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Multiscale Nature of Complex Fluid-Particle Systems Jinghai Li and Mooson Kwauk* Institute of Chemical Metallurgy, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Most multiphase chemical reactors are complex systems. This paper explores the multiscale nature of complex systems in chemical engineering and the intrinsic mechanism of the scale-up effect. Two approachessmultiscale and pseudoparticlesare discussed, which we believe to be effective and promising in revealing underlying mechanisms and extracting simplicity in complex systems. Heterogeneous structure, the most important characteristic of complex systems and a crucial aspect in scaling up reactors, is analyzed as the focus of this paper. The multiscale approach follows the strategy of structure resolution in describing the heterogeneous structure and mechanism compromise in defining the variational criterion. The pseudoparticle approach describes the heterogeneous structure by discretizing the fluid into pseudoparticles with a size much smaller than that of the real particles, that is, the particle-fluid interaction is treated as the interaction between real particles and pseudoparticles. This paper outlines the respective advantages and prospects of these two approaches and explores some concepts in understanding complex systems. 1. Complex systems challenge chemical engineers and scientists, as research progress with them might stimulate revolutionary changes in chemical engineering. However, complexity science is yet a field of perplexity with respect to its definition, unification, and methodology. Therefore, chemical engineers should not wait for ready tools from complexity science, but rather should establish their own methodologies for predicting and controlling their complex systems and, in turn, contribute to complexity science. Scale-up of chemical reactors has been a significant problem for engineers and scientists for decades, and there is a perennial need for methods other than traditional approaches. Many phenomena in nature are nonlinear and nonequilibrium, dominated generally by at least two submechanisms and most likely showing heterogeneous structures. Because of the nonexistence of a single and general variational criterion1 for such systems, quantification of such phenomena is difficult and has become a bottleneck problem and a research focus in many fields, leading to the appearance of a new scientific disciplinescomplex systems or complexity sciences which has received great attention not only in the natural sciences, but also in engineering and even in the social sciences.2 Multiphase reactors in chemical engineering are likewise complex systems, characterized by a multiscale heterogeneous structure and state multiplicity, which make scaling-up difficult. It is but natural, therefore, to relate the scale-up of chemical reactors to complex systems research. Unfortunately, no unifying theory governing all complex systems yet exists, and complexity science itself is a field of perplexity.3 Likewise, insufficient attention has been paid to the complex systems in chemical engineering despite the profusion of general concepts of related nature, such as fractals, chaos, and pattern formation. Although complexity science is considered to be an important science of the 21st century, there are as yet * To whom correspondence should be addressed. Tel.: 0086-10-6255-4050. Fax: 0086-10-6255-8065. E-mail: Mooson@ Lcc.icm.ac.cn.
Figure 1. Local heterogeneity and its dynamic change in gassolid two-phase systems.
no ready tools for the specific needs of chemical engineers. This paper discusses the importance of complex systems for chemical engineers, examines how to find and extract simplicity in complexity, and suggests approaches that have appeared effective and promising. 2. Dynamic heterogeneous structure is the dominant feature of complex systems in chemical engineering. Such a structure is critical to transport and reaction, and an understanding of it is therefore the key to scaling up reactors. Heterogeneous structure leads to complexity in transfer phenomena and brings about state multiplicity in chemical systems. Scale-up effects, as evidenced by the dependence of reactor behavior on scale, originates intrinsically from the effect of scale on structure. Structural heterogeneity can be classified into the following types: Local heterogeneity and dynamic changes, as shown in Figure 1, illustrate how solids and gases can self-
10.1021/ie0011021 CCC: $20.00 © 2001 American Chemical Society Published on Web 05/23/2001
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Figure 2. Overall heterogeneity in gas-solid two-phase systems.
