Chemical Engineering Science 58 (2003) 2515–2518www.elsevier.com/locate/ces

Entropy of !uidized bed—a measure of particles mixing

Miroslav Pun*coch+a*r∗, Ji*r+. Draho*sInstitute of Chemical Process Fundamentals, AS CR, 165 02 Prague 6, Czech Republic

Received 7 October 2002; received in revised form 5 February 2003; accepted 10 February 2003

Abstract

The paper presents an attempt to de6ne the physical entropy of the dense phase of a !uidized bed, based on liquid-like propertiesof !uidized systems. The quantity U was used as the analogy of temperature in classical thermodynamics. The obtained expression forphysical entropy was compared to the correlation suggested for the Kolmogorov entropy in bubbling bed.? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Entropy; Fluidization; Kolmogorov entropy; Powders

1. Introduction

Fluidized beds exhibit very complex hydrodynamics dueto the non-linear interactions between the two independentmedia with their own individual movement tendencies—theparticles and the !uid. Thus, the state and dynamics of par-ticle motion is one of the most important characteristics ofthe bed. It is however di>cult to measure directly the mo-tion of particles (see e.g. the review by Yates & Simons,1994) and therefore the main eCort has been aimed at ex-planation of the behaviour of !uidized bed on the basis ofbubbles measurements. The time-averaged quantities as su-per6cial gas velocity, bed voidage or average bubble diam-eter have been used in modelling the !uidized beds. Theproblem is that this approach did not take into considerationthe time-dependent dynamical behaviour of the bed. Onlysince the early 1980s, the statistical analysis of time seriesof pressure and/or voidage !uctuations has been applied forcharacterization of !uidized beds. However, all these anal-ysis techniques use the linear theory of random variables orFourier analysis of signal, which is also linear in principle.Stringer (1989) was the 6rst to suggest that !uidized bedis in its nature a non-linear, chaotic system and thus chaosanalysis can deliver a relevant mathematical apparatus fordescription of its dynamics. Since that time a lot of work hasbeen done in the chaos analysis of !uidizing systems (seee.g. Zhao, Chen, & Yang, 2001 and references therein).

∗ Corresponding author. Tel.: +420-2-2039-0298;fax: +42-02-2092-0661.

E-mail address: punc@icpf.cas.cz (M. Pun*coch+a*r).

One of the most important chaos characteristics is theKolmogorov entropy, which measures the rate of loss of in-formation and quanti6es the limited predictability of chaoticsystem. Kolmogorov entropy is large for very irregular dy-namic behaviour, while it is small in case of more regular,periodic like behaviour. Formal de6nition of Kolmogoroventropy is given e.g. in Hilborn (1994). Schouten, Van derStappen, and Van den Bleek (1996) presented an interest-ing attempt to relate the Kolmogorov entropy to bubblingbed design parameters. They used the time series of pres-sure !uctuations in bubbling bed for direct calculation ofKolmogorov entropy and derived the following equation:

K = 19:3(U − Umf)0:4 DTH 1:6s: (1)

The correlation of experimental data using Eq. (1) was fairlygood for a given type of particle in a wide range of super6cialgas velocities and bed ratios (Hs=DT ). The investigation ofdata from various particle systems however revealed thatcorrespondence only could be obtained when the correlationwas changed in the following way:

K = 10:7(U − UmfUmf

)0:4 D1:2T

H 1:6s: (2)

The connection between the Kolmogorov and physicalentropy is not straightforward, see e.g., Latora and Baranger(1999). Moreover, according to authors’ knowledge, the def-inition of entropy in !uidized bed does not exist yet. Thiscontribution presents an attempt to 6nd the physical entropyof the emulsion phase of !uidized bed and its relation toEq. (2).

0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0009-2509(03)00115-5

2516 M. Pun)coch*a)r, J. Draho)s / Chemical Engineering Science 58 (2003) 2515–2518

2. Theory

Let us start with characterization of transition to !uidiza-tion. In a 6xed bed the particles are in constant positionsand dissipation of energy is due to friction forces only. Byincreasing the gas velocity we reach the threshold of !u-idization where the weight of the bed is balanced by frictionforces. From this point, the suspension of particles exhibitssome features analogous to the liquids, e.g. expansion, vis-cosity or miscibility. When we further increase gas velocity,the bed starts to expand. In such a way we arrive at the crit-ical value, the minimum bubbling velocity Umb, where thebubbles commence. In the !uidized systems consisting ofthe particles of Geldart B group (Geldart, 1973), the expan-sion is negligible and the bubbles appear immediately afterthe threshold of !uidization.

Furukawa and Ohmae (1958) showed that the inter-particlepotential forces causing motion of particles in !uidizedsystems may be compared with the intermolecular forcesin liquid systems. While in gases and liquids, temperatureis a good measure of the intensity of thermal motion, fora !uidized system the quantity �U can serve as a measureof the extent of particle motion. Later on Gelperin andEinstein (1971) stated that the kinetic energy of solid par-ticles in a bed is a following function of linear velocity U :

QWk = 12 mp Qu2

p = f(U ): (3)

It can be seen from Eq. (3) that in homogenous !uidizedbed the gas velocity plays the same role as temperature inliquids.

In what follows we shall try to de6ne the entropy of emul-sion phase in a bubbling bed. According to two-phase the-ory of !uidization (Fan & Zhu, 1998), the emulsion phaseof bubbling !uidized bed can be approximately describedby the linear velocity Umf and voidage �mf. Thus, we shalluse the analogy between liquid and suspension of particlesin homogenous !uidized beds.

