Advances in Chemical Engineering, Volume 26

CONTRIBUTORS Numbers in parentheses indicate the pages on which the author’s contributions begin.

N. S. DESHPANDE, Department of Chemical Technology, University of Bombay, Matunga, Mumbai-400 019, India (1) M. DINKAR, Department of Chemical Technology, University of Bombay, Matunga, Mambai-400 019, India (1) J. B. JOSHI, Department of Chemical Technology, University of Bombay, Matunga, Mambai-400 019, India (1) MICHAEL NIKOLAOU, Department of Chemical Engineering, University of Houston, Houston, Texas 77204 (131) D. V. PHANIKUMAR, Department of Chemical Technology, University of Bombay, Matunga, Mumbai-400 019, India (1)

vii

PREFACE The field of chemical engineering is advancing on many fronts at the same time, and is in greater danger of fragmentation than ever before. The hot new fields of biotechnology, information technology, novel materials, and environmetal protection all have their own specialized journals and meetings. This fragmentation is happening in all fields in America, and not only in chemical engineering. In the 1930s, everyone read Life magazine, which appealed to a wide audience and gave millions of people a set of information that was generally considered important to share. Any two people getting together could talk about what they had read in this magazine. Its preeminence and readership has eroded over the years, to magazines devoted exclusively to sports, fashion, entertainment, health, etc. This year, we hear that this venerable magazine will cease publication. Now, there are fewer things in common that any two people can share. Fortunately, Advances in Chemical Engineering is still being published in the hope of bringing out something that all chemical engineers should know, in addition to the specialized journals that they read. This time we have two topics that have made giant strides both in theory and in applications. The simple single phase flow reactors have received most of the attention in the past, in academia due to their simplicity which is amenable to analysis, and in industry due to their predictability in that they will perform as designed. However, industry has increasingly turned to gas– liquid bubble columns, solid–liquid fluidized beds, gas–solid fluidized beds, and three phase beds for their heavy lifting. There is a great need for an update on what we know now, and how that knowledge can be used to help in design. The chapter on ‘‘Hydrodynamic Stability of Multiphase Reactors’’ by J. B. Joshi et al. of the University of Bombay analyzes the investigations of the past fifty years, critiques them on their contributions, and provides unified criteria on stability and transition. It will be heavily used to make further advances, both in understanding and in practice. Model predictive control (MPC) is a very widely used process control technology for process plants. They have been in use for many years before there was a sound understanding of how they work, and what their limitations are. There is now a firm foundation on MPC theory, which promises to push the envelope in plant design and implementation. We should not

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PREFACE

automatically assume that better knowledge will lead to better practice. The surgeon–researcher Judah Folkman once remarked that medical science has known the cause of sickle cell anemia for fifty years, but there is still no cure; and we do not know the cause of appendicitis, but it is very easy to cure. It takes outstanding engineering creativity to go from knowledge to better practice. The chapter on ‘‘Model Predictive Controllers: A Critical Synthesis of Theory and Industrial Needs’’ by Michael Nikolaou of the University of Houston presents an overview of some of the most important recent developments in MPC theory, and their implications. JAMES WEI

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS J. B. Joshi,* N. S. Deshpande,1 M. Dinkar,2 and D. V. Phanikumar3 Department of Chemical Technology, University of Bombay, Matunga, Mumbai-400 019, India I. Introduction 2 II. Generalized Criterion for Unbounded Dispersions 6 A. Solid–Liquid Fluidized Beds 7 B. Gas–Solid Fluidized Beds 15 C. Gas–Liquid Bubble Columns 18 D. Liquid–Liquid Spray Columns 18 18 E. Typical Behavior of f1 III. Review of Stability Criteria Based on Fundamental Approach 22 IV. Review of Stability Criteria Based on Heuristic Approach 28 A. Relationship between Fundamental Approach and Heuristic Approach 32 V. Model Predictions and Experimental Observations 40 A. Estimation of Model Parameters 40 B. Stability Maps 46 C. Comparison of Experimental Results with Unbounded Bed Analysis 65 VI. Generalized Criteria for Bounded Dispersions 69 A. Mathematical Model 73 B. Estimation of Model Parameters 90 C. Results and Discussion 91 D. Comparison with Experimental Data 99 VII. Comparison of Bounded and Unbounded Analysis 100 VIII. Three-Phase Fluidization 103 A. Introduction 103 B. Heuristic Models 105 C. New Criterion for the Prediction of Contraction/ Expansion 111 * Corresponding author. Present addresses: 1 Hindustan Lever Research Center, Chakala, Andheri (East), Mumbai 400; 2 Principal Research Scientist, Honeywell Technology Center, 3660 Technology Center, Minneapolis, MN 55418, U.S.A.; and 3 GE India Technology Center, Innovation Building, International Technology Park, Whitefield Rd., Banglore-560066 India. 1 ADVANCES IN CHEMICAL ENGINEERING, VOL. 26

Copyright  2001 by Academic Press. All rights of reproduction in any form reserved. 0065-2377/01 $35.00

2

J. B. JOSHI ET AL. IX. Conclusions X. Suggestions for Future Work Appendix A: Modeling of Correlation of Fluctuating Pressure and Hold-up Appendix B: Forces Acting on a Particle in a Fluidized Bed Nomenclature References

113 114 115 115 122 127

Gas–liquid bubble columns, solid–liquid fluidized beds, gas– solid fluidized beds, gas–liquid–solid fluidized beds, and transport reactors are very widely used multiphase reactors. These reactors operate in either of the two characteristic regimes: particulate or homogeneous, and aggregative or heterogeneous. The rates of heat, mass, momentum transfer, and mixing are quite different in these two regimes. Therefore, it is important to know the range of operating and design parameters over which the two regimes prevail and the conditions under which the transition occurs. This subject has been extensively investigated during the past 50 years and numerous fundamental, semiempirical, and empirical approaches have been reported for the prediction of transition. All these studies have been analyzed in this monograph. Further, unified and generalized criteria have been developed. In view of these, the past published results have been discussed. Stability maps have been presented. For several multiphase reactors, comprehensive comparison has been presented between the predicted and the experimental conditions of transitions. The characteristic differences among the various multiphase systems have been brought out. Suggestions have been made for future work.  2001 Academic Press. I. Introduction In multiphase reactors, the dispersed phase moves in one of two characteristic regimes, depending upon the nature of dispersion. The two regimes are homogeneous and heterogeneous. These regimes are commonly known as particulate and aggregative, respectively. In bubble columns, the gas phase exists as the dispersed phase in the continuous liquid phase. The homogeneous regime is characterized by almost uniformly sized bubbles. Further, the concentration of bubbles is uniform, particularly in the transverse direction. Therefore, bulk liquid circulation is practically absent. If the gas is sparged uniformly at the column bottom, it remains uniformly distributed all over the column. All the bubbles rise virtually vertically with minor transverse and axial oscillations. For all

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

3

bubble sizes, there is practically no coalescence or redispersion. Hence, size of the bubbles in homogeneous regime is almost entirely dictated by the type and design of the sparger and the physical properties of the system. In contrast, the heterogeneous regime is characterized by nonuniform bubble concentration, especially in the transverse direction, and the existence of gross liquid circulation. There exists a wide bubble size distribution in the main bulk region, and the average bubble size in the bulk region is governed by coalescence and redispersion phenomena, which in turn are controlled by the energy dissipation rate in the bulk. The intensity of turbulence is also much higher. Such highly turbulent recirculation results into substantially high values of eddy diffusivities for mass, heat, and momentum. As a result, the rates of heat and mass transfer and mixing are quite different in homogeneous and heterogeneous regimes. Therefore, it is important to know the range of physical properties and operating parameters over which the two regimes prevail. As the transition from the homogeneous to the heterogeneous regime starts, there is an onset of liquid circulation that is upward in the central region and downward near the column wall. So more bubbles enter the central region, as it is the path of lower resistance. As a result, a transverse holdup profile begins to build up, which in turn intensifies the liquid circulation. Therefore, the beginning of transition regime is very important, and further development is selfpropagating. Joshi (1981, 1983) and Shnip et al. (1992) have discussed the pertinent details of these two regimes in bubble columns. In solid–liquid fluidized beds the particle phase is the dispersed phase and the bed usually operates in the particulate (homogeneous) regime. However, for heavy particles (large size and density or high terminal settling velocity), heterogeneity sets in. In gas–solid fluidized beds, the particle phase is the dispersed phase up to a certain critical superficial gas velocity. Above this, the excess gas is considered to move in the form of bubbles, and the bubble phase forms the dispersed phase. The critical superficial gas velocity usually equals the minimum fluidization velocity. However, in the case of fine powders or at high pressure, the bed expands homogeneously even above the incipient fluidization velocity without the formation of bubbles. The bubbles are formed at much higher velocity where the void fraction of the emulsion phase is greater than that at the incipient fluidization. It may be pointed out at this stage that the dispersed bubble phase (which is formed either at the incipient fluidization velocity or at a higher velocity) can also remain in the homogeneous regime up to a certain superficial gas velocity, and the formation of bubbles need not be considered as equivalent to the beginning of the heterogeneous regime. The term ‘‘three-phase fluidization’’ refers to that operation in which an upward cocurrent flow of liquid and gas fluidizes the solid particles. The

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J. B. JOSHI ET AL.

gas flows as bubbles. For such operations, knowledge of the expansion properties of the bed as well as holdups of the individual phases is important in determining the size and efficiency of the equipment. In three-phase fluidized beds, the particles and the bubbles form the dispersed phases. However, the behavior of bubbles plays a dominant role in deciding the regime of operation and the regime transition can be viewed as similar to that in bubble columns. There is a small difference in that the bubble size (and velocity) depends upon the particle size, density, and solid loading (Pandit and Joshi, 1984, 1986; Khare and Joshi, 1990). The three-phase reactors pose yet another interesting problem. When the three-phase dispersion is obtained by the introduction of gas phase in a solid–liquid fluidized bed, the bed either contracts or expands. This observation is now very well documented and carefully analyzed by many investigators, including Turner (1964), Stewart and Davidson (1964), Ostergaard (1965), Darton and Harrison (1975), Epstein and Nicks (1976), Epstein (1976), and ElTemtamy and Epstein (1979). In a solid–liquid fluidized bed, the particle settling velocity is less than the terminal settling velocity because of hindered settling. When the gas is introduced in a particulate fluidized bed, the gas bubbles extract energy from the liquid phase. As a result, the drag on each particle is reduced the particle settling velocity increases and bed contraction occurs. When gas is introduced in an aggregative fluidized bed, the gas supplies energy to the liquid phase. This energy is reflected in higher turbulence intensity in the liquid phase. Because of this, the particle settling velocity gets further hindered and bed expansion occurs. This initial behavior of contraction/expansion has been shown to depend upon the regime transition (Joshi, 1983). The behavior needs further reexamination in view of recent developments in the understanding of multiphase phenomena. In the past, some criteria have been developed using both empirical and relatively fundamental approaches. The latter approach can be classified into two categories. In the first category, the multiphase system is assumed to have no bounds. Therefore, this approach does not consider the existence of either the sparger or the column wall. Such beds may be termed as unbounded beds. In the second approach, the existence of the sparger plate and column wall is considered; such beds may be termed bounded beds. In both these cases, the method of linear stability has been used for obtaining the transition criterion. The state variables such as the local holdup and the phase velocities are given small perturbations. In the homogeneous regime these small disturbances die out with time, indicating that the beds are stable to small fluctuations. If the regime of operation is heterogeneous, then the small perturbations that are superimposed in the homogeneous regime grow exponentially with time, and nonuniformity sets in. This makes the bed heterogeneous, indicating that the beds are unstable.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

5

The terms unstable bed and heterogeneous regime are therefore used synonymously. Jackson (1963a, 1963b, 1964) was one of the pioneers in formulating the set of Navier–Stokes equations to describe the fluid dynamics of fluidized beds (both bounded and unbounded beds). Jackson and co-workers (Anderson and Jackson, 1967a, 1967b, 1968, 1969; Jackson, 1985; Medlin et al., 1974; Medlin and Jackson, 1975; Agarwal et al., 1980) have also developed the stability criterion. Homsy and co-workers (El-Kaissy and Homsy, 1976; Homsy and El-Kaissy, 1980; Didwania and Homsy, 1981a, 1981b; Green and Homsy, 1987a, 1987b; Ham et al., 1990) performed a comprehensive analysis of unbounded beds and presented their dynamic behavior. They have also reported a systematic set of experimental data for the dynamic behavior. Rietema (1973) and Musters and Rietema (1977) have also presented unbounded bed analysis for gas–solid fluidized beds. Batchelor (1988) systematically derived the particle phase force balance and attributed physical significance to all the terms in the force balance equation. In addition to these theoretical approaches, Wallis (1969), Verloop and Heertjes (1970), and Gibilaro, Foscolo, and co-workers (Foscolo and Gibilaro, 1984; Gibilaro et al., 1984, 1986, 1987a, 1987b, 1989, 1990) have presented very interesting criteria on the basis of heuristic arguments. All the just-mentioned analyses focus more or less on fluidized beds and under unbounded conditions. The analysis of all these investigators concentrated on whether a periodic disturbance in the axial direction grows with respect to time, leading to instability, or decays with time, indicating a stable system. The treatment is strictly for axial direction and the axial nonuniformities are the source of transition. Analyses such as these have the limitation that the real fluidized bed is not of infinite extent, and that some account must be taken of the boundaries at the upper and lower surface of a bed, the finite depth, and also the walls that bound the beds laterally. Further, the one-dimensional unbounded description does not generally explain all the experimental observations at transition. Shnip et al. (1992) have used the theory of linear stability for the analysis of bounded beds. The present work aims at the following: (i) To derive the stability criterion for gas–liquid, solid–liquid, gas–solid, and gas–liquid–solid unbounded dispersions. (ii) To understand the physical significance of all the terms in the stability criterion. (iii) To present the relative merits of all the previous approaches. (iv) To make an attempt to give physical significance to the unbounded and bounded bed analyses. (v) To analyze all the published experimental information on the stability of gas–liquid, solid–liquid, gas– solid, and gas–liquid–solid systems. (vi) To bring out the relative utility of the unbounded and bounded bed analyses. (vii) To present a generalized

6

J. B. JOSHI ET AL.

procedure for checking the stability of multiphase systems. (viii) To present stability maps for all types of multiphase reactors from which the stable regions can easily be identified.