organize to form a complicated two-phase structure consisting of a solid-rich dense phase and a gas-rich dilute phase, the configuration of which changes dynamically. Overall heterogeneity consists principally of the coreannulus structure of radial nonuniformity, as shown in Figure 2a, and the coexistence of two different axial sections, as shown in Figure 2b, depending on the operating and boundary conditions. Jump changes in structure, called choking in engineering, occur in relation to state multiplicity, that is, there could be more than one state prevailing in a system at the slightest change of certain specified operating conditions, e.g., of gas flow as shown in Figure 3a or of solids inventory as shown in Figure 3b. Chemical reactors can be quantitatively scaled up only when these structure changes can be predicted because of the critical dependence of transfer behavior on structure, as illustrated in Figure 4, which compares,
Figure 3. Jump change of flow structure.
as an example, the average drag coefficients for three different structures in the same volume containing the same number of particles: Cd/Cdo ) 0.032, 1, and 15.4, in panels a-c, respectively. It is evident that the scaleup of reactors calls for an understanding not only of the changes in structure but also of the dependence of transfer rate on structure. 3. Analyses of heterogeneous structure in complex systems belong to three categories: (1) an average approach, which will become less important because of its inadequacy in describing structures and transport phenomena; (2) a multiscale approach, which is promising because of its simplicity and effectiveness in approximating structures; and (3) a discrete approach, which should receive more attention with increasing computational capacity. Two types of heterogeneity are possible: single-scale, generally prevailing in liquidsolid systems as shown in Figure 5, and multiscale, mostly occurring in gas-solid systems as shown in Figure 6. Multiscale structure is complicated by the occurrence of inflective changes at several characteristic scales, whereas single-scale structure shows only gradual changes. Most heterogeneous structures in chemical reactors are multiscale, and they can be analyzed following two strategies, reasoning either from observed phenomena to intrinsic mechanism or from intrinsic mechanism to phenomena, leading to three different approaches. The first approach, involves averaging all parameters over a specific volume by considering the system to be uniform. This is the simplest approach, and the one most commonly used, but it cannot correctly represent particle-fluid interactions in heterogeneous structures. In fact, as long as there are more than two particles in a control volume, multiple arrangements of these particles in the volume are inevitable, leading to uncertainty in calculations. The second approach involves taking the multiscale structure into account and considering the disparity of
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Figure 4. Dependence of transfer rate on structure.
Figure 5. Single-scale structuresthere is no inflective change of observation results with increasing observation scale.25
Figure 6. Multiscale structuresobservation results show inflective changes at several characteristic scales.25
the particle-fluid interaction at different scales and in different phases. This approach calls for variational criteria to identify the prevailing steady state or preferred particle arrangement of the heterogeneous structure. The third approach involves tracking the movement of all individual particles to elucidate the details of particle-fluid interactions. This is an idealized approach, and it is not realistic at present because of limitations in experimental techniques and computer capacity. Such a discrete simulation of particle-fluid systems has recently been applied by many researchers including Tsuji.4
Considering the inadequacy of the average approach, the scale-up of reactors with heterogeneous structure could follow either the multiscale approach or the discrete approach, which are outlined below. 4. All complex systems are dominated by at least two submechanisms. Compromise between these dominant submechanisms is the origin of complex structures, and therefore, it provides the key to studying complex systems. For example, the variational criterion could be deduced from the compromise between dominant submechanisms so that we extract simplicity from complexity. From thermodynamics, we know that the equilibrium of a system is always related to maximum entropy, which, is however, not followed by nonequilibrium systems according to nonequilibrium thermodynamics.5 Nonequilibrium systems close to equilibrium are called linear nonequilibrium systems, whereas those far from equilibrium are named nonlinear nonequilibrium systems. The variational criterion for linear nonequilibrium systems could be formulated by the theorem of minimum entropy production,5 but there is no single and general variational criterion for nonlinear nonequilibrium systems.1 Although the so-called evolution criterion was established by Glansdorff and Prigogine,6 variational criteria for most nonlinear nonequilibrium systems in engineering are still difficult to formulate, thus stimulating extensive research in this aspect.7 The difficulty in this aspect arises from the nongenerality of extremum behavior for complex systems because there are different dominant mechanisms for different systems, which are characterized by different extremum tendencies without a unified rule. However, structure is the common nature of complex systems, being induced by compromise between dominant submechanisms. Therefore, common nature could be deduced by analyzing the compromise. Multiphase reactors, such as gas-solid fluidized reactors, are typically nonlinear and nonequilibrium, as the particles and fluid have to compromise with each other in following their own respective movement tendencies. As shown in Figure 7, for fixed-bed operation when particles dominate the fluid, that is, when the fluid cannot move the particles, the movement tendency of the particles to attain a minimum potential energy, that is, ) min, could be exclusively realized. In contrast, for dilute transport when the motion of particles is dominated by the fluid, the particle movement tendency is suppressed by the fluid in realizing the fluid movement tendency, expressed as Wst ) min.8 Fluidiza-
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Figure 7. Three cases of particle-fluid interactions and their relationship (modified from Li et al.22).
