the exploding bubbles does not depend on the

configuration of the air-supply system. The position (in the

frequency spectrum) of these bubbles cannot be predicted

at present. In some cases the exploding bubbles seem to be

superimposed on normal bubbles and they obtain the same

frequency. In high velocity cases, the effects from the air-

supply system are not felt in the bed, because of the

increased resistance of the air-distributor.

It seems that the understanding of the interaction of the pressure waves, generated in the bed, with pressure and flow

pulsations from the air-supply system is a key to explain the

dynamic behavior of Case c problems. Hence, an analysis

should include all relevant components inside the boundary

defining the dynamical system (Fig. 3). Modeling of the

interaction between certain parts of a fluidized bed system

will then involve the coupling between phenomena, such as

two-phase turbulent flow, hydrodynamic instabilities and

the wave propagation.

Notation

Aap cross-sectional area of the air-plenum (m2)

A bed cross-sectional area of the bed (m2)

AB cross-sectional area of a bubble (m2)

c0 speed of sound (m/s)

C i (i = l– 4) coefficients (expressions (3) and (4))

C xy coherence between measurement positions

COP xy coherent part of the cross spectral density (Pa2/Hz)

d p particle diameter (m)

f 0 frequency of nondamped oscillations (Hz)

f frequency of oscillations (Hz), damping included

F (t ) external excitations (Eq. (2))

g gravity (m/s2)

G transfer function of the system

G i (i = 1,2) transfer functions between the output and the i -th input, Eqs. (16a), (16b)

G p mass flow rate of bursting particles (kg/s/m2)

H bed bed height (m)

H mf bed height at minimum fluidization velocity (m)

H eq= V ap / Aap effective height of the plenum (m)

H eq= (V ap+ V pipe)/ Aap effective height of the entire air-

supply system (m)

IOP xy incoherent part of the cross spectral density

(Pa2/Hz)

K parameter (Eq. (A3))

m parameter (Eq. (A3)), (1 < m <2), m =1.5 used in

simulations




P pressure (Pa), (Eq. (A4))

P ap pressure in the air-plenum (Pa)

b P ap pressure in the air-plenum (Pa), mean value

pap pressure in the air-plenum (Pa), fluctuating part

P ex pressure, external disturbances (Pa)

b P ex pressure, external disturbances (Pa), mean value

pex pressure, external disturbances (Pa), fluctuating part

P fb pressure in the bed (Pa)

b P fb pressure in the bed (Pa), mean value

pfb pressure in the bed (Pa), fluctuating part

P in pressure pulse from internal disturbances (Pa),

Eq. (1)

P V

pressure in the freeboard (Pa), Fig. (A2)

P xy, P yy power spectral density of pressure signals from the

air-plenum and the bed, (Pa2/Hz)

P xy cross spectral density of pressure signals from the

air-plenum and the bed, (Pa2/Hz)

Qap volume flow rate of air in the air-plenum (m3

/s)bQap volume flow rate of air in the air-plenum (m3/s),

mean value

qap volume flow rate of air in the air-plenum (m3/s),

fluctuating part

Qex volume flow rate of air, external disturbances

(m3/s)

bQex volume flow rate of air, external disturbances

(m3/s), mean value

qex volume flow rate of air, external disturbances

(m3/s), fluctuating part

Qfb volume flow rate of air in the bed (m3/s)

bQfb volume flow rate of air in the bed (m3/s), mean

value

qfb volume fl ow rate of ai r i n t he bed (m3/s),

fluctuating part

Qmf volume flow rate of air at minimum fluidization

velocity, (m3/s)

s Laplace operator

t time (s)

t n period of oscillations (s), Eq. (6)

uc velocity of pressure waves (m/s), compressive

waves

u1, u2 inputs to a linear dynamical model, Eq. (15)

U eq equivalent gas velocity (m/s)

U mf gas velocity at minimum fluidization (m/s)U s solids velocity (m/s), Eq. (A4)

V ap volume of the air-plenum (m3)

V pipe volume of the air-supply ducts (m3)

V B volume of the bubble (m3)

x vertical coordinate (s), Figs. (A1), (A2)

x(t ) property of a mechanical system, Eq. (2)

Y output of a linear dynamical model, Eq. (15)

Greek symbols

D pfan, D pv pressure drop across fan and valve, respectively(Pa), Fig. 4a and b

D P dist pressure drop over the air-distributor (Pa)