II. Generalized Criterion for Unbounded Dispersions The transition from homogeneous to heterogeneous regime in case of bubble columns was qualitatively described in the introduction. At a certain critical superficial gas velocity, a small disturbance in the hold-up profile sets in liquid circulation, which in turn magnifies the hold-up disturbance. In this way, the disturbance grows, and finally an approximately parabolic gas hold-up profile and intense liquid circulation develop, which are the characteristics of the heterogeneous regime. Now, consider the case of a solid–liquid fluidized bed operating in the particulate regime. If the liquid flow rate is increased, the bed still remains in the particulate regime up to a certain liquid flow rate. After this, alternate bands of high and low density start appearing in the bed. If the liquid flow rate is increased still further, the bed starts operating in the aggregative or heterogeneous regime characterized by nonuniform hold-up profiles. In the following section, a criterion has been developed on the basis of linear stability theory. This approach has been described by Jackson (1963a). It will consist of the following steps: (i) The starting point is the equations for continuity and motion for solid–liquid dispersion under turbulent conditions. These equations will be derived in detail, using the time averaging of instantaneous equations and introducing usual turbulent modeling. (ii) The equations of motion of both the phases will be combined to eliminate the pressure term. This step is very important and useful. (iii) Perturbations will be introduced and equations of continuity and motion will be written in terms of perturbation variables. (iv) The resulting equations are linearized. (v) The velocity perturbations will be eliminated using the equations of continuity, and a final linearized equation will be obtained in terms of perturbation in fractional hold-up. (vi) Under the homogeneous regime, any perturbation to the flat hold-up profile decays with respect to time. In contrast, if the small perturbations grow with time, transition to the heterogeneous regime occurs. In the linear stability analysis, perturbations are given in form of a periodic function and the solution to the perturbation equation is obtained for neutral stability. (vii) Initially, the solid–liquid dispersion will be assumed to have no bounds (unbounded case). In other words, the existence of the sparger or the column wall will

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

7

not be considered. Later on, the analysis will be extended to a case where the existence of the sparger and column wall is considered (bounded case in Section VI). (viii) The following analysis is for solid–liquid fluidized beds. However, the same analysis applies to other multiphase systems N such as gas–solid fluidized beds and gas–liquid bubble columns.

A. SOLID –LIQUID FLUIDIZED BEDS Let us begin with the unbounded case. The following additional assumptions are made: 1. The concept of interpenetrating continua (continuum approach) has been used. It involves treating the multiphase dispersion as a continuum in which the variables are related to local averages over regions that are large enough compared with the dispersed particles (bubbles or solid particles) and small enough compared with the length scales over which the variables can be assumed to remain constant. 2. The continuous phase (liquid) is incompressible. This implies that the density is not a function of both time and space. However, the dispersion is compressible because the local hold-up is not a constant. The compressibility is accounted for by the spatial variation of the fractional solid holdup, ␧S . 3. Particles are perfectly spherical and monosized (no size distribution). 4. The flow is assumed to be turbulent. Whenever there is a gradient in solid or liquid holdup, the transport of liquid and solid phases has been assumed to occur by dispersion. Therefore, these terms appear in the equations of continuity. 5. The theory of linear stability has been used. Jackson (1963a) has pioneered this method for the analysis of multiphase dispersions. The pertinent details have been given by Shnip et al. (1992). Using these assumptions, the fundamental transport equations for solid– liquid fluidized beds are given next.

1. The Governing Equations The equation of continuity for solid phase in a one-dimensional coordinate system can be written as ⭸␧S ⭸ ⫹ (␧Svz) ⫽ 0. ⭸t ⭸z

(1)

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J. B. JOSHI ET AL.

Following the procedure of Reynolds averaging, the instantaneous quantities are written in terms of time-averaged and fluctuating quantities as ␧S ⫽ ␧¯ S ⫹ ␧⬘S , vz ⫽ v¯z ⫹ v⬘z ,

(2)

where the overbars indicate the time-averaged quantities and the primes indicate the fluctuating quantities. Substituting Eq. (2) in Eq. (1) and noting that time average of fluctuating quantities is zero, we get ⭸␧¯ S ⭸ ⭸ ⫹ (␧¯ Sv¯z) ⫽ ⫺ (␧⬘Sv⬘z). ⭸t ⭸z ⭸z

(3)

The third term of Eq. (3) contains ␧⬘Sv⬘z, which is generally modeled in terms of turbulent dispersion in a manner analogous to the well-known gradient hypothesis of Boussinesq, as proportional to the gradient of holdup in the z direction, the constant of proportionality being referred to as the turbulent dispersion coefficient: ␧⬘Sv⬘z ⫽ ⫺DS

⭸␧¯ S . ⭸z

(4)

Substitution of Eq. (4) in (3) gives the equation of continuity as (after removing overbars for convenience)

冉

冊

⭸␧S ⭸ ⭸ ⭸␧ ⫹ (␧Svs) ⫺ DS S ⫽ 0. ⭸t ⭸z ⭸z ⭸z

(5)

The equation of continuity for the liquid phase can be derived in a similar manner and is given by

冉

冊

⭸␧ ⭸ ⭸␧L ⭸ ⫹ (␧Luz) ⫺ DL L ⫽ 0. ⭸t ⭸z ⭸z ⭸z

(6)

Equations (5) and (6) were derived for solid–liquid dispersions. These can be easily adapted to any other multiphase system such as gas–solid and gas–liquid systems. The one-dimensional equation of motion for solid phase in terms of instantaneous quantities is given by

␳S

冋

册

⭸ ⭸ ⭸p (␧Svz) ⫹ (␧Svzvz) ⫽ ⫺␧S ⫹ ␧S␳Sg ⫺ ␧L fz ⫺ ⵜ.␶S , ⭸t ⭸z ⭸z

(7)

where fz is the interaction force. It is a function of the slip velocity between the two phases and the relative acceleration of the two phases. It consists of drag force and the virtual mass force. The drag force per unit volume is given by fzd ⫽ C ⬘D␧S(vz ⫺ uz)

(8a)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

9

where C ⬘D ⫽ (␧S ⫺ ␧L)g/(v¯ z ⫺ u¯ z). It may be noted that C ⬘D is not conventional dimensionless drag coefficient. It has dimensions of kg/m 3s or N · s/m 4. The virtual mass force is given by fzv ⫽ ␧SCv␳L

D (uz ⫺ vz), Dt

(8b)

where the term Duz /Dt is based on the continuous phase velocity and Dvz /Dt is based on the dispersed phase velocity. Therefore, the total interaction force is fz ⫽ C ⬘D␧S(vz ⫺ uz) ⫹ ␧SCv␳L

D (vz ⫺ uz). Dt

(9)

It is known that Cv is either considered as a constant (Cook and Harlow, 1986; Drew et al., 1979) or as a function of voidage (Ham et al., 1990). In the following analysis it will be assumed to be constant. Using Reynolds’ averaging procedure for Eq. (7), and neglecting the laminar stresses, we get:

␳S

冋

⭸ ⭸ (␧¯ Sv¯z ⫹ ␧⬘Sv⬘z) ⫹ (␧¯ Sv¯zv¯z ⫹ ␧¯ Sv⬘zv⬘z ⫹ v¯z␧⬘Sv⬘z ⫹ v¯z␧⬘Sv⬘z) ⭸t ⭸z

冉

⫽ ⫺ ␧¯ S

冊

册

⭸p¯ ⭸p⬘ ⫹ ␧¯ S␳Sg ⫺ C ⬘D␧¯ L␧¯ S(v¯z ⫺ u¯z) ⫺ C ⬘D␧⬘L␧⬘S(vz ⫺ u¯z) (10) ⫹ ␧⬘S ⭸z ⭸z

⫺ C ⬘D(1 ⫺ 2␧L)(␧⬘Sv⬘z ⫹ ␧⬘Lu⬘z) ⫺ Cv␳L␧¯ L␧¯ S ⫺ Cv␳L␧⬘L␧⬘S

D (v¯z ⫺ u¯z) Dt

冋

册

D ⭸v ⭸u¯ (v¯z ⫺ u¯z) ⫺ Cv␳L(1 ⫺ 2␧L) ␧⬘Sv⬘z z ⫺ ␧⬘Lu⬘z z . Dt ⭸z ⭸z

In writing the preceding equation, we have neglected all the triple and quadruple products of the fluctuating quantities. Also, the following double products are assumed to be zero: ␧⬘S

⭸v⬘z ⭸v⬘z ⭸v⬘z ⫽ ␧⬘S ⫽ v⬘z ⫽ 0. ⭸t ⭸z ⭸z

(11a)

Further, v⬘zv⬘z ⫽ ⫺2␯t

⭸v¯z ⭸␧¯ ⭸␧¯ , ␧⬘Sv⬘z ⫽ ⫺DS S , ␧⬘Lu⬘z ⫽ ⫺DL L , ⭸z ⭸z ⭸z

(11b)

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J. B. JOSHI ET AL.

where vt represents the turbulent kinematic viscosity. DS and DL are the solid-and liquid-phase dispersion coefficients. ␧⬘L␧⬘S(v¯z ⫺ u¯z) ⫽ ⫺␧⬘Ll

⭸␧¯ S (v¯z ⫺ u¯z) ⭸z

⫽ ⫺␧⬘L l

⭸␧¯ S (v¯z ⫺ u¯z) ⭸z

⫽ ⫺ ␰D S

⭸␧¯ S . ⭸z

(12)

It has been assumed that ␧⬘Ll(v¯z ⫺ u¯z) ⫽ lvS␧⬘L ⫽ ⫺ ␰Ds, i.e., some proportion of the dispersion coefficient. The pressure coupling term has the same magnitude but the opposite sign in the continuous and the dispersed phase momentum equations, which implies a transfer of momentum between the phases. The correlation between the fluctuating hold-up and fluctuating pressure can be modeled as follows (Appendix A):

冉

冊

⭸p⬘ ⭸␧ ⭸v¯ ⭸ ⭸␧¯ 1 ␧⬘ DS S . ⫽ ⫹DS S z ⫹ v¯z ␳S S ⭸z ⭸z ⭸z ⭸z ⭸z

(13)

Substitution of Eq. (11)–(13) in Eq. (10), using the continuity equation (5) for simplification, assuming DS and DL to be independent of z, and removing the overbars for convenience, we get

冉

冊

⭸vz ⭸v ⫹ vz z ⭸t ⭸z ⭸p D ⫽ ⫺␧S ⫹ ␧S␳Sg ⫺ C ⬘D␧L␧S(vz ⫺ uz) ⫺ Cv␳L␧L␧S (vz ⫺ uz) ⭸z Dt ⭸␧ ⭸v ⭸ ⭸v ⭸␧ ⭸v ⭸␧ ⫹ 2␳S␯t S z ⫹ 2␧S␳S ␯t z ⫹ ␳SDS S z ⫹ C ⬘D␰DS S (14) ⭸z ⭸z ⭸z ⭸z ⭸z ⭸z ⭸z

␧S␳S

冉 冋

⫹ C ⬘D(1 ⫺ 2␧L) DS

冉 冊 冊

冉

⭸␧S ⭸␧ ⭸␧ Dvz Duz ⫹ D L L ⫹ C v ␳ L ␰ DS S ⫺ ⭸z ⭸z Dz Dt Dt

⫹ Cv␳L(1 ⫺ 2␧L) DS

册 冉

冊

冊

⭸␧S ⭸vz ⭸vz ⭸␧ ⭸u ⭸␧ ⫺ DL L z ⫹ ␳S DS S . ⭸z ⭸z ⭸z ⭸z ⭸z ⭸z

If we assume the equality of turbulent kinematic viscosities of the two phases, the z component of equation of motion for the liquid phase can

11

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

be written as ␧L␳L

冉

⭸uz ⭸u ⫹ uz z ⭸t ⭸z

⫽ ⫺␧L

冊

⭸p D ⫹ ␧L␳Lg ⫹ C ⬘D␧L␧S(vz ⫺ uz) ⫹ Cv␳L␧L␧S (vz ⫺ uz) ⭸z Dt

⫹ 2 ␳L␯t

冉 冊 冊

⭸␧L ⭸uz ⭸ ⭸u ⭸␧ ⭸u ⭸␧ ␯t z ⫹ ␳LDL L z ⫺ C ⬘D␰DS S ⫹ 2␧L␳L ⭸z ⭸z ⭸z ⭸z ⭸z ⭸z ⭸z

冉 冋

⫺ C ⬘D(1 ⫺ 2␧L) DS

冉 冉

⭸␧S ⭸␧ ⭸␧ Dvz Duz ⫹ DL L ⫺ Cv␳L␰DS S ⫺ ⭸z ⭸z ⭸z Dt Dt

⫺Cv␳L(1 ⫺ 2␧L) DS

册

冊

(15)

冊

⭸␧S ⭸vz ⭸␧ ⭸u ⭸ ⭸␧ DL L . ⫺ D L L z ⫹ ␳L ⭸z ⭸z ⭸z ⭸z ⭸t ⭸z

The equations of motion of the two phases can be combined by eliminating the pressure gradient, ⭸p/⭸z. This is achieved by multiplying Eq. (15) by (1 ⫺ ␧L)/␧L and subtracting the resulting equation from Eq. (14). The resulting equation is (1 ⫺ ␧L)[␳S ⫹ ␳LCv]

冋

册

⫽ (1 ⫺ ␧L)(␳S ⫺ ␳L) g ⫺ 웁(vz ⫺ uz) ⫹ 2␳S␯t

冉 冊 冉 冊

⫺ 2((1 ⫺ ␧L)/␧L)␳L␯t ⫹ 2(1 ⫺ ␧L)␳S

冉

⫹ ␳S DS

冋

⭸vz ⭸v ⭸uz ⭸u ⫹ vz z ⫺ (1 ⫺ ␧L)␳L[1 ⫹ Cv] ⫹ uz z ⭸t ⭸z ⭸t ⭸z

冉 冊

册

⭸␧S ⭸vz ⭸z ⭸z

⭸␧L ⭸uz ⭸z ⭸z

冉 冊 冊 冉

⭸ ⭸v ⭸ ⭸u ␯t z ⫺ 2(1 ⫺ ␧L)␳L ␯t z ⭸z ⭸z ⭸z ⭸z

⭸␧S ⭸vz ⭸␧ ⭸u ⫺ [(1 ⫺ ␧L)/␧L]␳LDL L z ⭸z ⭸z ⭸z ⭸z

⫹ [1 ⫹ (1 ⫺ ␧L)/␧L]C ⬘D␰DS

(16)

冊

⭸␧S ⭸␧ ⭸␧ ⫹ C ⬘D[(1 ⫺ 2␧L)/␧L] DS S ⫹ DL L ⭸z ⭸z ⭸z

⫹ [1 ⫹ (1 ⫺ ␧L)/␧L]Cv␳L␰DS

冉

Dvz Duz ⫺ Dt Dt

冊

12

J. B. JOSHI ET AL.

冋

⫹ Cv␳L[(1 ⫺ 2␧L)/␧L] DS ⫺[(1 ⫺ ␧L)/␧L]␳L

冉

册

冉

⭸␧S ⭸vz ⭸␧ ⭸u ⭸ ⭸␧ DS S ⫺ D L L z ⫹ ␳S ⭸z ⭸z ⭸z ⭸z ⭸t ⭸z

冊

冊

⭸ ⭸␧ DL L , ⭸t ⭸z

where 웁 ⫽ C ⬘D␧S ⫽

(␳S ⫺ ␳L) g(1 ⫺ ␧L) . (vz ⫺ uz)

2. Solution under Steady-State Conditions Under conditions of the homogeneous regime, the steady-state conditions for solid–liquid dispersion are given as follows. At time t ⫽ 0: v z ⫽ v0 ⫽ 0

(17a)

uz ⫽ u0 ⫽ constant

(17b)

␧L ⫽ ␧L0 ⫽ constant.

