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Pergamon Chemical Engineering Science, Vol. 51, No. 4, pp. 667-669, 1996 Copyright © 1996 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0009-2509/96 $15.00 + 0.00

0009-2509(95)00321-5

G a s - s o l i d f l u i d i z a t i o n : a t y p i c a l d i s s i p a t i v e s t r u c t u r e

(Received 22 June 1995)

INTRODUCTION

In accordance with the controlling role of the particles and/or the fluid, Li et al. (1992) characterized the three major regimes in particle-fluid two-phase flow as: particle domina- ting (PD) for the fixed bed, particle-fluid compromising (PFC) for fluidization and fluid dominating (FD) for dilute trans- port. Subsequent efforts have been focused on transition and structural difference between regimes (Li and Kwauk, 1994).

In modeling particle-fluid two-phase flow by the energy- minimization multi-scale (EMMS) method, Li et al. (1988) attributed the stability of the heterogeneous two-phase struc- ture of the PFC regime to the inherent tendency of the system toward minimal energy expended in suspending and transporting the particles Nst. For the FD regime, on the other hand, the EMMS method revealed that N,t tends toward a maximum. Such contrary criteria point to the extremum behavior (Li et al., 1993) unique to particle-fluid two-phase flow.

According to the theorem of minimum entropy produc- tion (Prigogine, 1967), steady states of linear nonequilibrium systems prevail only when the entropy production rate is minimized. There is however no single and general varia- tional theorem for nonlinear steady-state dissipative systems (Gage et al., 1966; Nicolis, 1994). This paper purports to examine the extremum behavior of fluidized systems in the light of nonequilibrium thermodynamics in order to explore valid approaches to elucidate the bifurcation phenomenon and dissipative structure for such complicated systems.

EXTREMUM ENERGY DISSIPATION In a particle-fluid system with superficial fluid velocity

Ug and solids flow rate Gs, the energy supplied to the system is spent not only in suspending and transporting the particles but also as dissipated energy in particle collision, circulation, acceleration, etc. (Li et al., 1988). With respect to unit mass of particles in unit cross-sectional area normal to the direction of flow, the total energy consumed is

APUg Nr (1)

AL(1 - e)pv where

Therefore

AP ~-£ = (1 - ~)(p, - pz)g. (2)

U ~ g ( P v - P I ] N~ = \ ~ / (3)

The energy consumption transformed into particle potential energy is

N, = G~#/(1 - e)Pv. (4)

Therefore, the total dissipated energy is

Ndi s = N T - - N t. (5)

Figure 1 shows the variation of different energy terms with gas velocity, all computed according to the EMMS model (Li et al., 1988; Li and Kwauk, 1994). The figure indicates that in the PFC regime for fluidization, the dissipated energy Ndis occupies a considerable portion of the total energy Nr, while as the flow regime transits to dilute transport, it drops dramatically.

In the EMMS model, the energy consumption for sus- pending and transporting unit mass of particles N~ is shown to be minimal in the PFC regime and maximal in the FD regime, and it can therefore be used as the criterion for stability conditions. Further work (Li et al., 1993) indicated that Nt can be used as a substitute for N~, to define stability conditions of these two regimes, that is, Nt = rain in the PFC regime and Nr = max in the FD regime. Noting the con- stancy of Nr at any specified value of Ug from eq. (3), it thus follows from eq. (5) that Nais = max in the PFC regime with a two-phase structure and that Nals = rain in the FD regime with a uniform structure.

TWO BIFURCATIONS The PD regime for the fixed bed essentially corresponds to

single-phase fluid flow through a maze of channels com- posed of packed particles. Particles determine the configura- tion of these channels, and the fluid merely seeks these channels and distributes itself through paths with minimal resistance, resulting in minimal energy dissipation. The pres- sure gradient AP/AL in this regime is known to vary with superficial fluid velocity Ug, and in the case of fine particles, it is proportional to Ug and is affected by an exponential function e-4'7 of the voidage of the packed solid particles (Li and Kwauk, 1994):

2 4 7 d v e • AP U9

18(1--e)# I AL

implying linear relationship between the driving force AP/AL and the superficial fluid velocity Ug, as shown in Fig. 2 by experiments carried out with FCC particles fluidized with air, with local voidages monitored with an optical fiber probe.

As the superficial fluid velocity increases to that for min- imum fluidization Uml, particles start to move and expand with increasing voidage e, leading to nonlinearity between Ug and AP/AL. The system is said to have entered the PFC regime. With increasing superficial fluid velocity, nonlin- earity of the system increases and reaches a critical extent at the minimum bubbling velocity Umb at which the fluid organ- izes itself into bubbles and particles aggregate into the emul- sion phase, resulting in a dramatic change in flow structure which is now dissipative with ordered behaviors as Nicolis and Prigogine (1977) described. Such a change is known as the first bifurcation corresponding to nonequilibrium phase transformation in thermodynamics. For particle-fluid sys- tems with large particles, corresponding to high Rev, the dissipative structure appears immediately beyond Urns, that

667

668 Shorter Communication

is, Umf = U=b. This dissipative structure is brought about jointly by the particles and the fluid, both of which are mobile. Its nonlinear nature is characterized by a highly heterogeneous two-phase structure with significant energy dissipation which is necessary to maintain its steady state.