D P fb pressure drop over the bed (Pa)

e, emf voidage, voidage at minimum fluidization

j ratio of specific heats

k the wavelength of the oscillations (m)

qap, qfb density of air in the air-plenum and in the bed

(kg/m3)

q bed bulk density of the bed (kg/m3)

qex density of air in the air-supply duct (kg/m3),

Eq. (A1)

qg density of gas (kg/m3), Eq. (7)

qmf density at minimum fluidization (kg/m3)

qs density of the solid material (kg/m3), Eq. (7)

sP particle relaxation time (s)

x0 natural frequency of a mechanical system (rad/s)

x angular frequency of oscillations (waves) (rad/s)

n damping factor ( –)

Acknowledgement

This work was financed by the Swedish Energy Agency.

Appendix A

A.1. Mathematical model of the response of a fluidized bed

to disturbances

The dynamical system of interest is presented in Fig. 2.

The main assumptions of the model are:

& Pressure fluctuations in the bed and in the air-plenum are

the result of external disturbances from the air-supply

system (flow fluctuations) and from the action of

bubbles.

air distributor

Pex, Qex

Pap, Qap

air plenum

x

Fig. A1. Mass balance over the air-plenum.

air distributor

Qfb x

fluidized bed

Poo

+ Pin

Hmf

Fig. A2. Momentum balance-fluidized bed.




& The entire bed mass is assumed to move vertically or to

move and expand at the same time, whereby the mass of

the bed remains constant.

& The model assumes a periodic eruption of bubbles (or of

a single bubble) at the surface of the bed.

The derivation procedure comprises the following balances.

1) Mass balance over the air-plenum (Fig. A1)

V ap

dqap

dt ¼ qexQex qapQap ðA1Þ

If isothermal flow is assumed, Eq. (A1) becomes

V ap

d P ap

dt ¼ P exQex P apQap ðA2Þ

By introducing mean and fluctuating quantities:

( Qex= bQex+ qex; P ex= b P ex+ pex) and ( Qap= bQap+ qap;

P ap=b P ap

+ pap), assuming that

b P ex

=b P ap

, b

Qex

=bQap

(mean values) and neglecting the product of fluctuating

components, Eq. (A2) finally becomes

V ap

d pap

dt ¼ b P ap qex qap

þ bQap pex pap

: ðA3Þ

2) Momentum balance-fluidized bed (Fig. A2)

A 1-D model is adopted

q bed

BU s

Bt þ U s

BU s

B x

¼ B P

B x q bed g ðA4Þ

where q bed is defined as: q bed= qs(1 q).

Neglecting the inertial forces due to the fluid motion in

Eq. (A4), we have (BU s / Bt )Y(dU s / dt )=(d/dt )(( QfbQmf ) / A bed)). This expression indicates that the bed is

accelerated because of the bubble activity ( QfbQmf is

the difference of the flow of gas above the distributor in the

bed and that at minimum fluidization velocity. According to

the two-phase theory of fluidization, this flow is transported

as bubbles). Strictly speaking, because of the assumptions

made, the integration (with respect to the coordinate ‘‘ x’’,

where ‘‘d x’’ is understood as the displacement of the bed

because of bubbles) of such an equation will be valid only if

the bed height is close to the one at incipient fluidization

( H mf ). In that case the bed density becomes qmf .

We have: qmf (1 / A bed)(dQfb / dt ) = (d P / d x)qmf g andafter integrating from 0 to H mf , the following expression is

obtained

qmf

H mf

A bed

dQfb

dt ¼ P fb P V þ P inð Þ qmf gH mf :

By introducing mean and fluctuating components for

pressure and volume flow rate in the bed ( Qfb= bQfb+ qfb;

P fb= b P fb+ pfb) and assuming that b P fb P V

= qmf gH mf ,

we finally come to

qmf H mf

A b

dqfb

dt ¼ pfb P in: ðA5Þ

The term P in in Eq. (A5), which represents the pressure

disturbance originating from the bubble eruption at the bed

surface [3], acts like a decelerating force in the model. The

model for P in is presented below.