(17c)

Substitution of Eqs. (17a), (17b), and (17c) in Eqs. (1), (2), and (16) gives the corresponding equations at steady state. The continuity equations are trivially satisfied and Eq. (16) gives 웁0 ⫽ ⫺

(␳S ⫺ ␳L) g(1 ⫺ ␧L0) , u0

(18)

where 웁0 is the steady-state drag coefficient evaluated at the steady state, ␧L ⫽ ␧L0 . It may be pointed out that 웁0 is dimensional having units of kg/m 3 · s. 3. Linearization According to the theory of linear stability analysis, infinitesimally small perturbations are superimposed on the variables in the steady state and their transient behavior is studied. At this stage the difference between turbulent fluctuations and perturbations may be noted. Turbulence is the characteristic feature of the multiphase flow under consideration; the mean and fluctuating quantities were given by Eq. (2). The fluctuating components result in eddy diffusivity of momentum, mass, and Reynolds stresses. The turbulent fluctuations do not alter the mean value. In contrast, the perturbations are superimposed on steady-state average values and another steady

13

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

state is obtained. Therefore, in our procedure, we first time-smooth to erase turbulent fluctuations and then examine the larger scale instabilities in averaged variables. The time and length scale of perturbations is usually much higher than those for turbulent fluctuations. Using Eq. (17) we get the variables in their perturbed state as ␧L ⫽ ␧L0 ⫹ ␧L1

(19a)

uz ⫽ u 0 ⫹ u 1

(19b)

vz ⫽ v 1 .

(19c)

The drag coefficient term (웁) is linearized with respect to the steady-state value 웁0 as 웁 ⫽ 웁0 ⫹ 웁⬘0␧L1 .

(19d)

We now introduce perturbations in the equations of continuity (5) and (6). We get ⭸␧L1 ⭸2␧L1 1 ⭸v1 . (20a) ⫽ ⫺ DS ⭸z (1 ⫺ ␧L0) ⭸t ⭸z2

冋

册

The continuity equation for the liquid phase can be written in a similar fashion:

冋

册

⭸u1 1 ⭸␧L1 ⭸␧ ⭸2␧L1 . ⫽⫺ ⫹ u0 L1 ⫺ DL ⭸z ␧L0 ⭸t ⭸z ⭸z2

(20b)

Similarly, we introduce the perturbations into the equation of motion (16) and neglect the nonlinear terms arising out of it. We also note two more points. First, ⫺C ⬘D␰Ds ⭸␧s /⭸z ⫽ C ⬘D␧⬘L␧⬘S(v¯z ⫺ u¯z) is also nonlinear, and we neglect it. Further, ␧S ⫹ ␧L ⫽ 1 and ⭸␧S /⭸z ⫽ ⫺⭸␧L /⭸z. Simplification of Eq. (16) gives (1 ⫺ ␧L0)[␳S ⫹ ␳LCv]

冉 冊

冋

⭸v1 ⭸u1 ⭸u ⫺ (1 ⫺ ␧L0)␳L[1 ⫹ Cv] ⫹ u0 1 ⭸t ⭸t ⭸z

⫽ (␳S ⫺ ␳L)(⫺␧L1) g ⫺ 웁0(v1 ⫺ u1) ⫺ 웁⬘0(␧L1)(v0 ⫺ u0) ⫹ (1 ⫺ ␧L0)␳S␯t ⫺ ␳S

冉

冋 冉 冊册 冊 冉 ⭸ ⭸v 2 1 ⭸z ⭸z

册

冋 冉 冊册 冊

⫺ (1 ⫺ ␧L0)␳L␯t

⭸ ⭸␧ (1 ⫺ ␧L0) ⭸ ⭸␧ DS L1 ⫺ ␳L DL L1 . ⭸t ⭸z ⭸L0 ⭸t ⭸z

⭸ ⭸u 2 1 ⭸z ⭸z

(20c)

14

J. B. JOSHI ET AL.

Now we take the divergence of Eq. (20c) and simplify the resulting equation using Eqs. (20a) and (20b) to get A

⭸2␧L1 ⭸2␧L1 ⭸2␧L1 ⭸3␧L1 2 ⫹C 2 ⫹B 2 ⫹ Zⵜ ␧L1 ⫹ E ⭸t ⭸t⭸z ⭸z ⭸t⭸z2

(21)

⭸4␧L1 ⭸␧ ⭸␧ ⫽ 0. ⫹ F L1 ⫹ G L1 ⫹ Hⵜ4␧L1 ⫹ I ⭸t ⭸z ⭸z4

Equation (21) is similar to the general expression obtained by Anderson and Jackson (1968) and Liu (1982), where A, B, C, Z, E, F, G, H, and I are constants given as follows: A⫽1⫹ B⫽2 C⫽

冋

␳L (1 ⫹ Cv) ⫺1 ␳S ␧L0

册

(22a)

1 ⫺ ␧L0 ␳L (1 ⫹ Cv)u0 ␧L0 ␳S

(22b)

1 ⫺ ␧L0 ␳L (1 ⫹ Cv)u20 ␧L0 ␳S

(22c)

웁0 DS D ⫹ L ␳S (1 ⫺ ␧L0) ␧L0

(22d)

Z⫽⫺

冋 冋

冉

E ⫽ ⫺2 1 ⫹

冊 册 册

冉

(1 ⫺ ␧L0)␳L ␳ C DS␧L0 ⫹ (1 ⫺ ␧L0)DL ␯t ⫺ L v ␧L0␳S ␳S ␧L0

冊

(22e)

F⫽

웁0 1 1 ⫹ ␳S ␧L0 (1 ⫺ ␧L0)

G⫽

웁0u0 웁⬘0u0 (␳S ⫺ ␳L) ⫺ ⫹ g ␧L0␳S ␳S ␳S

(22g)

H⫽

2 ␯t [DS␳S␧L0 ⫹ (1 ⫺ ␧L0)␳LDL] ␧L0␳S

(22h)

I⫽ ⫺

2␳Lu0(1 ⫺ ␧L0) [(1 ⫹ Cv)DL ⫹ ␯t]. ␧L0␳S

(22f)

(22i)

We seek a solution to the foregoing partial differential equation to be of the form ␧L1 ⫽ ␧L0 eikzest,

(23)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

15

where k is the wave number in the z direction and s is the growth rate constant in time. If the real part of s is positive, then there is positive growth. This indicates that the disturbance or the perturbation will grow in time. For neutral stability the real part of s will be zero and for stability the real part of s is negative. Substituting Eq. (23) in Eq. (21), we get a quadratic equation in s. Equating then the real and the imaginary parts of the quadratic equation, we get the final stability criterion as [A(G/F ) ⫺ B/2]2 ⬍ 1. A(Z ⫺ C ) ⫹ B2 /4

(24)

Equation (24) gives the stability criterion for the solid–liquid fluidized beds. Equation (24) can be rewritten as f1 ⫽ 1 ⫺

[A(G/F ) ⫺ B/2]2 . A(Z ⫺ C ) ⫹ B2 /4

(25)

The solid–liquid fluidized bed is stable when f1 is positive, unstable when f1 is negative, and neutrally stable when f1 ⫽ 0. It may be noted that the viscous terms given by constants E, H, and I appear with second-order derivative terms and these have a stabilizing effect for large wave number disturbances. Liu (1982) has recast Eq. (21) in a form interpretable in terms of wave hierarchies. He has shown that the terms involving A, B, C, F, Z, and G appear in the expressions for lower order kinematic wave velocity and higher order elastic wave velocity, and that the stability criterion depends on the relative magnitudes of kinematic wave velocity and the elastic wave velocity. He has shown that for a humplike initial disturbance, the exponential growth is made milder by the viscous modification through the viscous term (E) by a factor 1/(␯et)1/2. The importance of viscous correction depends on the wavelengths relative to the effective viscous length scale (␯e(A/F))1/2, where A and F are given by Eqs. (22a) and (22f), respectively. This length scale will be still larger if we use eddy viscosity instead of molecular viscosity. The terms such as H and I do not appear in Liu’s analysis, but these are the terms with higher order similar to E, and the stability criterion is still decided by the relative magnitudes of kinematic and elastic wave velocity. B. GAS –SOLID FLUIDIZED BEDS The preceding mathematical analysis also holds for gas–solid fluidized beds. In this case, the gas phase is the continuous phase and the solid phase is the dispersed phase. The criterion given by Eq. (24) holds where the values of the constants are given in Table I. It may be noted that the terms

16

J. B. JOSHI ET AL. TABLE I PARAMETERS APPEARING IN THE GENERALIZED STABILITY CRITERION (UNBOUNDED ANALYSIS) FOR VARIOUS MULTIPHASE SYSTEMS

Solid–liquid fluidized beds A⫽1⫹ B⫽2 C⫽

册

冋

␳L (1 ⫹ CV) ⫺1 ␧L ␳S

u (1 ⫺ ␧L) ␳L (1 ⫹ CV) sup ␧L ␳S ␧L

冉 冊 冊

u (1 ⫺ ␧L) ␳L (1 ⫹ CV) sup ␧L ␧L ␳S

Z⫽⫺

冉

2

D DS 웁0 ⫹ L ␳S (1 ⫺ ␧L) ␧L

冋

册

F⫽

웁0 1 1 ⫹ ␳S ␧L (1 ⫺ ␧L)

G⫽

웁0 u0 웁⬘0u0 (␳S ⫺ ␳L) ⫺ ⫹ g ␳S ␧L ␳S ␳S

웁0 ⫽

(␳S ⫺ ␳L)(1 ⫺ ␧L) g vs

vS ⫽ v0 ⫺ u0 ⫽ ⫺u0 ⫽ ⫺VS앝␧Lm⫺1 Gas–solid fluidized beds A⫽1⫹ B⫽2 C⫽

冋

册

␳G (1 ⫹ Cv) ⫺1 ␳S ␧G

(1 ⫺ ␧G) ␳G u (1 ⫹ Cv) sup ␧G ␳S ␧G

冉 冊 冊

(1 ⫺ ␧G) ␳G u (1 ⫹ Cv) sup ␧G ␳S ␧G

Z⫽⫺

冉

DS 웁0 D ⫹ L ␳S (1 ⫺ ␧G) ␧G

冋

册

F⫽

웁0 1 1 ⫹ ␳S ␧G (1 ⫺ ␧G)

G⫽

웁0 u0 웁⬘0u0 (␳S ⫺ ␳G) ⫺ ⫹ g ␳S ␧G ␳S ␳S

웁0 ⫽

(␳S ⫺ ␳G)(1 ⫺ ␧G) g vs

vS ⫽ v0 ⫺ u0 ⫽ ⫺u0 m⫺1 ⫽ ⫺VS앝␧G

2

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS TABLE I (continued) Gas–liquid systems 1. Liquid phase velocity ⫽ 0 A⫽

␳G (1 ⫹ Cv) ⫺1 ⫹ ␧L ␳L

冋

B⫽2 C⫽

冋

册 册冉

␳G vsup ⫹ Cv (1 ⫺ ␧L) ␳L

␳G ⫹ Cv ␳L

Z⫽⫺

冊 册

vsup (1 ⫺ ␧L)

冋

2

DG 웁0 DL ⫹ ␳L ␧L (1 ⫺ ␧L)

冋

册

F⫽

웁0 1 1 ⫹ ␳L ␧L (1 ⫺ ␧L)

G⫽

웁⬘0v0 (␳G ⫺ ␳L) 웁0 v0 ⫹ ⫹ g ␳L (1 ⫺ ␧L) ␳L ␳L

웁0 ⫽

(␳G ⫺ ␳L)(1 ⫺ ␧L) g vs

vS ⫽ v0 ⫺ u0 ⫽ v0 ⫽ VB앝␧Lm⫺1 2. Liquid phase velocity ⫽ u0 (cocurrent or countercurrent) A⫽

␳G (1 ⫹ Cv) ⫹ ⫺1 ␳L ␧L

B⫽2 C⫽

冋

冋

册 册冉

␳G vsup (1 ⫺ ␧L) usup ⫹2 ⫹ Cv Cv ␳L (1 ⫺ ␧L) ␧L ␧L

␳G ⫹ Cv ␳L

Z⫽⫺

冊 册

vsup (1 ⫺ ␧L)

冋

2

⫹

冉 冊

(1 ⫺ ␧L) u Cv sup ␧L ␧L

2

웁0 DL DG ⫹ ␳L ␧L (1 ⫺ ␧L)

冋 冋

册 册

F⫽

웁0 1 1 ⫹ ␳L ␧L (1 ⫺ ␧L)

G⫽

웁⬘0 웁0 v0 u (␳ ⫺ ␳L) ⫹ 0 ⫹ (v0 ⫺ u0) ⫹ G g ␳L (1 ⫺ ␧L) ␧L ␳L ␳L

웁0 ⫽

(␳G ⫺ ␳L)(1 ⫺ ␧L) g (v0 ⫺ u0)

17

18

J. B. JOSHI ET AL.

␧S0 , ␧L0 , u0 , v0 , 웁0 , etc., in Eq. (22) indicate the initial condition required for the perturbation analysis. Now, after obtaining the constants, the subscript 0 (which indicates initial condition) has been deleted. The important use of Table I is to discern the operating regime of a given multiphase system. For instance, if for a given gas–solid fluidized bed (␳S , ␳L , ␧S , usup , dP) we are interested in finding the operating regime, then the stepwise procedure is as follows: (i) For given ␳S, ␳L, ␧S, usup, dP, etc., find the values of A, B, C, F, G, Z by using Table I. (ii) Find the operating regime using Eq. (24).

C. GAS –LIQUID BUBBLE COLUMNS The preceding mathematical analysis also holds for gas–liquid bubble columns. In this case, the gas phase is the dispersed phase and the liquid phase is the continuous phase. The criterion given by Eq. (24) holds where the values of the constants are given in Table I. For the procedure of using Table I, refer to Sections V.A.2 and V.B.2.

D. LIQUID –LIQUID SPRAY COLUMNS When the dispersed phase is the light phase, the constants are similar to those for bubble columns. The case of a heavy dispersed phase can also be handled in a similar manner.