At U o > U,.b, as shown in Fig. 2, the A P / A L ~ U 9 curve bifurcates into two branches both satisfying the mass and momentum conservation equations, corresponding to the heterogeneous and the homogeneous structure, respectively. For gas-solid systems, only the two-phase heterogeneous structure, however, is physically stable, in which the particles organize themselves into a dense phase with voidage e~ and

'$ E

0

,,5

PD Fixed Bed

40

30

20

10

0

PFC FD F lu i d t za t i on T ranspo r t

U=f U

O~k" ~ Nd,, =min

Nd i = max

_/_ . . . . ,.,:~_,.. . . . . . ~. . . . . . J i ~ i J

1 2 3 4 5

Gas Velocity Ug (m/s)

Fig. 1. Energy consumptions in different regimes of a par- ticle-fluid system, calculated from the EMMS model (Li et al., 1988), showing maximum dissipation in fluidization regime but minimum dissipation in transport regime

(FCC/air: pp = 930 kg/m 3, dp = 54 #m, Gs = 50 kg/m2s).

the fluid into a dilute phase with voidage ef. Although thermodynamics does not specify any single and general variational theorem for such a steady-state dissipative struc- ture, the E M M S analysis indicates that for this highly dissi- pative structure, Nd~s = max.

As superficial fluid velocity further increases, a second critical velocity Up~ is reached, resulting in a second bifurca- tion, at which the nonlinear two-phase PFC regime suddenly gives way to the single-phase dilute transport (generally known as "choking"), as shown by experimental data in Fig. 2. With the onset of this second bifurcation, the self- organization of particles is suppressed, and the dense phase disappears, although the dilute phase becomes somewhat denser. Such a change corresponds to a switch of stability conditions from Nd~s = max to N d i s = min. In dilute trans- port, the fluid determines the flow structure, and it is there- fore designated fluid dominating (FD).

Figure 3 shows the bifurcation phenomena in terms of local voidages in the fluidized FCC bed. In the particle- dominating fixed-bed regime, local voidage takes a constant value. If the first bifurcation occurs at minimum fluidization (case for large Rep), local voidage changes suddenly to alter- nate between a value close to unity and a value approaching era f , showing an ordered dissipative structure and extremum behavior of the system. With increasing fluid velocity, greater perturbation prevails in such a two-phase structure, resulting in more extensive scatter of voidage values between unity and e,,/, attesting to intensification of chaotic and/or random behaviors. With the onset of the second bifurcation at choking, the probability of the existence of these two voidage values narrow down to a very limited range (a single value in theory), implying a more or less homogeneous structure.

Because of the predominance of the dissipative structure in gas solid ftuidization, the conventional time- and space- averaged approaches do not suffice for adequate elucidation of the mechanisms and for quantifying the process, future research should be focused on the non-linear behaviors of the system.

Q. X

J

10

0 0.00

• • • ~j~~i~A N°nlineaz

/ U,b "', P, am

/ I [ - . e ~ " B / L .1. "~B . / ,.r£,.

Fixed Bed Fluidization , . Fixld B i i a t o ,

0.01

Upt Second Bifurcation

0.01 0.0~ 0.~ 0.5 1.0 1.5 2.0 2.5 8.0 $.5

V note charge of scale far L~

Gas Velocity Ug (m/s)

Fig. 2. Experimental results (shown as dots) in a fluidized bed of 90 mm ID with FCC particles, showing linear characteristics in the fixed-bed regime but nonlinear behaviors in the fluidization regime. The first bifurcation at U,,b results in two branches of fluidization: one is stable with an aggregative structure due to self-organization of particles and the fluid; another is unstable with particulate structure as shown by the dashed line. The second bifurcation at Upt, known as "choking", leads to a second sudden change in flow

structure.

10

0 4.0

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Shorter Communication

73

o

P ~ x e d B e d ,

1 .O0 - - -

0.89

0.78

F ILIIdl Z8 t tON r T r a n s p o r

• ../,-~,-.~.-, : ; ~ - . ~ . ' : ~ ~ ~

" :" Fdllll I ' . . : . . " i ' i:"..' ,:v..',':'~.:. ..