Figure 8. Three cases of interactions between the viscous effect and the inertial effect in pipe flow (s, calculation; ‚ ‚ ‚, Karman’s correlation) (modified from Li et al.25).
tion is substantially different from these two extreme cases, as the movement tendencies of both the particles and the fluid play important, although not exclusive, roles, calling for mutual compromise and leading to only partial realization of either tendency, that is, Nst ) Wst/ (1 - ) ) min. Such a compromise results in the formation of the heterogeneous flow structure of fluidization, the only structure that satisfies both movement tendencies simultaneously. The recognition of the important role of particle-fluid compromise and its formulation became the important basis of the so-called energy-minimization multiscale (EMMS) model.8-10 A similar strategy based on the compromise between two submechanisms was also applied to analyzing the variational criterion for the velocity distribution in turbulent pipe flow,11 as indicated in Figure 8. As we know, with increasing Reynolds number, single-phase fluid flow in the pipe would develop from the viscositydominated state (laminar flow) to the turbulent state, which is jointly dominated by viscosity and inertia, and would finally approach the limiting state dominated exclusively by inertia. In the laminar regime, the viscosity effect exclusively dominates the fluid velocity distribution, which satisfies the minimum of viscous
dissipation Wv because of its linear nonequilibrium feature, leading to a parabolic velocity profile, as is commonly known. In contrast, when the fluid flow is exclusively dominated by inertia and the viscosity effect is fully suppressed in the whole pipe except at the wall, the fluid distributes its velocity uniformly over the pipe cross section, leading to maximum of the total dissipation WT related to high viscous effects only at the wall. However, in the turbulent regime, neither viscosity nor inertia can exclusively dominate the system, and they have to compromise with each other in realizing their respective intrinsic tendencies, resulting in an inertiadominated core region coexisting with a viscositydominated wall layer. Such a velocity distribution satisfies Wv ) min|WT)max, which, according to applied mathematics, could be expressed as F ) WTR - Wv(1 R) f max. With increasing R, the inertial effect is intensified, whereas the viscosity effect is suppressed, causing a flatter velocity distribution, as shown in Figure 8, in which the calculated curve (s) is compared with the classical empirical correlation of Karman.12 From the above two examples, it was deduced that, for systems dominated by two submechanisms, the variational criterion of the system could be physically
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Figure 9. Important role of compromise between submechanisms in inducing structural changes.
expressed as a mutually constrained extremum between the two tendencies, that is
(
variational extremum tendency ) criterion of mechanism 1
)(
extremum tendency of mechanism 2
)
Obviously, the compromise between the two submechanisms plays a critical role in the formation of the dissipative structure, as was verified by computer-aided experiments on the interaction between two discrete media, particles A and B,13 as shown in Figure 9. Media A and B, with identical diameters but different densities, move countercurrently with the same velocity. Therefore, the density ratio FA/FB represents the relative dominance between A and B. Figure 9 shows that a complex structure appears only when neither A nor B can dominate the other, indicating the importance of their mutual compromise in generating complexity. Considering the prevalence in nature of processes dominated by two or more submechanisms, the important role of compromise between the submechanisms in analyzing system stability and establishing the corresponding variational criterion should be given sufficient attention. Compromise is important not only for the dynamic structure but also for the formation process of the stationary structure, as shown in Figure 10. If material C is compounded from materials A and B, its structure could be critically subject to the compromise between A and B in terms of the formation process and their properties. By changing the properties of A and/or B during the formation process, the final structure of C could be in the form of either C1 or C2. If the property of A were different from that of B, the property of C1 could be tremendously different from that of C2, and the properties of the compound material could be optimized. 5. The multiscale approach provides a useful tool for finding simplicity in complexity. However, because of its approximations, some information related to structure might be lost. Therefore, as a key step for the multiscale approach, a variational criterion is needed to retrieve the lost information. The multiscale approach was adopted in studying the two-phase structure of gas-solid fluidization,9,10 and its potential has been indicated in describing the heterogeneous structure, defining the choking point,14 and calculating the radial heterogeneity.15 Figure 11 shows the physical concept of the so-called multiscale analysis that resolves the system into three basic
Figure 10. Comprise between two materials in the process of forming a compound material.