E. TYPICAL BEHAVIOR OF f1 1. Fluidized Beds Figures 1 and 2 show typical behavior of f1 as a function of liquid holdup and gas hold-up for solid–liquid fluidized beds and gas–solid fluidized beds, respectively. In both these cases, three distinct regions can be observed. In region I, up to some critical value of hold-up (point P), the function is positive. In region II, the function is negative up to point Q, and in region III, the function again becomes positive. Region I represents the particulate regime for solid–liquid and gas–solid fluidized beds, with point P representing the liquid (or gas) hold-up up to which the solid–liquid (or gas–solid) fluidized bed remains in the particulate regime. To the right of the point P, the function f1 is negative and the region II represents the aggregative regime, up to the point Q. At point Q the function f1 again

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

19

FIG. 1. Typical behavior of f1 : solid–liquid fluidized beds. dP ⫽ 655.0 애m, 움 ⫽ 3.0; CV ⫽ f(␧); ␳L ⫽ 1000 kg/m3; 애L ⫽ 1.0 mPas.

becomes positive, and thus region III on the right of the point Q represents a homogeneous regime of a different sort. For solid–liquid and gas–solid fluidized beds, Region III may be considered as a homogeneous regime under dilute conditions, i.e., the solid hold-up is low (say 3–40%), and again there are no axial/radial hold-up variations. From Figs. 1 and 2, it can be seen that the transition at point Q (between regions II and III) is very sharp as compared to the transition at point P (between regions I and II). This difference in nature can be attributed to the different terms through which the function becomes negative. It can be seen from Eq. (25) that the numerator in the second term on the RHS is always positive, whereas its denominator can be either negative or positive. When the denominator becomes zero, f1 will become discontinuous. The other way in which the function f1 can become negative is when the denominator is smaller than the numerator. In the case of the transition from region I to region II, which is gradual, the function f1 becomes negative as the numerator exceeds the denominator (which remains positive). This generally happens when the inertial terms B and C are small, term A approaches unity, and thus the transition is decided by the comparison between G/F and 兹Z. It will be shown later

20

J. B. JOSHI ET AL.

FIG. 2. Typical behavior of f1 : gas–solid fluidized beds. dP ⫽ 40.0 애m, 움 ⫽ 3.0; CV ⫽ f(␧); ␳G ⫽ 1.15 kg/m3; 애G ⫽ 2 ⫻ 10⫺2 mPas.

that G/F represents the voidage propagation velocity and 兹Z represents the restoring or homogenizing velocity in case of fluidized beds. Thus transition from region I to region II occurs when the inertial terms are negligible, and the transition is decided by the relative magnitudes of the disturbance propagation velocity (G/F) and the restoring velocity (兹Z). In the case of transition from region III to region II, which is very sharp, the reason is that the denominator changes from negative to positive, and when the denominator approaches zero, the function f1 approaches ⫺ 앝. From Eq. (25), it can be seen that the sign of the denominator is governed by relative magnitudes of Z (the dispersion term) and C (the inertial term). Thus, for the transition from region III to region II, the relative magnitude of inertia and dispersion is generally the deciding factor and the gravity and interaction terms have little, if any, bearing on the transition from region III to region II. The sharp transition at point Q means that the function f1 is more sensitive to small changes in the inertial as well as the stabilizing dispersion term. A small change in the physical properties or in the value of other parameters, such as virtual mass coefficient, dispersion coefficient, or the slip velocity–terminal velocity relation, can cause a considerable change in the value of transition from region III to region II.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

21

2. Bubble Columns Figure 3 gives the typical behavior of f1 in the case of bubble columns as a function of liquid phase hold-up. Figure 3 is similar to Figs. 1 and 2 as far as its nature is concerned. However, the stable operating regime of fluidized beds is commonly defined in terms of continuous phase hold-up and that for bubble columns is commonly defined in terms of the dispersed phase (gas) hold-up. Thus, the homogeneous regime at low gas hold-up is represented by region III (and not region I). As the gas hold-up increases (liquid hold-up decreases), the point Q is approached. The point Q represents the critical gas hold-up at which the transition from homogeneous to heterogeneous regime occurs. Region II, to the left of point Q, represents the heterogeneous regime up to point P, at which the function again becomes positive. Region I (to the left of point P) has such a high gas holdup that the column may be thought as a stable gas–liquid froth (gas holdup is high and there are no axial or radial gradients in the gas hold-up). Any gas hold-up to the right of point P (region II) may be thought of as the gas hold-up below which the froth is unstable (gas hold-up becomes low, liquid hold-up becomes high, and liquid draining may cause axial gas hold-up gradients). In the present work, we have analyzed only one transition for all the multiphase systems. For bubble columns, we have considered the transition from region III to region II (point Q). In region I, gas hold-up is typically

FIG. 3. Typical behavior of f1 : gas–liquid bubble columns. dB ⫽ 40.0 애m, 움 ⫽ 0.5, CV ⫽ 1.0, ␳L ⫽ 1000 kg/m3; 애L ⫽ 1.0 mPas.

22

J. B. JOSHI ET AL.

greater that 70%, which appears difficult in bubble columns. For fluidized beds, we have considered the transition between region I and region II (point P). Other transitions (point P in case of gas–liquid flows and point Q in the case of gas–solid and solid–liquid systems) will be analyzed in future.

III. Review of Stability Criteria Based on Fundamental Approach In the previous section, stability criteria were obtained for gas–liquid bubble columns, gas–solid fluidized beds, liquid–solid fluidized beds, and three-phase fluidized beds. Before we begin the review of previous work, let us summarize the parameters that are important for the fluid mechanical description of multiphase systems. The first and foremost is the dispersion coefficient. During the derivation of equations of continuity and motion for multiphase turbulent dispersions, correlation terms such as ␧⬘Sv⬘z appeared [Eqs. (3) and (10)]. These terms were modeled according to the Boussinesq hypothesis [Eq. (4)], and thus the dispersion coefficients for the solid phase and liquid phase appear in the final forms of equation of continuity and motion [Eqs. (5), (6), (14), and (15)]. However, for the creeping flow regime, the dispersion term is obviously not important. The second important term is the virtual mass coefficient (Cv). When the dispersed phase accelerates (or decelerates) with respect to the continuous phase, the surrounding continuous phase has to be accelerated (or decelerated). For such a motion, additional force is needed, which is called ‘‘added or virtual mass force.’’ This force was given by the second term in Eq. (8). The constant Cv is called the virtual or added mass coefficient. It is difficult to estimate the value of Cv with the present status of knowledge. Therefore, many recommendations are available in the published literature. In an extreme case of potential flow, the value of Cv is 0.5. As regards the pressure, at a given location the pressure has been considered to be the same for all the phases. Further, when any area is occupied by different phases, the pressure has been assumed to be shared by the phases proportional to the fractional area occupied by the phases. This approach has been used by several investigators in the past (Kataoka, 1986; Kataoka and Serizawa, 1989; Elgobashi and Abou-Arab, 1983). As regards to the estimation of force due to buoyancy, there have been two schools of thought. One considers the force equal to vp␳fg where ␳r is the continuous phase density and vp is the particle volume (Clift et al., 1987; Clift, 1993; Joshi, 1983; Epstein, 1984). The other school considers the buoyancy force equal to vp␳Dg where ␳D is the average density of dispersion (Foscolo and Gibilaro, 1984; Gibilaro et al., 1987; Astarita, 1993).

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

23

The details pertaining to the debate are given in Appendix B. It has been concluded that the buoyancy force is governed by the fluid density. The drag force depends upon the particle Reynolds number. Different CD –Re relationships have been used in the past. Further, several investigators have attempted fluid-mechanical description of multiphase systems. It was thought desirable to analyze all these attempts in view of the discussion given earlier. Such an analysis is summarized in Table II. Some additional points are given next. Jackson and co-workers were probably the first to embark on the theoretical analysis of the regime transition. They were primarily interested in describing a fluidized bed with the help of equations of motion. Stability analysis was carried out as a test to support their model equations. They presented the analysis of unbounded as well as bounded beds. While comparing the model predictions with experimental observations, they needed the values of particle phase viscosity and particle phase pressure. They selected some values and predicted that the gas–solid and solid–liquid fluidized beds are always unstable. This was probably because the stabilizing effect of dispersion was not included in the model. Jackson (1985) assigned the significance of elasticity to particle phase pressure and considered this to be responsible for the bed stability. The method for the estimation of particle phase pressure was, however, not given. The order of magnitude for particle phase pressure was given by p⬘S 앜 u2sup ,

(26)

where usup is the superficial liquid velocity. Dorgelo et al. (1985) have reported the following correlation for the particle phase dispersion coefficient in solid–liquid fluidized beds: Ds 앜 u2sup .

(27)

The similarity between Eqs. (26) and (27) is worth noting. Batchelor (1988) has proposed that the dispersion is the principal cause for the bed stability. Homsy and co-workers consolidated the theory proposed by Jackson and co-workers. More importantly, they (El-Kaissy and Homsy, 1976; Didwania and Homsy, 1981a, 1981b, 1982; Ham et al, 1990) developed an elegant experimental setup and procedure for noting the transition. Their earlier experiments (El-Kaissy and Homsy, 1976; Didwania and Homsy, 1981a) showed transition at the condition of minimum fluidization itself. It was clearly shown later by Ham et al. (1990) that this observation was due to the selection of large particles. Further, Ham et al. (1990) systematically investigated the effect of particle size, liquid viscosity, and the density ratio on transition. They attributed the stability to the bed elasticity and showed

24

J. B. JOSHI ET AL.

that the elasticity is the combined effect of Reynolds stress and hydrodynamic dispersion. From the experimental results, they found the values of bed elasticity over a wide range of particle size, liquid viscosity, and density ratio. The resulting values of elasticity were empirically correlated. However, they have not suggested any procedure for the independent measurement of bed elasticity, on the basis of which the transition can be predicted. Rietema (1973) and Musters and Rietema (1977) presented the stability analysis of gas–solid fluidized beds. They have presented results of ingenious experiments where the effect of bed angle on the stability was investigated. It was convincingly shown that the bed becomes unstable only at a certain angle of tilting. Up to this angle, there is some force that stabilizes the bed and prevents it from becoming heterogeneous. This stability was

TABLE II SUMMARY OF TWO-PHASE FORMULATIONS USED BY VARIOUS INVESTIGATORS

Dispersion in the equation of continuity Dispersion in the equation of motion Definition of pressure Pressure sharing among the two phases Definition of buoyancyb Virtual mass effect Formulation of particle phase drag Any other feature a

JCa

HC

MR

BT

BG

LF

PB

SC

NC

NC

NC

NC

NC

NC

NC

NC

NC

NC

C

C5

C5

C

C

NC

␳s

␳s

—

8

11

NC

NC

NC

C

C

C

C

C

␳f

␳f

␳f

eD

eD

␳f

␳f

␳f

C

C

NC

C

C

C

C

NC

1

3

6

9

12

14

3

2

2

7

10

13

15

17

4

16

JC, Jackson and co-workers; HC, Homsy and co-workers; MR, Musters and Rietema; BT, Batchelor; BG, Biesheuvel and Gorrisen; LF, Lisseter and Fowler; PB, Pauchon and Banerjee; SC, Shnip and co-workers; C, considered; NC, not considered. b ␳f means the buoyancy is decided by the fluid density, whereas eD means the buoyancy is decided by the disperse phase mixture density. 1 웁, drag coefficient, depends only on hold-up and slip velocity. 2 Particle phase pressure has been assumed to provide the stabilizing mechanism. 3 움3 ⫽ ␧0(1 ⫺ ␧0)(␳s ⫺ ␳F) g/vS, where 움3 is the drag coefficient and Vs is the slip velocity. 4 Interparticle cohesion force was considered to provide the stabilizing mechanism.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

25

attributed to the cohesion between the particles, which imparts the property of elasticity to the bed. However, this hypothesis needs reexamination. Further, the analysis was restricted to beds where the particle Reynolds number was in the creeping flow region. The constant used for the evaluation of the drag term is for packed bed conditions and the changes due to fluidization are not included (for example, see Joshi, 1983, and Foscolo and Gibilaro, 1984). Other investigators have also argued in favor of interparticle forces. Tsinontides and Jackson (1993) have argued that stability in gas–solid fluidized beds is due to yield stresses of particle assemblies and the particle–particle contact forces are responsible for stability of fluidized beds. They have used the well-known hysteresis behavior that is observed when the flow rate is first increased (particulate–aggregative transiTABLE II (continued) 5

The dispersion in equation of motion (12) appears in the second-order derivative, while Batchelor (1988) has considered mobility to derive the dispersion term, which is a first-order derivative. 6 Drag coefficient has been obtained under creeping flow conditions. 7 Particle diffusion considered to stabilize while particle inertia forces promote the amplitude. 8 Kinetic contribution to the effective pressure is considered. 48 9 CD ⫽ ⫹ O(Re⫺3/2). Re 10 Uniform bubbly flow is unstable to void fraction disturbances above some critical value of the void fraction (⬎45%). 11 Liquid side pressure is considered mainly due to hydrostatic head. 6/7 2 kDb [1 ⫹ 17.67(␧3/2 L ) ] 12 CD ⫽ . 3/2 2 2 (18.67␧L ) 13 Wall friction has been considered. 14 Frictional force was assumed to be constant in the first case, while it was assumed to be a function of void fraction in the second case. 15 Turbulence has been assumed to provide the stabilizing mechanism through axial dispersion. 16 ⌬P ⫽ kVVG ⫹ kTV2G . 17 Transition from the homogeneous to the heterogeneous regime occurs as follows: Semibatch operation: gL v2B앝 앟 sinh(앟h) ⬍ . [kV ⫹ 2␧G0kTVS](1 ⫺ ␧G0)m⫺1[1 ⫺ (m ⫹ 1)␧G0] cosh(앟h) ⫺ 1 Continuous operation: gL v2B앝 [kV ⫹ 2␧G0kT(W0 ⫹ VS)](W0 ⫹ VS ⫹ ␧G0V⬘S)

⬍ 앟 coth(앟h).

26

J. B. JOSHI ET AL.

tion) and then decreased (aggregative–particulate transition). Also, the hysteresis is perhaps an inherent characteristic of transitions. For example, Maruyama et al. (1981) have observed similar hysteresis behavior in the case of gas–liquid bubble columns. They have argued that the hysteresis indicates that once the transition to the heterogeneous regime occurs, the heterogeneous regime with ordered liquid circulation is more stable than flow without ordered liquid circulation. In any case, the bubble–bubble contact forces or yield stresses may not be able to explain the observed hysteresis behavior in case of gas–liquid bubble columns. Clift (1993) has also argued that the hydrodynamic models are not sufficient to explain the stability of fluidized beds and that the interparticle forces that determine the elasticity of the bed are important to explain the stability. He has given expressions of Abdel-Ghani et al. (1991) for the mean elasticity modulus of the bed, E* ⫽

冉

9앟⌫E 2sd5P 16(1 ⫺ m2P)2

冊

1/3

,

(28)

where Es is the elasticity modulus of the material of the bed, ⌫ is the interfacial energy, and mp is the Poisson ratio of the material of the bed. For particles that contact at asperities with radius of curvature rc , the mean elasticity modulus is given by E* ⫽

冉

f(␧) 9앟⌫E 2sr5c dP 4(1 ⫺ m2P)2

冊

1/3

.

(29)

However, the measured values of elasticity modulus are several orders of magnitude lower than the values predicted by these equations. Perhaps the bed elasticity may have a role to play for fine cohesive particles at high solid hold-ups. In that case, this can be included in future in the equation of motion for the particle phase as an extra force arising out of particle–particle interaction, without affecting the fundamental approach of linear stability analysis. Batchelor (1988), for the first time, bridged the gap between stability and instability in fluidized beds in a more cogent manner. The elusive stabilizing term was given the significance of dispersion. In fact, the previous theories had to resort to some fictitious terms that stabilize the bed. For example, Jackson and co-workers attributed it to the particle phase pressure, Homsy and co-workers to the collisional pressure, Musters and Rietema (1977) to the cohesion forces of the particle network, and Foscolo and Gibilaro (1984) to the residual force acting on the particles when a pressure disturbance propagates. It is likely that particle–particle interactions are important under certain conditions, such as fine particles and/or

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

27

high solid hold-ups. However, such interactions may not be important when the solid hold-up is small (⬍ 30%). Further, in gas–liquid dispersion, such interactions are unlikely to play any significant role in governing the transition. Batchelor (1988) developed a force balance equation for the solid phase in fluidized beds using the ensemble average method. With the help of the resulting equation and the theory of linear stability the following criterion was obtained by Ham et al. (1990): Nm ⫽

[A(G/F ) ⫺ B/2]2 ⬍ 1. A(Z ⫺ C ) ⫹ B2 /4

(30)

The values of A, B, C, G, and F for various multiphase equipment are given by Table I. Z is defined as Z⫽

K0 ␯0(␳S ⫺ ␳f)DS , ⫹ u20 ␳SFr

(31)

where Fr ⫽ u20 /gdP . The first term on the LHS of Eq. (31) represents the Reynolds stress associated with particle fluctuations, and ␯0 is the mobility. Thus, the contribution to Z may be considered to be made up of two parts: one arising from the mean square velocity fluctuations, and the other due to dispersion. Batchelor (1988) has derived the particle phase governing equation directly without any need for writing a liquid phase momentum equation. This enabled elimination of the pressure term. The dispersion term was not included in the equation of continuity. However, it was included in the equation of motion. A revisit to the derivation of governing equations is expected to be useful in future development. Ham et al. (1990) used Eqs. (30) and (31) and estimated the values of the solid phase dispersion coefficient using the experimental results on transition in solid–liquid fluidized beds. However, the estimated values of Ds deviate from the experimental values of Ds obtained by Dorgelo et al. (1985). It may be noted that the RTD based experimental Ds values includes gross nonidealities in addition to the turbulent dispersion. Pauchon and Banerjee (1988), in their analysis of bubbly flows, have shown that the kinematic wave velocity based on a constant interfacial friction is weakly stable. They have also obtained a functional dependence of the interfacial friction factor on the void fraction by assuming the kinetic wave velocity equal to the characteristic velocity (kinetic waves are neutrally stable). They have assumed that turbulence provides the stabilizing mechanism through axial dispersion of the void fraction.