. . • . . . . : ~ : - - . . ~

I • .' .' . : . " . - ~ ' , ~ , ~ : " : I . " " - " . " " ". • " "" " - ? . " : ' - ' 1

dp g G, Nst

NT

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i Ur.f . "': ' :." ' " . k# ! " . Upt 0.66 I . ":.: .. " . , : . ' . " . . ; " Nt I

• . . . : . - . . . . " ' . : . ' . ~ , . ' - I

056 i " . . : , ." : , . " . V ' , ' . : " . : " I N d i s

. . . . . ..'~~'.~,.::-'. ~¢', f . . ".- AP/AL 0.44

001 0.1 1 10 Umf Upt

~_~as:-- V e l o c i t y LJo ( m / s ) Ug Umb

Fig. 3. Experimental results (shown as dots) in the same bed p: as Fig. 2, showing bifurcations of local voidage: pp

ef • a single constant voidage in fixed-bed regime, ~c • biextremum voidages with increasing chaotic and ran- s

dora changes in fluidization regime, sa • convergence to another single-valued voidage in trans-

port regime, e*

CONCLUSIONS (1) Major regime transition in particle-fluid two-phase

flow (notably for gas-solid systems) corresponds to bifurca- tion in thermodynamics. The first bifurcation marks the occurrence of bubbling with sudden appearance of dissi- pative structure; the second, PFC/FD transition, from fluidization to dilute transport, with sudden reversion to homogeneous structure, as particle self-organization be- comes suppressed.

(2) The stability of the nonlinear dissipative PFC regime is governed by Ndis = max. Whether such a variational cri- terion is applicable to some other nonlinear nonequilibrium systems is to be explored further.

(3) Fluidization is a typical dissipative structure with non- linear and nonequilibrium behaviors. Multiple resolution appears to be a promising approach toward understanding such a complicated system: total energy resolved into revers- ible and irreversible energies, the whole process resolved into ordered and disordered branches, particle-fluid movement resolved into extremum and chaotic and/or random move- ments, and global structure resolved into subsystems of different scales.

Acknowledgements The authors wish to thank Prof. Mooson Kwauk for his encouragement and help. Valuable discussion with Prof. Weikang Yuan and Prof. Jiuli Luo and financial support from National Natural Science Founda- tion of China and Academia Sinica are also gratefully ac- knowledged.

JINGHAI LI* GUIHUA QIAN LIXIONG WEN

Multi-Phase Reaction Laboratory Institute of Chemical Metallurgy Academia Sinica, Beijing 100080, P.R. China

NOTATION

particle diameter, m gravity acceleration, m/s 2 solids flow rate, kg/m 2 s energy consumed for suspending and transporting unit mass of particles, J/kg s total energy consumption with respect to unit mass of particles, J/kg s transporting energy consumption with respect to unit mass of particles, J/kg s total dissipated energy with respect to unit mass of particles, J/kg s pressure gradient, kg/m 2 s 2 minimum fluidization velocity, m/s minimum velocity for dilute transport, m/s superficial velocity, m/s minimum bubbling velocity, m/s fluid density, kg/m 3 particle density, kg/m 3 voidage in dilute phase voidage in dense phase average voidage average voidage in the PFC regime at critical point average voidage in the FD regime at critical point fluid viscosity, kg/m s

R E F E R E N C E S

Gage, D. H., Schiffer, M., Kline, S. J. and Reynolds, W. C., 1966, The non-existence of a general thermodynamic variational principle. In Non-Equilibrium Thermodynamics Variational Techniques and Stability (edited by R. J. Don- nelly, R. Herman and I. Prigogine). The University of Chicago Press, Chicago.

Li, J., Chen, A., Yan, Z., Xu, G. and Zhang, X., 1993, Particle- fluid contacting in circulating fluidized beds, presented at the 4th International Conference on Circulating Fluidized Beds, Pittsburgh, 1 5 August 1993.

Li, J. and Kwauk, M., 1994, Particle-Fluid Two-Phase Flow--the Energy-minimization Multi-scale Method. Me- tallurgical Industry Press, Beijing.

Li, J., Kwauk, M. and Reh, L., 1992, Role of energy minimi- zation in gas/solid fluidization. In Fluidization V l l (edited by O. E. Potter and D. J. Nicklin), p. 83. Engineering Foundation, New York.

Li, J., Tung, Y. and Kwauk, M., 1988, Multi-scale modeling and method of energy minimization in particle-fluid two- phase flow. In Circulating Fluidized Bed Technology, Vol. II (edited by P. Basu and J. F. Large), p. 89. Pergamon, Oxford.

Nicolis, G., 1994 (personal communication). Nicolis, G. and Prigogine, I. 1977, Self-Organization in

Nonequilibrium Systems, for Dissipative Structures to Or- der Through Fluctuations. J. Wiley, New York.

Prigogine, I., 1967, Introduction to Thermodynamics of Irre- versible Processes. Interscience Publishers, New York.

* Corresponding author.