scales: particle scale, cluster scale, and unit scale, that is, micro-, meso-, and macroscale, respectively: The microscale considers discrete individual particles inside either the dense or the dilute phase. Totally different mechanisms of gas-solid interaction prevail in these two phases: particle-dominated (PD) inside the dense-phase cluster and fluid-dominated (FD) inside the dilute-phase broth. Such a disparity of gas-solid interaction mechanisms in the two phases can be described only with phase-specific parameters when the tracking of all individual particles is not practical. The mesoscale corresponds to the cluster size, involving interactions between the dilute broth phase and the dense cluster phase. The disparity of gas-solid interaction mechanisms inside the two phases gives rise to particle-fluid-compromising (PFC) interaction between the dense-phase cluster and the dilute-phase broth on a mesoscale. This scale of interaction is characterized not only by ordered behaviors but also by irregular changes. It is the irregular changes that lead to frequent violations of the particle-dominated condition inside the dense phase and the fluid-dominated condition inside the dilute phase, thus giving rise to complicated time series of fluctuations with intermediate states. This scale of interaction is of fundamental significance in heterogeneous systems, generally showing unique interfacial phenomena between phases, as will be discussed below.
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Figure 11. Eight parameters and three scales of interaction in heterogeneous two-phase flow.22
The macroscale encompasses the global system of the particle-fluid suspension within its boundaries. This scale of interaction reflects the effect of boundaries on particle-fluid interactions or on the particle-fluid suspension as a whole, by changing the relative dominance between the particles and the fluid. Boundaries (including reactor internals) lead to disparities in the particle-fluid compromise with respect to space, including axial and radial heterogeneity. Because of the multiscale behavior of gas-solid flow, phase-specific parameters have to be defined to consider the disparity of gas-solid interactions in the two phases and to represent their mutual compromise. Altogether eight parameters, X ) (f, c, f, Ugf, Ugc, Upf, Upc, dcl), are needed to describe such a heterogeneous flow structuresUgc, Upc, c, f, and dcl for the dense phase and Ugf, Upf, and f for the dilute phase, as shown in Figure 11. The multiscale concept facilitates an analysis of the compromise between the particles and the fluid in realizing their respective movement tendencies and enables the formulation of different particle-fluid interactions (particle-dominating, fluid-dominating, and particle-fluid-compromising) in the system. For example, the occurrence of interfacial interaction between the dense-phase cluster and the dilute-phase broth arises from the appearance of a heterogeneous twophase structure, as shown in Figure 12. The pressure drops in the dense and dilute phases are ∆Pdense and ∆Pdilute, respectively, and Finter represents the overall interfacial interaction between the two phases. Therefore, pressure balance gives
∆Pdilute + Finter/(1 - f) ) ∆Pdense Obviously, Finter ) 0 would lead to ∆Pdense ) ∆Pdilute
Figure 12. Interfacial interaction within a heterogeneous structure.25
when the heterogeneous structure degenerates to the uniform structure, meaning that the heterogeneous structure cannot be divorced from the interfacial interaction. It is this interaction that induces the irregular dynamic changes of the flow structure. By considering the disparity of the particle-fluid interactions between the dense phase and the dilute phase, the multiscale analysis extracts the interfacial interaction between the two phases, which can obviously not be formulated by any averaging method. By generalizing the above analysis for gas-solid systems, chemical processes could be categorized in
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Figure 13. Examples of processes occurring at different scales for gas-solid two-phase processes (modified from Li et al.25).