28

J. B. JOSHI ET AL.

Biesheuvel and Gorrisen (1990) have derived one-dimensional conservation equations for void fraction disturbances in a uniform bubbly fluid. They have studied the features of the propagation of void fraction disturbances and investigated the stability of uniform bubbly flows. They observed that the voidage fraction waves are unstable for void fractions above some critical value (⬎45%). Their method of approach was similar to that of Batchelor (1988) in many respects. Lisseter and Fowler (1992) have derived a simple set of equations for bubbly flow through a vertical tube. They have shown that under steady flow conditions, the void fraction will relax from its inlet value to an asymptotic value within only a short distance from the inlet. They have obtained a relationship between the inlet void fraction and the imposed pressure drop and derived a simple expression for the equilibrium void fraction. They have also considered the wall friction in their analysis of bubbly flows. Shnip et al. (1992) have developed criteria for predicting the transition from the homogeneous to the heterogeneous regime in two-dimensional bubble columns, using the theory of linear stability. They analyzed both for semibatch and continuous modes of operation. They found good agreement between the predicted and experimental values of the critical gas velocity over a wide range of hole diameters and numbers of holes. They obtained 0.42 as the maximum value of the predicted critical gas hold-up. The transition criterion developed by them was found to be independent of the viscosity. Sangani and Didwania (1993) have derived averaged equations for largeReynolds-number, laminar-flow gas–liquid dispersions accounting for slowly varying spatial and temporal fields. In particular, they obtained an exact expression for the dispersed-phase stress tensor to be used in the force balance equation for gas bubbles and illustrated its application by evaluating the stress tensor for a few special cases. It is shown that the dispersed-phase stress tensor gradient with respect to the mean relative motion or the void fraction for the uniformly random bubbly liquids under conditions of large Reynolds number laminar flows is negative and thus has a destabilizing influence on the dynamics of void fraction waves in bubbly liquids.

IV. Review of Stability Criteria Based on Heuristic Approach Wallis (1969) proposed a very elegant and simple criterion for the transition in fluidized beds. The criterion is based on the concept of elasticity of fluidized beds. It assumes that a fluidized bed resists deformation like any

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

29

other elastic material. Also, it possesses an upper limit of elasticity beyond which the bed can no longer sustain the strain and collapses to become heterogeneous. Mathematically, it states that the bed becomes heterogeneous when the propagation velocity of voidage disturbance (also called continuity wave velocity) exceeds the velocity of elastic waves (also called the dynamic wave velocity) in the bed. The theoretical justification behind such a criterion has been given by Whithman (1974). The physical significance of these two velocities is given next. Slis et al. (1959) investigated the voidage propagation velocity. The authors not only provided the theoretical basis but provided an excellent experimental support for u⑀. Therefore, this investigation has been used extensively by the subsequent workers in this area. Following is the mathematical derivation of u⑀ in its simplest form. For more comprehensive derivation, the original work of Slis et al. (1959) is recommended. Consider a gas–solid fluidized bed operating in the homogeneous regime. Let the gas flowrate be reduced suddenly. The adjustment to this new flowrate starts from the bottom of the bed. Visually we can see a line of demarcation that separates the two regions of equilibrium voidage. This sharp front travels up through the bed. The unsteady behavior ends when this sharp front and the falling bed surface confront each other and the voidage becomes completely uniform according to the new flowrate. It is this sharp front velocity that is commonly termed the voidage propagation velocity. Considering the mass balance on the dispersed phase, if in time ⌬t the distance traveled by the sharp front is ⌬h, then ⌬h(␧GB ⫺ ␧GA) ⫽ ⌬t⌬usup(1 ⫺ ␧GA),

(32)

where ⌬usup is the change in the continuous phase superficial velocity. Rearranging Eq. (32), we get (1 ⫺ ␧GA) ⌬h ⫽ ⌬usup . ⌬t (␧GB ⫺ ␧GA)

(33)

In the limiting case of infinitesimally small change in the gas velocity, we get the following relation: u␧ ⫽ (1 ⫺ ␧G)

dusup . d␧

(34)

We have the equation m usup ⫽ Vs앝␧G ,

where Vs앝 is the terminal settling velocity and ␧G is the voidage.

(35)

30

J. B. JOSHI ET AL.

Substituting Eq. (35) in (34) gives ⫺1 u␧ ⫽ Vs앝m(1 ⫺ ␧G).␧m G .

(36)

It has been mentioned earlier that the voidage propagation velocity is the same as the sharp front velocity. Thus, the equation for u␧ is the same in gas–solid fluidized beds and liquid–solid fluidized beds. However, for bubble columns, the sharp front velocity is given by the following equation: ⫺1 ⫺1 u␧ ⫽ VB앝␧m ⫺ m␧GVB앝␧m . L L

(37)

While presenting a comprehensive analysis of two-phase flow, Wallis (1969) has discussed the problem of transition. He has given a very systematic derivation for the voidage propagation velocity and takes the form of Eq. (36) for the case of small variations about the steady state. Wallis (1969) also derived an equation for the elastic wave velocity: ue ⫽ (⭸p/d␳)0.5.

(38)

The value of ue was evaluated from equation of motion and for the case of solid–liquid fluidized beds, and under some simplifying conditions it is given by ue ⫽

冋

v2S ␧L / ␳L ⫹ ␧ S / ␳S

册冋 1/2

␧S ␧L ⫹ ␳S ␳L

册

⫺1/2

.

(39)

Wallis (1969) compared the values of u␧ and ue for solid–liquid fluidized beds and gas–solid fluidized beds and concluded that all the beds are unstable. Verloop and Heertjes (1970) used Eq. (36) for the voidage propagation velocity. The elastic wave velocity was calculated on the basis that the fluidized bed was considered to be an elastic substance: ue ⫽ (ES / ␳)1/2,

(40)

ES ⫽ (⌬F/⌬X )(4/앟d).

(41)

where ⌬F is the increase in drag force when the distance between two particles is decreased by ⌬X. The following relationship of Rowe (1984) was assumed: FD 0.68 ⫽ . FD 앝 웃r

(42)

FD and F앝 are the values of drag force on a particle in a fluidized bed and on a particle in an infinite medium, respectively. 웃r is the dimensionless distance (x/dp) between the particles. The following equations were derived for the elastic wave velocity:

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

ue ⫽ [128gdP(Vmf /vS)a]1/2,

31 (43)

when ReP ⬍ 2

where a ⫽ 1

when 2 ⬍ ReP ⬍ 500.

a ⫽ 1.4

This analysis suffers from the following limitations: 1. Equation (42) indicates that the drag force depends upon the interparticle distance. In fact, for a fluidized particle, the drag force equals the net force due to gravity and buoyancy, FD ⫽ vP(␳S ⫺ ␳L) g,

(44)

and the value of FD is independent of voidage. It may be noted that the drag coefficient increases with an increase in the solid hold-up. However, the slip velocity decreases with an increase in the solid hold-up. However, the overall effect of drag coefficient and slip velocity is that the value of FD remains constant irrespective of value of ␧s . 2. The structure of fluidized bed is highly idealized. For deriving equation for elastic wave velocity, Foscolo and Gibilaro (1984) designed an ingenious thought experiment. A fluidized bed was considered to be supported by a frictionless piston that also acted as a gas distributor. In the steady state, the piston was maintained in position by means of an external pressure, p, acting on its bottom face. A small increase ⌬p in this pressure caused an upward movement of the piston and the compression of the bottom portion of the bed. A net force ⌬F acts on each of the NS particles in the bottom layer in contact with the distributor. The consequent upward displacement reduces the local voidage in such a way that each particle in the next layer experiences the same net force, which in turn is transmitted to the layer above. In the way, the pressure wave was considered to travel longitudinally through the bed. The relations obtained were ⌬F ⫽ AS⌬p/NS ,

(45)

where NS is the number of particles, NS ⫽

4␧SAS . 앟d 2P

(46)

Substitution of Eqs. (45) and (46) in (38) gives uc ⫽

冋 册冋 册 冋 册冋 4␧S 앟d 2P

0.5

dF d ␳¯

0.5

⫽

4␧S 앟d 2P

0.5

dF/d␧C d ␳¯ /d␧C

册

0.5

,

(47)

where ␳¯ is the average density of the bed and ␧C is the fractional hold-up of continuous phase,

32

J. B. JOSHI ET AL.

␳¯ ⫽ ␧C␳C ⫹ ␧D␳D .

(48a)

For a gas–solid fluidized bed,

␳¯ ⫽ ␧G␳G ⫹ ␧S␳S .

(48b)

For a solid–liquid fluidized bed,

␳¯ ⫽ ␧L␳L ⫹ ␧S␳S .

(48c)

Foscolo and Gibilaro (1984) assumed the continuous phase density (␳C) to be negligible. Though this is true for gas–solid fluidized beds, ␧C␳C should be considered in liquid–solid fluidized beds and particularly in gas–liquid bubble columns. The net force (F) was considered to be the difference between the drag force (FD) and the effective weight (We). FD was derived on the basis of particle bed model. For instance, for the case of a solid–liquid fluidized bed, F ⫽ FD ⫺ W e ⫽ F D 앝

冉 冊 usup VS 앝

4.8/m

⫺3.8 ␧G ⫺

앟d 3P (␳S ⫺ ␳G) g␧G . 6

(49)

While estimating dF/d␧, the value of the superficial velocity was assumed constant. Substitution of Eqs. (48) and (49) in (47) results in the following equation for the elastic wave velocity: ue ⫽ [3.2gdP(1 ⫺ ␧G)(␳S ⫺ ␳G)/ ␳S]1/2 .

(50)

In the preceding thought experiment and derivation where there are two regions of ␧, some explanation is needed for the physical significance of F and dF/d␧C . In fact, if The Richardson-Zaki correlation is substituted in Eq. (49), F works out to be zero. This point becomes particularly relevant because, in the derivation of elastic wave velocity, Verloop and Heertjes (1970) have used the drag force directly [Eq. (41)], whereas Foscolo and Gibilaro (1984) have used the difference between the drag force and the effective weight of particles [Eq. (49)]. Foscolo and Gibilaro (1984) have claimed excellent agreement between the model predictions and the experimental observations. However, Tsinontides and Jackson (1993) have expressed some reservations regarding such an agreement. A. RELATIONSHIP BETWEEN FUNDAMENTAL APPROACH AND HEURISTIC APPROACH 1. Solid–Liquid and Gas–Solid Fluidized Beds In all the criteria developed using the heuristic approach (discussed earlier in this section) a comparison is made between the dynamic (elastic)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

33

wave velocity (ue) and the continuity wave (voidage propagation) velocity (u␧). The transition is said to occur when u␧ exceeds ue . It will be worth while if this criterion is derived from the fundamental approach. Further, it will be good to know the range of parameters over which the heuristic criterion can be safely used. Let us start with the criterion developed using the fundamental approach [Eq. (24)]: [A(G/F ) ⫺ B/2]2 ⬍ 1. A(Z ⫺ C ) ⫹ B2 /4 Simplifying, A(G/F )2 ⫺ B(G/F ) ⫹ C ⬍ Z.

(51)

The values of A, B, C, F, G, and Z for gas–solid fluidized beds, solid–liquid fluidized beds, gas–liquid bubble columns, and three-phase fluidized beds are given in Table I. Let us consider these parameters for gas–solid fluidized beds (refer to Table I). Since ␳G / ␳S can be considered negligible, A ⫽ 1,

B ⫽ 0,

C ⫽ 0.

(52)

Substitution of Eq. (52) in (51) gives the stability criterion G/F ⬍ 兹Z,

(53)

G ⫺1 ⫽ mVS앝␧S␧m G F

(54)

where

which is the same as Eq. (36) and represents the voidage propagation velocity. Z⫽

冋

册

␧S(␳S ⫺ ␳G) g DS DG ⫹ . ␳SvS ␧S ␧G

(55)

Further, as a first approximation, if we neglect gas phase dispersion as compared with solid phase dispersion, Eq. (55) takes the following form: Z⫽

冉 冊

( ␳S ⫺ ␳ G ) ⫺웁0 DS DS g ⫽ . ␳SvS ␳S ␧S

(56a)

There is little information regarding the form of DG . A possible alternative is to assume that DG ⫽ DS . If we had taken DG ⫽ DS, the expression for Z would be Z⫽

冉 冊

⫺웁0 DS . ␳S ␧S␧G

(56b)

34

J. B. JOSHI ET AL.

The assumption of equality of dispersion coefficients of both the phases implies that the root mean square velocity and length scale of turbulence of the gas phase is the same as that of the solid (dispersed) phase. We have taken the length scale of the solid phase to be equal to twice the particle diameter. It is not clear whether the same length scale can be assumed for the gas phase, as the gas phase eddies can be smaller than the particle diameter. Also, the root mean square velocities of both the phases may not be equal. Furthermore, because of the slip velocity, sufficient time may not be available for the local gas phase eddies to be in equilibrium with the dispersed phase eddies. Thus, both of these assumptions are restrictive, and in the absence of additional information, one of these assumptions needs to be made. For the case of gas–solid fluidized beds and solid–liquid fluidized beds, if we assume DG ⫽ DL ⫽ 0, we get a criterion [Eq. (61)] that is almost same as the criterion of Gibilaro and Foscolo (1984), given in Eq. (50). Therefore, this assumption was made. The solid phase dispersion coefficient can be written as the product of the length and velocity scale of turbulence: DS ⫽ u⬘zl.

(57)

Handley et al. (1966) measured the solid-phase rms velocity in solid–liquid fluidized beds. These values can be correlated on the basis of energy balance developed by Joshi and Lali (1984): u⬘z ⫽ 1.5␧SvS

(58a)

l ⫽ 2dP .

(58b)

DS ⫽ 3␧SvSdP .

(59)

Therefore,

The proportionality constant of 3 may be considered as preliminary estimate. Substitution of Eq. (59) in (56) gives Z ⫽ 3dP␧S(␳S ⫺ ␳G) g/ ␳S .

(60)

Substitution of Eq. (54) and (60) in (53) gives the following condition for the transition: ⫺1 兹3dP␧S(␳S ⫺ ␳G) g/ ␳S ⬎ m␧SVS앝␧m G .