3) Pressure drop over the air-distributor (Fig. A3)

P ap P fb ¼ K Q

m

ap: ðA6ÞAfter introducing mean and fluctuating values and

linearizing Eq. (A6) ( K (bQ ap+ q ap )m K bQmap+ mK

bQm1ap qap), we arrive to

pap pfb ¼ mK bQm1ap qap: ðA7Þ

The expression that will close the system of equations is

the link between the flows on both sides of the air-

distributor (qap and qfb being the density of air in the

corresponding positions):

qfb

¼

qapqap

qfb

:

ðA8

ÞTherefore, the system of equations, which describes the

dynamic behavior of the fluidized bed under external and

internal disturbances, is

V ap

d pap

dt ¼ b P ap qex qap

þ bQap pex pap

:

qmf H mf

Ab

dqfb

dt ¼ pfb pin:

pap pfb ¼ mK bQm1ap qap:

qfb ¼ qapqap

qfb

:

Finally, from the system of equations Eqs. (A3), (A5),

(A7), and (A8), the following linear differential equation for

the fluctuating pressure in the air-plenum is derived

qap

qfb

H mf qmf

A bed

V ap

b P ap

d2 pap

dt 2 þ V ap

P ap

mK bQm1ap

d pap

dt þ pap

¼ qap

qfb

H mf qmf

A bed

dqex

dt þ mK bQm1

ap qex þ P in;

Having in mind that V ap= Aap H eq and D P dist = K bQmap, the

coefficients (expressions (3) and (4)) are obtained. Ingeneral, the ratio of the air densities (qap / qfb) is assumed

to be independent of time in the derivation procedure. For

ambient conditions this term is close to one.

Qfb

Qap

air distributor

Fig. A3. Pressure drop over the air-distributor.




A.2. Modeling of the internal disturbances

Internal disturbances are represented by the effect of a

bursting bubble at the surface of the bed. In this work, a

model similar to the one of [6] is used. By doing so, we do

not claim that bubble eruption is necessarily the only and

dominant source of pressure fluctuations in fluidized beds.The exact origin of pressure fluctuations in freely bu bbling

beds is still not straightforward in literature. In Ref. [16] it

has been shown that bubble formation and bubble coales-

cence are the dominant effects taking place in a bed with

respect to pressure fluctuations. On the other hand,

numerous publications (e.g., [1,21–26], among others)

claim that the main sources of fluctuations are significantly

closer to the surface of the bed (the conclusions are

supported by presenting the dependence of the amplitude

of pressure fluctuations on the bed height). Furthermore, it

is well known that the information obtained significantly

depends on the type of the probe used (single vs. differential pressure measurements, see [25] or [8]). In summary,

supported by this evidence, bubble eruption was chosen as

representative of internal effects in the bed. The existence of

other sources within the bed does not change the nature of

the model proposed.

The following quantities are introduced:

Bubble eruption at the surface (Fig. A4) generates the

pressure fluctuation, and the pressure pulse, as the differ-

ence between the pressure drop of the bed before and after

the event ( P in=D P beforeD P after ), is assumed to have a

following form:

P in ¼ G PjG Pjq b

AB

A bed

ðA9Þ

The event of the bubble eruption in the time domain will

look according to the Fig. A5, where t 1 (s) is the duration of

the bubble eruption, while t 2 (s) represents the time interval

between two bubble eruptions.

Finally, assuming that only one bubble erupts at the timeat the surface of the bed (or many bubbles simultaneously),

the mass flow rate of the bursting particles is modeled as:

G P ¼ v q bedV B

AB

1

t 1 F t ð Þ; ðA10Þ

where the function F (t ) controls that: G Pm0 while 2t 1 and

G P= 0 ‘‘outside’’ 2t 1.This is achieved by using the Heaviside step function in

the following way:

H t nt 2ð Þ H t nt 2 2t 1ð Þ ¼ 1; while 2t 10; otherwise:

The second part of the function F (t ) from (A10) simply

has to reproduce the behavior represented by the graph

G P= f (t ) (Fig. A5). Therefore, we have:

G P ¼ v q bedV B

AB

1

t 1 H t nt 2ð Þ H t nt 2 2t 1ð Þf g

p

2 sin p

t nt 2

t 1

: ðA11Þ

References

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AB cross-sectional area of a bubble (m2);

G P mass flow rate of bursting particles (kg/s/m2);

V B volume of the bubble (m3)

m the volumetric ratio of bursting particles to a bubble.

air distributor

fluidized bed ∆Pbefore

bursting

bubble

∆Pafter

Fig. A4. Bubble eruption at the bed surface.

t

GP

2t1

t2

Fig. A5. Model: the event of bubble eruption in the time domain.




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