terms of subprocesses and scales,16 as tabulated in Figure 13, which shows the relationship between representative subprocesses, including flow, transfer, phase separation, and reaction, and six scales for reactors. Figure 13 shows that a chemical reaction that occurs at the molecular scale can be realized commercially, with high conversion and high selectivity, on an apparatus scale to output qualified products on a factory scale only when appropriate control and optimization were carried out on related intermediate scales, e.g., nanometer/micrometer for particles and clusters, with respect to the related subprocesses. In fact, in all industrial operations, reaction conditions at the molecular scale can be adjusted only by conducting control at the apparatus scale. Understanding the behaviors at different scales and their interrelationships is the key to realizing successful design and operation of process engineering. For gas-solid two-phase reactors, in particular, the cluster scale plays a key role because the particle-fluid compromise prevails at this scale. Multiscale methods, in general, have been used in different fields since the 1970s or even earlier, but recognition of its significance only occurred in recent years. A recent literature search on Engineering Index indicates a dramatic increase of the number of papers on multiscale methods (6 in the 1970s, 24 in the 1980s, 488 in the 1990s), as shown in Figure 14. These papers presented two categories of multiscale methods: descriptive, only to note the multiscale nature of various structures and signals, and analytical, to describe the structure and to reveal the dominant mechanisms of the structure formation. Most papers published are descriptive, and only a few are analytical. Special conferences on multiscale methods were organized in Germany in 199617 and in China in 2000.18 It is a pity that no systematic theory has yet been established for multiscale approaches in chemical engineering, although it has been emphasized by Villermaux,19 Lerou and Ng,20 and Charpentier21 and explored in the so-called energyminimization multiscale (EMMS) model,8,10 as outlined in Figure 15. In the EMMS model, the gas-solid two-phase flow system is first resolved into three scales, the microscale of particle size, the mesoscale of cluster size, and the
Figure 14. Literature search result for multiscale method from Engineering Index.
macroscale of apparatus size, as indicated in Figure 11. The analyses of mass and momentum conservation in such a multiscale structure yielded six equations with respect to the eight variables enumerated in the previous section.To solve for these eight variables, correlating different scales of interactions and considering the multiplicity in multiscale systems, a variational criterion was established by resolving the global mechanism into two submechanisms, analyzing the extremum tendencies of the submechanisms, and considering the compromise between them. The EMMS model integrates the mass and momentum conservation equations with the variational criterion to calculate the local and mesoscale heterogeneous flow structures,10 the saturation carrying capacity,14 and the macroscale radial15 and axial distributions.8 The EMMS model has recently been coded into a software package linked to the package of Siegen University, as summarized by Li et al.22 Furthermore, the model has recently been extended to gassolid-liquid three-phase flow, resulting in reasonable agreement with experimental data available in the literature.23 We believe that the complexity and number of the characteristic scales of a complex system increase with increasing number of dominant mechanisms. This deduction was supported by the experiments of Chirone
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Figure 15. Algorithm for EMMS modeling (modified from Li et al.22).
Figure 16. Number of characteristic scales of complex systems increases with increasing number of dominant mechanisms.