(61)

It can be readily seen by comparison with Eq. (50) that the left-hand side of Eq. (61) is the elastic wave or dynamic wave velocity of Gibilaro Foscolo and co-workers. Thus, Eq. (61) forms the basis of comparison of dynamic wave velocity

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

35

with continuity wave velocity. This is a very useful result. At this state, the following additional points may be noted. 1. The stabilizing property of the dynamic or elastic wave velocity is due to the dispersion in the solid phase. The higher the dispersion coefficient, the greater the ability of the bed to remain homogeneous. In earlier work, this property was not clearly recognized (Section III). Jackson attributed the stability to particle phase pressure; Homsy and co-workers attributed it to collisional pressure, whereas Musters and Rietema (1977) attributed it to the elastic property of the bed. In all these cases, it is difficult to measure the respective properties. Batchelor (1988) was the first to recognize the ability of dispersion to keep the bed homogeneous. 2. Equation (61) is the transition criterion provided the conditions given by equation (52) are satisfied. From Table I it can be seen that these conditions are satisfied only in the case of gas–solid fluidized beds and in some cases of solid–liquid fluidized beds where ␳S Ⰷ ␳L . Therefore, for other multiphase dispersions [such as gas–liquid (bubble columns) and solid–liquid fluidized beds (where ␳L is not negligible)] the comparison of dynamic wave velocity with continuity wave velocity is not valid for deciding the bed stability. Further, the above analysis holds for transition from region I to II (point P in Fig. 1) and not for III to II (point Q). Therefore, the criterion does not hold for bubble columns and dilute dispersions. 3. Equation (36) gives the voidage propagation velocity (sharp front or continuity wave velocity) for gas–solid and solid–liquid fluidized beds. However, for the other multiphase dispersions, the procedure given by Eqs. (32) to (37) should be used. Thus, for gas–liquid dispersions, the sharp front velocity is given by Eq. (37). 4. The stability criterion given by Eq. (61) can be simplified even further. Rearranging equation and considering ␳S Ⰷ ␳G , V S앝 兹gdP

⬍

1 ⫺1 m␧m G

冪␧3 .

(62)

S

The left-hand side is Froude number. For gas–solid fluidized beds, in many cases, the aggregative fluidization occurs at the condition of incipient fluidization. Substituting ␧S ⫽ 0.6, ␧G ⫽ 0.4, m ⫽ 2.4, the right-hand side works out to be 3.3. This is again an interesting result. As early as 1948, Wilhelm and Kwauk were probably the first to provide some insight in the transition criterion. Wilhelm and Kwauk concluded that beds with small Froude numbers do not bubble easily. It may be pointed out that Eq. (62) was derived for gas–solid fluidized beds with a simplyfying assumption. The bubbling condition may coincide

36

J. B. JOSHI ET AL.

with transition in gas–solid fluidized beds; however, it may not coincide with transition in solid–liquid fluidized beds. In fact, the experiments of Anderson and Jackson (1968) and Homsy and co-workers have revealed clearly that the liquid fluidized beds are not linearly stable and that voidage waves exist in them as well. Thus, the lack of presence of bubbles in solid–liquid fluidized beds may not be equivalent to stability. 2. Gas–Liquid Bubble Columns Lockett and Kirkpatrick (1975) have presented an interesting analysis for the prediction of critical gas hold-up (␧GC). In the absence of liquid flow, VG /␧G ⫽ VS . Therefore, VG ⫽ ␧G(1 ⫺ ␧G)m⫺1VB앝 .

(63a)

A plot of ␧G vs VG passes through a maximum and gives a value of maximum gas throughput which may be achieved. The maximum permissible gas hold-up can be obtained by setting dVG /d␧G ⫽ 0. The result is ␧GC ⫽

1 . m

(63b)

Ideally, it should be possible to reach the maximum and still remain in the bubbly flow regime. Thereafter, any further increase in the gas flow rate results in a continuity limitation on the gas. Flooding occurs, resulting in the formation of large bubbles and transition. Thus, ␧GC may be considered as the transition gas hold-up. In the preceding analysis, the velocity–hold-up relationship was expressed in terms of the Richardson–Zaki equation. Alternatively, the relationship can be derived from the Ergun (1952) equation. This latter approach was used by Molerus (1993). In the homogeneous regime, the hindered velocity of the dispersed phase is known to be due to an increase in the drag coefficient. Thus, CD 앝

앟d 2B ␳LV 2B앝 앟 3 ⫽ d B(␳L ⫺ ␳G) g 4 2 6

(64a)

앟d 2B ␳Lv 2S 앟 3 ⫽ d B(␳L ⫺ ␳G) g, 4 2 6

(64b)

CD

where CD앝 is the drag coefficient under the terminal rise conditions, CD is the drag coefficient in the presence of other particles (or bubbles), and vS is the hindered velocity. For solid–liquid fluidized beds, CD is given by CD ⫽

4 (␳S ⫺ ␳L)dP g . 3 ␳Lv 2S

(65)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

37

Molerus established the following correlation for CD on the basis of bed expansion characteristics: CD ⫽

冋

冉

r2o 24 r 1 ⫹ K1 o ⫹ 0.5 2 ReP 웃 웃

冊册

⫹

冋

冉冊册

4 r 1 ⫹ K2 o 웃 兹ReP

1.5

r 1 ⫹ 0.4 ⫹ K3 o n . 웃 ReP

(66)

Here, ro 1 ⫽ 웃 ␨ /␧L1/3 ⫺ 1

and

ReP ⫽

usupdP␳L ␧L애 L

␨ ⫽ 0.9, K1 ⫽ 0.341, K2 ⫽ 0.07, K3 ⫽ 0.907, n ⫽ 0.1,

(67) (68)

and CDRe2P ⫽

冉

冊

4 ␳S ⫺ ␳L d 3P g . 3 ␳L ␯ 2L

(69)

It may be noted from Eq (66) that the value of CD for a single particle in an infinite, medium is given by 24/Re and 0.4 in the creeping and turbulent flow conditions, respectively. Molerus (1993) assumed that Eqs. (65)–(69) also hold for gas–liquid systems, though marked differences are known to exist. He defined the dimensionless parameters

冉

冊

␳ ⫺ ␳G d 3B g dˆ3B ⫽ L ␳L ␯ 2L

(70)

vSdB␳L 애L

(71)

␴ 3o ⫽

v3S ␳L ␳L ⫺ ␳G ␯L g

(72)

웂3 ⫽

␳L v3sup , ␳L ⫺ ␳G ␯L g

(73)

ReB ⫽

where dˆB defines the dimensionless bubble diameter, ␴0 the dimensionless relative velocity, and 웂 is the dimensionless gas throughput. A typical plot of ␧G vs ␴0 with dˆB and 웂 as parameters is shown in Fig. 4. The solid lines are lines of dimensionless bubble size dB ⫽ constant. Numbers indicate the bubble sizes dB (mm) for air in water. These lines end at the abscissa for dimensionless single bubble rise velocity (␧G ⫽ 0). The dotted lines are

38

J. B. JOSHI ET AL.

FIG. 4. Transition plot from Molerus (1993).

lines of constant gas throughput. The direction of increasing gas throughput is shown by an arrow. It can be seen from the figure that each line of 웂 ⫽ constant shows tangential contact with a particular line dˆB ⫽ constant. This behavior means that for a given bubble size, a maximum gas throughput exists that cannot be exceeded with homogeneous bubble flow. If gas throughput is increased beyond the limit of homogeneous bubble flow, Molerus suggested the following two different means of transition: 1. Froth formation can occur if coalescence of small bubbles is completely suppressed. 2. Liquid circulation will begin for accommodating the higher gas throughput. Ranade and Joshi (1987) have developed a criterion for small bubbles. The small bubbles rise upward without any oscillations. The liquid carried upward in the bubble wakes is released at the top liquid surface, which then flows downward in the bubble-free region. The downward liquid flow hinders the bubble rise. It was proposed that the transition will occur when the bubble rise velocity equals the downward liquid velocity. Under this condition, the bubble rise velocity with respect to the column wall is zero and the gas phase accumulates in the column, leading to transition.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

39

If ␨ is the fractional wake volume, the liquid phase mass balance gives

␨␧GVB앝 ⫽ (1 ⫺ ␧G ⫺ ␨␧G) UD .

(74)

The bracketed term corresponds to the bubble-free region where the liquid down flow occurs. UD is the downward liquid velocity. At transition, equating UD and VB앝 , we get

␨␧GC ⫽ (1 ⫺ ␧GC ⫺ ␨␧GC).

(75)

Selecting ␨ equal to 11/15 (Kumar and Kuloor, 1972), the value of critical gas hold-up at transition works out to be 0.42. This value, estimated for small bubbles, is in agreement with the experimental findings of Oels et al. (1978) and Koetsier et al. (1976). The difference between the analysis of Lockett and Kirkpatrick (1975), Molerus (1993), and Ranade and Joshi (1987) may be noted. In the former case [Lockett and Kirkpatrick (1975) and Molerus (1993)], flooding occurs because the bubble slip velocity gets reduced with an increase in the gas hold-up and at ␧GC flooding occurs. In the latter case (Ranade and Joshi, 1987), flooding occurs because of the downflow of liquid (which was carried upward through wakes). However, both the analyses give the upper limit that is observed by small bubbles (due to sparger design, low surface tension, high gas density or non-coalescing system). For relatively large bubbles, transition may occur below the flooding point because of either coalescence (formation of large bubbles) or intrinsic hydrodynamic instability (explained in the next subsection). For relatively large bubbles (⬎2 mm) and coalescing liquids, the value of ␧GC may be obtained by the following consideration. The large bubbles rise with oscillations due to periodic shedding of vortices behind the bubbles. Since the size of a vortex is of the order of dP /2, it may be assumed that the scale of oscillations is approximately dP /2. Using this value for the bubble clearance and the cubic lattice structure, the value of critical gas hold-up works out to be 0.16. This is in good agreement with the experimental values of 0.15 to 0.2 reported by Deckwer et al. (1977), Whalley and Davidson (1974), and Kelkar (1983). It may be pointed out that Taitel et al. (1980) have observed enhanced coalescence when ␧SC exceeds 0.18. This value supports the preceding heuristic arguments. The stability criterion can considerably be simplified for the case of gas–liquid dispersion. The constants in Eq. (24), for the case of semibatch operation (u ⫽ 0), are defined in Table I. Since the ratio ␳G / ␳L is negligible, the constants are simplified and are given by A⫽

(1 ⫹ Cv) ⫺1 ␧L

(76a)

40

J. B. JOSHI ET AL.

B ⫽ 2CvvS

(76b)

C ⫽ Cvv2S

(76c)

Z ⫽ 움dB␧G g,

(76d)

where 움 is the proportionality constant for dispersion. As discussed earlier, transition in bubble columns occurs in most cases because the denominator in Eq. (24) becomes zero, i.e., A(Z ⫺ C ) ⫹ B2 /4 ⫽ 0.

(77)

Equations (76) and (77) give the following simplified criterion: V B앝 ⫽ 兹dBg

⫹C ) 1 冪움(␧ C (1 ⫹ C ) ␧ G

v

v

v

m⫺1 . L

(78)

Again, the left hand side is the Froude number. For Cv ⫽ 0.5, m ⫽ 1 움 ⫽ 1.2, VB앝 ⫽ 200 mm/s, and dB ⫽ 3 mm, Eq. (78) gives ␧GC ⫽ 0.34.

(79)

Thus, different combination of parameters CV , 움 and m can lead to some maximum transition gas hold-up. This is again an interesting result. Equation (76) then provides a fundamental basis for many empirical criteria in the literature that simply give a maximum value of gas hold-up up to which the homogeneous regime can prevail. For example, Bach and Pilhofer (1978) gave the transition gas hold-up as 0.25, Mersmann (1978) gave ␧GC ⫽ 0.2, Taitel et al. (1980) gave ␧GC ⫽ 0.18, and Schugerl et al. (1977) gave ⑀GC ⫽ 0.42.

V. Model Predictions and Experimental Observations A. ESTIMATION OF MODEL PARAMETERS Various model parameters involved in the derivation of the stability criterion need to be specified in order to use the stability criterion for quantitative predictions. Model parameters essential for this purpose include the slip velocity, the virtual mass coefficient, and the dispersion coefficient. The procedure for estimation of these parameters is given for gas– solid (and solid–liquid) fluidized beds and bubble columns.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

41

1. Fluidized beds a. Estimation of Slip Velocity. In the case of fluidized beds, the slip velocity is given by (Vsup ⫽ 0) vs ⫽

usup , ␧c

(80)

where usup is the superficial fluid velocity. It is given by an equation similar to the Richardson–Zaki equation: usup ⫽ VS앝␧cm .

(81)

The terminal settling velocity was predicted using the following procedure. The Galileo number at given particle diameter dp was calculated as Ga ⫽

d 3P␳c(␳s ⫺ ␳c) g . 애2c

(82)

The Reynolds number (based on the terminal settling velocity) was estimated from the Galileo number using the correlations given by Lali et al. (1989). The terminal settling velocity was then calculated from the Reynolds number and the physical properties. The Richardson–Zaki index m was estimated using the correlations given by Richardson (1971). b. Estimation of the Dispersion Coefficient. An important parameter in the stability criterion is the dispersion coefficient. Correlations for the dispersion coefficient are mostly obtained by the one-dimensional model. The estimated value of the dispersion coefficient for solid–liquid and gas–solid fluidized beds was obtained from Eq. (59) given earlier: DS ⫽ 3dP␧SvS .

(59)

Equation (59) was used to estimate the dispersion coefficient with a proportionality constant of 3. However, the effect of extreme values of the proportionality constant of dispersion was also studied. The dispersion coefficient for the continuous phase was taken as zero for the model predictions. c. Estimation of Virtual Mass Coefficient. The virtual mass coefficient CV represents the acceleration reaction of the dispersed phase on the continuous phase. Virtual mass plays an important role in case of gas–solid fluidized beds and solid–liquid fluidized beds when ␳S Ⰷ ␳L . Jackson (1985) used a constant value of 0.5 as the virtual mass coefficient, which is for an isolated sphere in an infinite medium. Homsy et al. (1980)

42

J. B. JOSHI ET AL.

assumed the virtual mass coefficient to be a function of the continuous phase hold-up, given by Cv ⫽

3 ⫺ 2␧C , 2␧C

(83)

which, in the limiting case of ␧C ⫽ 1, gives Cv ⫽ 0.5. Equation (83) was used to estimate Cv for model predictions. The effect of Cv values of zero and 0.5 was also studied. 2. Bubble Columns In bubble columns, the estimation of parameters is more difficult than in the case of either gas-solid or solid–liquid fluidized beds. Major uncertainties in the case of bubble columns are due to the essential differences between solid particles and gas bubbles. The solid particles are rigid, and hence the solid–liquid (or gas–solid) interface is nondeformable, whereas the bubbles cannot be considered as rigid and the gas–liquid interface is deformable. Further, the effect of surface active agents is much more pronounced in the case of gas–liquid interfaces. This leads to uncertainties in the prediction of all the major parameters such as terminal bubble rise velocity, the relation between bubble diameter and terminal bubble rise velocity, and the relation between hindered rise velocity and terminal rise velocity. The estimation procedure for these parameters is reviewed next. a. Estimation of Terminal Rise Velocity of Bubbles. Clift et al. (1978) have critically reviewed the literature on terminal rise velocity of bubbles. One of the major factors influencing the terminal bubble rise velocity is the bubble shape, which can deviate from spherical to ellipsoidal or spherical cap because of the deformable interface. The presence of surface active agents also influences bubble shape. The bubble shape determines the drag force and, in turn, the terminal bubble rise velocity, since the terminal rise velocity of bubbles is decided by the balance among gravity, buoyancy, and drag. Another factor that affects the bubble rise velocity is the manner in which the bubble rises. The bubbles with diameter less than 1 mm travel in a straight line, whereas the bubbles with diameter greater than 1 mm oscillate horizontally while traveling upwards or follow a steady motion up a helix with a vertical axis. Depending on the manner of bubble travel (true vertical or zigzag), the distance covered by bubbles changes, thus affecting the terminal bubble rise velocity. The expressions for terminal rise velocities of bubbles in pure and contaminated liquids are given below.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

43

(i) Terminal Bubble Rise Velocity in Pure Liquids. For spherical bubbles with Re Ⰶ 1 (creeping flow), two extreme cases are possible depending on the nature of the gas–liquid interface. If the bubble is assumed to be similar to a rigid solid particle, the bubble rise velocity is given by Stokes relationship, V B앝 ⫽

(␳L ⫺ ␳G) gd2B . 18애L

(84)

However, if the bubble interface is assumed deformable and completely free from contaminants, the bubble rise velocity is given by the Hadamard– Rybczynski equation. When the viscosity ratio (애G /애L) is zero, the terminal bubble rise velocity is given by V B앝 ⫽

(␳L ⫺ ␳G) gd2B . 12애L

(85)

For ellipsoidal gas bubbles, Mendelsen (1967) has given the following correlation for terminal bubble rise velocity: VB앝 ⫽

冉

冊

gd 2␴ ⫹ B ␳LdB 2

1/2

.