If we consider the balance of energy in a homogenous bed(particulate !uidization), we obtain according to Molerus(1993) for the total power required to keep the system !u-idized,

Etot = �(1 − �)�s + ��f�gHAU: (4)

Of this, the part E! = �fgHAU is needed to lift the !uidover the height H . The term Edis = (1 − �)(�s − �f)gHAUrepresents the power needed for !uidization of particles. ForU ¿Umf we can write

REdis(Umf; U ) = (1 − �)(�s − �f)gHA(U − Umf): (5)

REdis represents the increase of power, which is con-sumed by the kinetic energy of the motion of particles inthe bed. We suppose that this motion is due to the risingbubbles, which cause the mixing of particles. As shownby Pun*coch+a*r, Draho*s, *Cerm+ak, and Seluck+y (1985) and

Draho*s, *Cerm+ak, and SchSugerl (1988), the appropriate mea-sure of bubbling intensity (i.e. also mixing intensity) is astandard deviation of pressure !uctuations �P , which is alinear function of gas velocity when measured in the lowerpart of the freely bubbling zone:

�P ≈ (U − Umf): (6)

Later on, Van der Schaaf, Schouten, Johnsson, and Van denBleek (1998) con6rmed that the standard deviation of pres-sure !uctuations correlates well with the bubble diameter.Thus, based on the above facts, we can consider the lineargas velocity as an analogous quantity to the temperature alsoin the case of the emulsion phase of bubbling beds.

The total drag force needed for !uidization is constantand is given by the expression:

G∗ = (1 − �)(�s − �f)gHA: (7)

Note that although the total drag force !uctuates in timefor U ¿Umf, we can use the averaged value given byEq. (7).

When considering the linear gas velocity U as the equiv-alent of temperature, we can directly de6ne the increase ofthe entropy of emulsion phase at the !uid velocity U ¿Umfas

RS =∫ U

Umf

d(REdis(Umf; U ))U

: (8)

The value REdis is supposed to be changed reversibly, whichcan be realised by a very slow increase ofU . The normalizedentropy RSN is given by

RSN =RSG∗ =

∫ U

Umf

dUU; (9)

where G∗ is the weight of the bed minus the buoyancy, seeEq. (7).

Eq. (9) gives the normalized value of the increase of en-tropy calculated using the value of Umf. Strictly speaking,it is the normalized increase of the rate of entropy, as thevalue RU is proportional to the increase of power (not en-ergy). For the sake of simplicity we shall however use asimpler notation of ‘normalized entropy’.

By solving the integral in Eq. (9) we obtain

R SN = lnUr; (10)

where

Ur =UUmf

: (11)

It was mentioned above that there exists relation betweenthe movement of bubbles and the mixing of bed particles.As the Kolmogorov entropy is a very useful tool for theanalysis of bubble behaviour, the idea arises to comparethe Kolmogorov entropy calculated on the basis of bubblesimpact (Schouten et al., 1996) with our concept of entropy.

M. Pun)coch*a)r, J. Draho)s / Chemical Engineering Science 58 (2003) 2515–2518 2517

Fig. 1. Comparison of the Kolmogorov and physical entropy as functionof Ur .

For this purpose we shall now compare the function lnUrwith the dependence of Kolmogorov entropy on the !uidiza-tion number given by the term (Ur − 1)0:4 in Eq. (2)—seeFig. 1. It can be seen from Fig. 1 that the curves are in ac-ceptable agreement within the range of experimental dataused by Schouten et al. (1996). It means that Eq. (10) in-deed is related to the Kolmogorov entropy of the bubblingbed. This fact seems to be very important for the consis-tency of our de6nition of entropy based on the use of U asan analogy of temperature.

3. Conclusions

De6nition of physical entropy of the dense phase of!uidized bed was given based on liquid-like propertiesof !uidized systems. We extended the analogy betweenthe linear gas velocity and liquid temperature (based onclassical thermodynamics) also for the case of bubblingbed.

Due to the fact that in our de6nition the increase ofpower of dissipated energy was used, we obtained rather theincrease of the rate of entropy. However, the considered in-crease of power is closely connected with bubble phenom-ena and it therefore re!ects a degree of disorder of bubblingsystems. This fact indicates the link to the Kolmogorov en-tropy, which is a measure of such disorder. We make noclaim of having a rigorous proof of identity of our de6ni-tion of physical entropy and the Kolmogorov entropy, butthe given concept can also be supported by the agreementof velocity functions in Eqs. (2) and (10).

From the practical point of view, we conclude that log-arithmic function of relative velocity is the very functiondescribing the measure of disorder of bed particles.

Notation

A bed cross-sectional area, m2

DT bed diameter, m

Edis power for !uidization, J/sREdis increase of Edis, J/sE! power for !uid, J/sEtot total power, see Eq. (4), J/sG∗ gravity of bed minus buoyancy, NH bed height, mHS settled bed height, mK Kolmogorov entropy, bits/smp mass of particle, kgRS increase of entropy of emulsion phase, see Eq. (7),

NRSN normalized increase of entropy, see Eq. (8)U linear velocity, m/sUmb minimum bubbling velocity, m/sUmf minimum !uidization velocity, m/sUr relative velocity (!uidization number)Qup mean velocity of particle, m/sQWk mean value of kinetic energy, J

Greek letters

� bed voidage� gas viscosity, Pa s�S particle density, kg=m3

�f gas density, kg=m3

�P standard deviation of pressure !uctuations, Pa

Acknowledgements

The authors acknowledge the support of the Grant Agencyof Czech Republic No. 104/97/S002.

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