and Russo,24 who studied sound-assisted fluidized beds and found that subclusters were generated in gas-solid fluidized beds, meaning that one more scale of interaction has to be considered in addition to the single particle scale and the cluster scale, as shown in Figure 16. Therefore, the dependence of characteristic scales on the number of dominant mechanisms appears to be another important subject for complexity science. 6. Multiple resolution is effective in extracting important information from the global behavior of complex systems, e.g., separating reversible from irreversible processes or ordered from disordered changes. Although global analyses cannot effectively distinguish the details of complex systems, multiple resolution can be used as an effective approach in such analyses. First, the heterogeneous structure as revealed by optical-fiber measurements could be resolved into a stationary heterogeneous structure consisting of two phases with voidages c and f and its accompanying dynamic changes. Because both the stationary (spatial) structure and the dynamic (mainly temporal) changes exhibit multiscale features, scale resolution is carried out for both, that is, the stationary structure is resolved into three scales of interactions microscale for both the dense and the dilute phases, mesoscale between the two phases, and macroscale
between the gas-solid suspension and its boundaries. Meanwhile, the dynamic changes are also resolved into the corresponding scales of fluctuations.25 Without going into details, suffice it to say that correlation of the above resolutions with energy resolution reveals that Nst corresponds approximately to the stationary structure and that Nd is roughly attributed to its dynamic changes. The importance of dynamic changes in gas-solid mass transfer can be demonstrated by Figure 17,26 in which the measured mass transfer rates represented by Shp in a real fluidized bed of 72-mm-i.d. with naphthalene particles fluidized with air are compared with those measured in a stationary structure consisting of artificial clusters of the same particles. Because the total rate of mass transfer can be considered to be the sum of the mass transfer in the stationary structure and the contribution of its dynamic changes, the difference between the curves, real (sbs) and stationary (sOs), shows the contribution of dynamic changes to the total rate. 7. Among the many computer fluid dynamics (CFD) approaches, the discrete approach is the most promising. Unlike the multiscale approach, the discrete approach does not need any variational criterion as it can describe the necessary details at a sufficiently small scale, but it calls for
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Figure 17. Comparison of mass transfer between stationary heterogeneous structure and real fluidized structure showing significant contribution of dynamic behavior.26
Figure 18. Physical model for pseudoparticle approach.
a physically accurate elemental model. The multiscale method focuses on the dominant nature of the flow structure in gas-solid fluidization to simplify the analyses and to avoid tracking all individual particles by grouping all particles into two groups, namely, particles in the dense phase and particles in the dilute phase, and then considering the interaction between these two groups. However, the dynamic changes and the intermediate states occurring in the dense-dilute alternation also have to be considered. For a detailed understanding of the particle-fluid interactions, the ideal approach is to track all individual particles throughout the duration of their movement. The current discrete approaches are limited by computer capacity. Although particle movement is considered individually for each particle, the particle-fluid
interaction is still treated on an average basis in a control volume containing more than one particle. This is not sufficient for understanding the details of the structure, particularly when both transport and reaction processes are involved. Consequently, discrete simulation on a microscale has been carried out by adopting the so-called pesudoparticle model in which the gas is treated as an assembly of fictitious particles with a size much smaller than that of the real particles involved,27 as shown in Figure 18. This model makes it possible to simulate the detailed flow fields around all individual particles, as shown in Figure 19, which is important in considering mass and heat transfer. Figure 2028 shows the simulated bubbling phenomenon in a gas-solid fluidized bed using the pseudoparticle model, which is in reasonable agreement with experimental observation, thus demonstrating the potential of this approach. However, the pseudoparticle approach is far from being mature. For example, the formulation of the interaction between pseudoparticles is yet to be improved. Although this approach has not been expected to be used in large-scale simulations, it is reasonable to apply it to elucidating the micro- or mesoscale behavior of fluidization and improving the effectiveness and quality of macroscale simulations such as the large eddy simulation of turbulent flow.29 In fact, a discrete nature is intrinsic in all phenomena in the universe, appearing at different levels and on different scales. For a specified problem, it is not necessary to deal with all levels of related phenomena as not all levels of mechanisms are substantially related to the objective performance in which we are interested. For example, electrons, protons, and neutron must be considered in studies of the chemical properties of a material, although they are not important in understanding the pressure exerted by a fluid on a wall, for which the molecular level is sufficient. This implies the possibility of developing a general computer algorithm for discrete phenomena, such as the finite element method for the continuum approach, that could be used jointly with a special element model for the specified problem. To realize such a paradigm, the following aspects of research are necessary: (a) a computer algorithm for dealing with a huge number of discrete elements, (b) physically accurate element models for describing the interactions between these elements for
Figure 19. Flow field around each particles simulated with the pseudoparticle model (vertical-up flow).