(86)

Here, ␳G has been assumed to be much less than ␳L . For spherical cap bubbles, Davies and Taylor (1950) have given the relationship VB앝 ⫽ 0.711(dB g)1/2.

(87)

(ii) Terminal Bubble Rise Velocity in Contaminated Liquids. For contaminated liquids, Clift et al. (1978) have given a generalized correlation in terms of dimensionless groups, which cover all bubble shapes. Based on this correlation, the relationship between bubble diameter and terminal rise velocity of bubble for an air-water system is given in Fig. 5. In the same Fig. 5, two more curves based on experimental data of Gaudin (1977) are given for air–water systems. From this figure, one of the key problems in estimation of terminal bubble rise velocity becomes apparent. For a widely used system such as air–water, if a bubble diameter of 2 mm is taken, the rise velocity can have any value between 150 mm/s and 300 mm/s. Conversely, if we take the terminal rise velocity of bubbles to be 200 mm/s, bubble diameter can be anywhere between 1 and 10 mm, depending upon the degree of contamination, which is difficult to quantify and can change over a period of time. In the present work, we have used correlations of Clift et al. (1978) for the predictions. We also have used the

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J. B. JOSHI ET AL.

FIG. 5. Various relations between bubble diameter and terminal rise velocity: air– water system.

predictions based on the data of Gaudin (1977) to demonstrate the effect of the relationship between the bubble diameter and the terminal rise velocity of bubbles. b. Estimation of the Slip Velocity for Gas–liquid Bubble Columns. The slip velocity can be given as vS ⫽ VB앝(1 ⫺ ␧G)m⫺1.

(88)

For the case of a semibatch bubble column (no liquid flow), the drift flux model takes the form J ⫽ vS␧G(1 ⫺ ␧G) ⫽ VB앝␧G(1 ⫺ ␧G)m.

(89)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

45

Various investigators have suggested different values of m: Davidson and Harrison (1966), m ⫽ 0 Turner (1966), m ⫽ 1 Wallis (1969), m ⫽ 0 for large bubbles and m ⫽ 2 for small bubbles Zuber and Hench (1962), m ⫽ 1.5 for large bubbles and m ⫽ 3 for small bubbles 5. Richardson and Zaki (1954), m ⫽ 2.39 6. Pal and Masliyah (1989), m ⫽ 2.39 1. 2. 3. 4.

Some investigators have given different forms of correlation. For instance, 5/3 7. Marucci (1965), vS ⫽ VB앝(1 ⫺ ␧G)/(1 ⫺ ␧ G ) 8. Lockett and Kirkpatrick (1975), vS ⫽ VB앝(1 ⫺ ␧G)1.39(1 ⫹ 2.55␧ 3G) 9. Zhou and Egiebor (1993), vS ⫽ VB앝(1 ⫺ 1.06␧G) when ⑀G ⬍ 0.3

The wide variation in the value of m may be attributed to the following reasons: 1. The different investigators have covered different regimes of operation. Any value of m less than 1.0 indicates the possibility of heterogeneous regime and Eqs. (87) and (88) are not applicable for such a regime. 2. The range of bubble size covered by different investigators is markedly different. 3. The most important reason for the wide variation in m seems to be due to the presence of surface active agents. These agents have two roles to play: (a) to retain the bubble size that is formed at the sparger, and (b) to reduce the interface mobility. In an extreme case, the interface becomes rigid enough so that the bubbles behave like solid particles in the same range of Reynolds number. 4. When ␧G ⬍ 0.2 the value of J becomes insensitive to the value of m. 5. For a given gas–liquid system, the value of m perhaps does not remain constant over a wide range of ␧G . The variation in m depends upon the coalescing nature of the bubbles. From the foregoing discussion, it is clear that the value of m depends upon the properties of the gas–liquid system under consideration. In the present work, the transition will be examined at four levels of m, namely, m ⫽ 1, m ⫽ 1.4, m ⫽ 1.9, and m ⫽ 2.4. c. Estimation of the Dispersion Coefficient. An equation similar to Eq. (59) was assumed to hold for gas–liquid bubble columns. For bubble columns, Eq. (59) is slightly modified as follows:

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J. B. JOSHI ET AL.

DL ⫽ 3dB␧GvS .

(90)

The gas phase dispersion coefficient was assumed to be zero. d. Prediction of the Virtual Mass Coefficient. In the case of bubble columns, the virtual mass coefficient was estimated in a manner similar to the method used in case of fluidized beds. Equation (83) was used for the estimation of virtual mass coefficient in case of bubble columns. Two other values of Cv were also examined. B. STABILITY MAPS 1. Gas–Solid and Solid–Liquid Fluidized Beds In order to study the effects of the fluid properties and particle sizes using the stability criterion developed in Section II, the following numerical calculations were undertaken. For a given set of physical properties of the system, two critical particle diameters were determined by employing the stability criterion derived earlier. The lower limit of the critical particle diameter, dPL , is the largest particle diameter up to which the particulate or homogeneous regime prevails. All the particle diameters less than the lower critical particle diameter are stable (i.e., they will exhibit particulate fluidization) even up to a hold-up of 95%. The upper limit of the critical particle diameter, dPU , is defined as the smallest diameter above which the heterogeneous regime prevails. All particle diameters greater than the upper limit will exhibit aggregative fluidization right at incipient fluidization conditions, i.e., at a hold-up value of 40%. Particle diameters between the two limits (dPL ⬍ dP ⬍ dPU) will exhibit transition from particulate to aggregative fluidization at progressively lower values of hold-up. The values of dPL and dPU were determined using f (dP) ⫽ 1 ⫺ Nm ,

(91)

where Nm is the parameter defined by Eq. (28) and the constants A, B, C, Z, G, and F for solid–liquid fluidized beds and gas–solid fluidized beds are summarized in Table I. Since these constants depend on the hold-up under steady-state conditions, the function f1 was evaluated for the entire range of ␧L, from packed bed condition (40% voidage) to 95% voidage, starting with an extremely small value of particle diameter, say 0.5 mm. The particle diameter was progressively increased and the function f1 was again evaluated for the entire hold-up range. The critical diameter dPL is that diameter for which the function f1 first becomes negative. The lower and upper limits mark the limits of the homogeneous and

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

47

heterogeneous regimes. All diameters of the particle below the lower limit will be stable, and therefore the area under the lower limit shows the stable area. Similarly, diameters above the upper limit will become unstable even at incipient fluidization, and therefore the area above this limit is called the unstable region and will always exhibit aggregative behavior. Typical transition plots for solid–liquid and gas–solid fluidized beds are shown in Fig. 6a and 6b, respectively. Physical properties of the fluid such as density, viscosity, and particle density and the model parameters such as dispersion coefficient and virtual mass coefficient have a substantial effect on the critical diameter. These effects are discussed systematically in the following paragraphs.

a. Effect of Fluid Viscosity. In the final stability criterion, viscosity of the fluid does not appear explicitly. However, it appears implicitly through the particle settling velocity and the Richardson–Zaki index. As the viscosity of the fluid increases, the terminal settling velocity of the particle decreases, and this leads to a reduction in the voidage propagation velocity, making the system more stable. This fact is shown in Fig. 7 for solid–liquid fluidized beds and in Fig. 8 for gas-solid fluidized beds. The stable area represented by the area under the lower critical diameter limit increases with an increase in the viscosity. Similarly, Figs. 9 and 10 show the effect of fluid viscosity on the upper critical diameter for liquid and gas fluidized beds, respectively. Figure 11a shows the effect of density difference on the particle Reynolds number, based on the terminal settling velocity, for solid–liquid fluidized beds. Reynolds number was evaluated at the lower critical particle diameter. In this plot, the stability limit is represented by taking the Reynolds number as ordinate, instead of the lower critical particle diameter itself. When represented in this manner, the plot exhibits a remarkable feature that the stability curve, plotted by taking density difference as abscissa, remains almost same even when there is a hundredfold change in the liquid viscosity. Figure 11b shows similar plot for gas-solid fluidized beds. Figures 12a and 12b show the effect of density difference on particle Reynolds number evaluated at the upper limit of the critical diameter. In all these figures (11a–12b), the similar unified curves, independent of liquid viscosity can be observed. An attempt was made to write the stability criterion in a suitable dimensionless form. This would clarify the reasons for the unified curves in Figs. 11a, 11b, 12a, and 12b. However, such an exercise was found to be difficult cause (i) the relationship between m and Rep is empirical, and further, m is an exponent on ␧C , and (ii) the relationships between the viscosity and VS앝 and also dP and VS앝 are empirical. Figure 11c is a generalized plot of particle Reynolds number vs density difference for solid–liquid

48

J. B. JOSHI ET AL.

FIG. 6. Typical stability maps. (a) Solid–liquid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), 애L ⫽ 1 mPas, ␳L ⫽ 1000 kg/m3]; (b) gas–solid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), 애G ⫽ 0.02 mPas, ␳G ⫽ 1 kg/m3].

fluidized beds, plotted for transitions occurring at intermediate values of hold-up. Figure 12c shows a similar plot for gas–solid fluidized beds. b. Effect of Fluid Density. Fluid density has an important role to play in deciding the transition in both solid–liquid and gas–solid fluidized beds, as shown in Fig. 13 and 14, respectively. An increase in the fluid density makes the bed more stable, as indicated by an increase in the stable area under the lower critical diameter limit. This can be explained as follows: The simplified stability criterion was given by Eq. (61). It states that for stability the following condition should be satisfied: ⫺1 兹3dP␧S(␳S ⫺ ␳G) g/ ␳S ⬎ m␧SVS앝␧m G .

With an increase in fluid density, the Reynolds number increases and the Richardson–Zaki index (m) decreases. This results in a decrease in the voidage propagation velocity (RHS) and brings in more stability to the system. There is yet another reason. Let us consider the two extreme cases of creeping and turbulent flows. In the former case, we know from Stokes’ equation that VS앝 varies linearly with (␳S ⫺ ␳G). Therefore, with an increase in the fluid velocity, the RHS (voidage propogation velocity) decreases

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

49

FIG. 6. (continued)

faster than the LHS (elastic wave velocity). This brings more stability to the system. For the case of turbulent flow, the preceding equation can be modified as ⫺1 兹3dP␧S(␳S ⫺ ␳G) g/ ␳L. ⭈ 兹␳G / ␳S. ⬎ m␧SVS앝␧m G .

(92)

The first part of LHS is VS앝 . Therefore, ⫺1 兹␳G / ␳S ⬎ m␧S␧m G .

(93)

From this equation, it is clear that, with an increase in fluid velocity, the LHS increases and imparts more stability to the system. c. Effect of Particle Density. An increase in the particle density leads to earlier transitions as shown in Figs. 6a and 6b. The critical diameter of a

50

J. B. JOSHI ET AL.

FIG. 7. Effect of liquid viscosity on lower critical particle diameter solid–liquid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), ␳L ⫽ 1000 kg/m3].

FIG. 8. Effect of gas viscosity on lower critical particle diameter: gas–solid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), ␳G ⫽ 1 kg/m3].

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

51

FIG. 9. Effect of liquid viscosity on upper critical particle diameter: solid–liquid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), ␳L ⫽ 1000 kg/m3].

FIG. 10. Effect of gas viscosity on upper critical diameter: gas–solid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), ␳G ⫽ 1 kg/m3].

FIG. 11. Effect of density difference at various liquid viscosities on particle Reynolds number evaluation at lower critical particle diameter. (a) Solid–liquid fluidized beds [움 ⫽ 3.0, Cv ⫽ f(␧), ␳L ⫽ 1000 kg/m3]. (b) Gas–solid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), ␳G ⫽ 1 kg/m3]. (c) Unified stability map of particle Reynolds number vs density difference for different values of transition hold-up: solid–liquid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), 애L ⫽ 1 mPas, ␳L ⫽ 1000 kg/m3].

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

53

FIG. 11. (continued)

lighter particle is much more than that of a heavier particle. As the particle density approaches that of the liquid, the critical diameter increases almost exponentially. d. Effect of Particle Diameter. For obtaining the dependence of particle diameter on transition, Fig. 15 was constructed for a particular solid–liquid system and noting the liquid hold-up at which the transition from homogeneous to heterogeneous regime occurs. As seen from Fig. 15, the transition occurs earlier as the particle diameter is increased. Figure 16 shows critical hold-ups for solid particles in gas–solid fluidized beds. e. Effect of Dispersion Coefficient. As pointed out earlier, the hydrodynamic dispersion is the main stabilizing parameter in the model. For estimating the effect quantitatively, the value of dispersion coefficient was varied over a wide range. For this purpose, the proportionality constant in Eq. (59) was varied. Figures 17 and 18 show the effect of dispersion coefficient on the lower limit of critical particle diameter for solid–liquid and gas–solid fluidized beds, respectively. The values of critical particle diameters were found to be very sensitive to the dispersion coefficient. An increase in the dispersion coefficient is seen to increase the stable region of fluidiza-

FIG. 12. Effect of density difference at various liquid viscosities on particle Reynolds number evaluated at upper critical diameter: (a) Solid–liquid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), ␳L ⫽ 1000 kg/m3]. (b) Gas–solid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), ␳G ⫽ 1 kg/m3]. (c) Unified stability map of particle Reynolds number vs density difference for different values of transition hold-up: gas–solid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), 애G ⫽ 0.02 mPas, ␳G ⫽ 1 kg/m3].

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

55

FIG. 12. (continued)

FIG. 13. Effect of liquid density on lower particle critical diameter: solid–liquid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), 애L ⫽ 1 mPas].

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J. B. JOSHI ET AL.

FIG. 14. Effect of gas density on lower particle critical diameter: gas–solid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), 애G ⫽ 0.02 mPas].

FIG. 15. Effect of particle diameter on critical liquid hold-up at different particle densities: solid–liquid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), 애L ⫽ 1 mPas, ␳L ⫽ 1000 kg/m3].

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

57

FIG. 16. Effect of particle diameter on critical gas hold-up at different density differences: gas–solid fluidized beds [움 ⫽ 3.0, CV ⫽ f(␧), 애G ⫽ 0.02 mPas, ␳G ⫽ 1 kg/m3].