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Figure 20. Simulated bubbling phenomenon with the pseudoparticle model.
specified problems, and (c) a common interface between the algorithm and different element models. We believe that the fully discrete approach will be increasingly attractive in analyzing complex systems with developing computer technology and increasing knowledge of microscale phenomena. 8. Prospects With increasing knowledge of complex systems and with developing computer technology, the scale-up of chemical reactors will become easier. To promote a breakthrough in chemical engineering, much attention should be paid to transdisciplinarity to quantify the various structures prevailing in reactors that are considered to be the main obstacles in scaling-up processes. Compromise between the movement tendencies of submechanisms in complex systems is the origin of heterogeneous structures. Multiscale analysis is a promising approach to cope with complicated structures; however, it calls for variational criteria. Alternatively, a discrete approach based on sufficiently small elements is intrinsically the solution to scaling up reactors, although a discrete approach at the microscale for systems at the industrial scale is still not realistic. Because transport behavior is critically dependent on the prevailing flow structure in the system, average approaches that do not consider structure are inadequate for scaling up reactors. At present, a combination of a multiscale approach for the macrostructure and a discrete approach for the microstructure is considered to be a practical solution. Complexity science is still a field of perplexity with respect to its definition and future, when considered in the light of establishing a general theory and a unified methodology, but it poses no hindrance to our understanding specific systems in our field. On the contrary, the understanding of chemical reactors will contribute to the progress of complexity science. Further development of chemical engineering depends on developing a better understanding of the complex structures that have become the focus of complexity science. In that respect, we are now face-to-face with a significant challenge, as well as a good opportunity, to make a breakthrough in chemical engineering. Nomenclature Cd ) drag coefficient Cdo ) drag coefficient in uniform structure dp ) particle diameter (m) dcl ) cluster diameter (m) f ) volume fraction of the dense phase F ) objective function for optimization
Finter ) total interfacial force between dense phase and dilute phase with respect to unit volume (kg m/s2) g ) gravity acceleration (m/s2) Gs ) solid flow rate [kg/(m2 s)] ms ) particle mass Nd ) difference between NT and Nst [J/(s kg)] Nst ) energy consumption for suspending and transporting with respect to unit mass of particles [J/(s kg)] P ) pressure (Pa) r ) radial coordinate (m) R ) pipe radius (m) Rep ) particle Reynolds number based on Vs Shp ) mass transfer Sherwood number t ) time step (s) u j ) average fluid velocity in single-phase pipe flow (m/s) u(r) ) local fluid velocity in single-phase pipe flow (m/s) Ug ) superficial gas velocity (m/s) Ugf ) superficial gas velocity in dilute phase (m/s) Ugc ) superficial gas velocity in dense phase (m/s) Umf ) minimum fluidization velocity (m/s) Up ) superficial particle velocity (m/s) Upf ) superficial particle velocity in dilute phase (m/s) Upc ) superficial particle velocity in dense phase (m/s) Us ) slip velocity (m/s) Usi ) slip velocity between phases (m/s) Usf ) slip velocity in dilute phase (m/s) Usc ) slip velocity in dense phase (m/s) Ut ) terminal velocity (m/s) Vs ) real slip velocity WT ) total dissipated energy in single-phase flow in unit volume [J/(m3 s)] Wv ) viscous dissipation in single-phase flow in unit volume [J/(m3 s)] Wst ) energy consumption for suspending and transporting particles in unit volume [J/(m3 s)] R ) weight factor for inertial effect ) voidage c ) voidage in dense phase f ) voidage in dilute phase Fp ) density of particle (kg/m3) Ff ) density of fluid (kg/m3) Fp/Ff ) particle/fluid real density ratio ∆Pdilute ) pressure drop in dilute phase (kg/m2‚s2) ∆Pdense ) pressure drop in dense phase (kg/m2‚s2)
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Received for review December 15, 2000 Revised manuscript received March 9, 2001 Accepted March 26, 2001 IE0011021