FIG. 17. Effect of proportionality constant for dispersion on lower critical diameter: solid– liquid fluidized beds [CV ⫽ f(␧), 애L ⫽ 1 mPas, ␳L ⫽ 1000 kg/m3].

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FIG. 18. Effect of proportionality constant for dispersion of lower particle critical diameter gas–solid fluidized beds [CV ⫽ f(␧), 애G ⫽ 0.02 mPas, ␳G ⫽ 1 kg/m3].

tion. This is true for the upper limit of critical diameter as well, as shown in Fig. 19 and 20, for solid–liquid and gas–solid fluidized beds, respectively.

f. Effect of Virtual Mass Coefficient. For studying the effect of the virtual mass coefficient, the value of Cv was taken as 0.0, 0.5, and as a function of hold-up given by Eq. (83). In the case of solid–liquid fluidized beds, the effect of virtual mass is to make the bed more unstable as shown in Fig. 21. This can be explained as follows: The effect of virtual mass is to increase the apparent density of the particle. As discussed in this section earlier, an increase in the particle density makes the system more unstable. This observation is consistent with Fig. 21. As the particle density increases, say, ␳S ⫽ 9000 kg/m3 and ␳L ⫽ 1000 kg/m3 , the effect of virtual mass is negligible and the curves are seen to merge irrespective of the formulation of virtual mass coefficient. In the case of gas–solid fluidized beds, the effect of virtual mass coefficient is negligible as shown in Fig. 22. However, as the gas density increases the effect of virtual mass becomes similar to that observed in solid–liquid systems as shown in Fig. 23.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

59

FIG. 19. Effect of proportionality constant for dispersion on upper particle critical diameter solid–liquid fluidized beds [CV ⫽ f(␧), 애L ⫽ 1 mPas, ␳L ⫽ 1000 kg/m3].

FIG. 20. Effect of proportionality constant for dispersion on upper particle critical diameter: gas–solid fluidized beds [CV ⫽ f(␧), 애G ⫽ 0.02 mPas, ␳G ⫽ 1 kg/m3].

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FIG. 21. Effect of formulation of virtual mass coefficient on lower particle critical diameter solid–liquid fluidized beds [움 ⫽ 3.0, 애L ⫽ 1 mPas, ␳L ⫽ 1000 kg/m3].

FIG. 22. Effect of formulation of virtual mass coefficient on lower particle critical diameter gas–solid fluidized beds [움 ⫽ 3.0, 애G ⫽ 0.02 mPas, ␳G ⫽ 1 kg/m3].

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

61

FIG. 23. Effect of formulation of virtual mass coefficient on lower particle critical diameter gas–solid fluidized beds [움 ⫽ 3.0, 애G ⫽ 0.02 mPas, ␳G ⫽ 1 kg/m3].

2. Gas–Liquid Bubble Columns Equation (25) was used to obtain the critical transition gas hold-up. Critical gas transition hold-up is plotted against terminal bubble rise velocity in a typical stability map. The effects of various parameters such as virtual mass coefficient, Richardson–Zaki index, and proportionality constant for dispersion are described next. a. Effect of the Relation of Bubble Diameter to Terminal Rise Velocity. Figure 24 shows the effect of various dB –VB앝 relationships on the transitions in gas–liquid bubble columns. It can be seen that the transitions are highly sensitive to this relationship. In contaminated systems, the contaminants tend to accumulate at the gas–liquid interface and reduce the mobility of the interface. Therefore, in contaminated systems, for the same bubble diameter, the rise velocity is more than that in the pure systems. b. Effect of Virtual Mass Coefficient. Three different prescriptions were used for the virtual mass coefficient. The virtual mass coefficient was taken as 0.5, 1.0, and as a function of ␧L as already given in Eq. (83). Figure 25 shows the effect of virtual mass coefficient on stability of bubble columns, with the proportionality constant for dispersion equal to 0.5 and the value

62

J. B. JOSHI ET AL.

FIG. 24. Effect of bubble diameter–terminal rise velocity on transitions: bubble columns [움 ⫽ 0.5, CV ⫽ 1.0, m ⫽ 1.9].

FIG. 25. Effect of virtual mass coefficient on transition gas hold-up: bubble columns [움 ⫽ 0.5, m ⫽ 1.9, dB –vB앝 Clift et al. relation].

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

63

of R–Z index m equal to 1.9. For this combination, it can be seen that the critical transition gas hold-up increases as the Cv increases from 0.5 to 1.0 and to f(␧L). The increase in stability with increase in the virtual mass coefficient in case of bubble columns may appear as opposite to the trend observed for solid–liquid and gas–solid fluidized beds. However, as described in Section IIE, the transition points are different for fluidized beds and bubble columns. It can be seen from Fig. 25 that the transition gas hold-up is quite sensitive to the formulation of virtual mass coefficient. Another interesting feature of Fig. 25 is the initial increase in the critical gas hold-up with the increase in bubble diameter upto about 3 mm, where the critical gas hold-up reaches a maximum value with respect to the bubble diameter and then starts decreasing as the bubble diameter increases above 3 mm. The maximum in the critical gas hold-up at a bubble diameter of about 3 mm reflects a similar maximum in the bubble rise velocity at a bubble diameter of 3 mm. The dependence of critical gas hold-up on the relation between dB and VB앝 can be explained using Eq. (78), which clearly indicates that the critical gas hold-up at transition is proportional to the ratio V2B앝 /dB . c. Effect of Dispersion Coefficient. To elucidate the effect of dispersion coefficient on stability, the proportionality constant for dispersion coefficient was varied in the range 0.5–3.0. However, it must be noted that once all the remaining parameters (physical properties and turbulence level) are fixed, the proportionality constant for dispersion will also be fixed, and in reality it cannot be changed as an independent variable. However, because the dependence of the dispersion coefficient on other parameters is unclear, the proportionality constant has been varied independently in the present work. Figure 26 shows the effect of the proportionality constant for the dispersion coefficient on the stability of bubble columns, for the value of Cv equal to 1.0. and m equal to 1.9. It can be seen that, as the proportionality constant increases, the bubble column becomes more unstable. Figure 26 also shows the maximum in the critical gas hold-up at a bubble diameter of about 3 mm. d. Effect of Richardson–Zaki index (m). Figure 27 shows the effect of the Richardson–Zaki index on the stability of bubble columns, with Cv equal to 1.0 and the proportionality constant for dispersion equal to 0.5. It can be seen that as the value of Richardson–Zaki index increases, the bubble column becomes more unstable (i.e., transition occurs at a lower value of gas hold-up). This is explained as follows: The condition for stability is given by Eqs. (76) and (77). Usually, the contribution of B2 /4 is negligible.

64

J. B. JOSHI ET AL.

FIG. 26. Effect of proportionality constant for dispersion on transition gas hold-up: bubble column [CV ⫽ 1.0, m ⫽ 1.9, dB –vB앝 Clift et al. relation].

FIG. 27. Effect of Richardson–Zaki index on transition gas hold-up: bubble column [움 ⫽ 0.5, CV ⫽ 1.0, dB –vB앝 Clift et al. relation].

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

65

Under these conditions, Eq. (77) gets simplified to Z⫺C⫽0 or ␧GC ⫽

Cvv2S . 움dB g

With an increase in m, the slip velocity (vs) decreases. Therefore, the critical gas hold-up also decreases.

C. COMPARISON OF EXPERIMENTAL RESULTS WITH UNBOUNDED BED ANALYSIS 1. Fluidized Beds Tables III and IV summarize the experimental details pertaining to the the solid–liquid fluidized beds and gas–solid fluidized beds, respectively. For the prediction of critical continuous phase hold-ups, the criterion developed in Section II is used. The value of dispersion coefficient was estimated using Eq. (59). The predicted values are given in Tables III and IV. It can be seen that the agreement is excellent when the continuous phase holdup is less than 60%. Beyond the hold-up of 60%, the agreement is fairly good with average deviation of 7% and maximum deviation of 9%. However, the error is systematic. Therefore, the value of 움 [proportionality constant in Eq. (59)] was adjusted so that the predicted value agrees with the experimental value. The values of 움 have been given in the last column. It can be seen that the 움 value is less than 3 when ␧LC ⬎ 60%. For such cases, the unbounded bed analysis is perhaps not sufficient and we need to consider the presence of the sparger and the side walls. These aspects are covered in Sections VI and VII. It may be pointed out that the values of transition hold-ups greater than 60% are observed in solid–liquid fluidized beds. In gas–solid fluidized beds, in most cases, the transition occurs at the condition of incipient fluidization. In some cases (fine particles or high-pressure fluidization), the transition may occur at hold-up values beyond minimum fluidization. In such cases, the bubble phase (which is formed at minimum fluidization) moves in the homogeneous regime up to the point of transition. Model predictions and experimental observations are compared in Figs. 28 and 29. The agreement was found to be excellent with a standard deviation of 12%.

␳S (kg/m3) 8710 8710 8710 8710 8710 8710 8710 8710 8710 8710 8710 8710

System Water–copper Water–copper Water–copper Water–copper Water–copper Water–copper Water–copper Water–copper Water–copper Water–copper Water–copper Water–copper

0.083 0.083 0.165 0.165 0.165 0.165 0.165 0.275 0.275 0.275 0.275 0.275

dP (mm) 0.7 0.6 1.3 1.0 0.85 0.7 1.0 0.85 0.7 0.6 1.3 1.0

애L ⫻ 103 (Pas) 47.6 50.9 66.7 82.9 97.2 105.9 115.5 110.3 131.2 152.2 173.4 190.0

V S앝 (mm/s) 5.30 5.11 5.01 4.91 4.89 4.71 4.67 3.62 3.52 3.42 3.40 3.30

m

75 72 74 71 69 66 64 66 60 56 54 51

␧LC experimental (%)

83 80 84 75 70 68 65 70 62 56 52 48

␧LC predicted (%)

1.87 1.82 1.81 2.30 2.73 2.71 2.76 2.57 2.72 3.01 3.48 3.64

Proportionality constant for dispersion

TABLE III COMPARISON OF EXPERIMENTAL RESULTS OF GIBILARO ET AL. (1986) WITH UNBOUNDED ANALYSIS: SOLID –LIQUID FLUIDIZED BEDS

67

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS TABLE IV COMPARISON OF EXPERIMENTAL RESULTS WITH UNBOUNDED BED ANALYSIS: GAS –SOLID FLUIDIZED BEDS

Investigator Rowe et al. (1982)

Rietema (1973)

King and Harrison (1982)

Jacob and Weimer (1987)

Musters and Rietema (1977)

Gas phase

␳S (kg/m3)

dP (애m)

␧GC experimental (%)

␧GC predicted (%)

CO2 CO2 CO2 Air Ar Ar Ar CO2 CO2 CO2 Air Air Air Ar Ar Ar H2 H2 H2 H2 Air Air Air Air Air N2 N2 N2 N2 N2 N2 CO/H2 CO/H2 CO/H2 CO/H2 CO/H2 CO/H2 Air Air Air

819 819 819 819 819 819 819 819 819 819 819 819 819 819 819 819 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 2400 850 850 850 850 850 850 920 920 920

70 70 70 70 70 70 70 82 82 82 82 82 82 82 82 82 70 70 70 70 70 70 70 70 70 61 61 61 61 61 61 44 44 44 112 112 112 39.7 78.0 103

57 59 61 61 62 63 64 56 57 58 58 59 60 60 61 62 38 43 48 55 42 45 47 57 62 44 44.5 44.5 44.5 44 44 76.5 76.7 77.8 57.8 60.4 63.3 77.6 74.2 70.1

56 59 63 63 65 67 70 52 55 56 57 59 62 61 64 66 42 42 46 52 42 43 50 62 67 45 45.8 45.8 46.3 46.8 47.2 79 80 81 64 62 60 76 68 65

68

J. B. JOSHI ET AL.

FIG. 28. Comparison of experimental and predicted critical hold-ups: solid–liquid fluidized beds.

2. Bubble Columns The transition in bubble columns has been the subject of interest for the past 30 years. Table V summarizes the previous work. The value of VB앝 has been estimated from the corresponding Vsup –␧G data on the basis of initial slope. Using this value of VB앝 and the physical properties, the bubble diameter was estimated using the correlation of Clift et al. (1978). The value of m was estimated using Vsup –␧G data. In the previous section, it was pointed out that the critical gas hold-up is very sensitive to the dB –vB앝 relationship and the values of m, Cv , and 움. It was thought desirable to calculate the value of 움 for all the experimental points shown in Table V. For each point, four values of m (1, 1.4, 1.9 and 2.4) and three levels of Cv were considered. The values of 움 are given in Table VI. The average value works out to be 3.14. Table V also gives the comparison between the experimental critical gas hold-up and the critical gas hold-up predicted using the transition criterion [Eq. (25)] with dispersion coefficient equal to 3. The comparison between predicted and experimental gas hold-up for bubble columns is shown in Fig. 30. It can be seen

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

69

FIG. 29. Comparison of experimental and predicted critical hold-ups: gas–solid fluidized beds.

that the agreement is fairly good, but not as good as that obtained for gas–solid and solid–liquid fluidized beds. The reason for this is the higher sensitivity of the transition criterion in the case of bubble columns to the Richardson–Zaki index, the virtual mass coefficient, and the dispersion coefficient. Also, the predictions depend on the accuracy of prediction of dB from the dB –VB앝 relationship. Hence, a small variation in any of these parameters can change the predicted transition gas hold-up by a great magnitude. From the foregoing discussion, it can be seen that accuracy of estimation of VB앝, the Richardson–Zaki index m, 움, and Cv is important for the prediction of ␧GC .

VI. Generalized Criteria for Bounded Dispersions All the analysis in Section II focused on the unbounded case. The analysis of all investigators including the present authors concentrates on whether a periodic disturbance in the axial direction grows with respect to time, leading to instability or decays with time indicating a stable system. The

70

Tetrabromo ethane Butanediol Pure water Tap water 10% glycerol 0.075% acetic acid Na2SO4 1% EtOH 1% Propanol Na2SO3 Dist. water Tap water Dist. water Dist. water 1% ethanol 1% n-propanol 1% n-propanol 1% n-butanol Water 1% glucose Water Water

Bach and Pilhofer (1978)

Kastanek et al. (1974)

Oels et al. (1978)

Kelkar et al. (1983)

Maruyama et al. (1981)

System

Investigator 267 500 167 217 187 250 243 224 238 253 229 222 242 267 96.3 85.1 95.2 87 250 222 283 417

VB앝 3.76 50.96 1.7 2.3 1.9 1.8 3.8 2.3 2.4 3.4 3.8 2.4 2.7 2.8 0.8 0.7 0.8 0.7 3.8 3.9 3.5 6.5

dB 3.15 0.8 3.05 3.35 3.0 5.65 2.55 4.1 3.9 3.45 2.3 3.45 3.55 4.15 2.6 3.25 3.85 3.3 2.85 2.10 3.8 3.8

Proportionality constant for dispersion 37.8 54.6 46.3 26.1 24.7 30.4 33.9 67.0 36.3 69.5 31.6 39.5 30.5 34.3 40.5 50.0 58.9 50.0 51 29.9 41.4 51.2

VGC experimental (mm/s)

TABLE V COMPARISON OF EXPERIMENTAL RESULTS WITH UNBOUNDED BED ANALYSIS: GAS –LIQUID BUBBLE COLUMNS

14.2 8.0 27.7 12.1 13.2 12.1 13.9 29.9 15.3 27.6 13.8 17.8 12.6 12.9 42.1 58.7 61.2 57.5 20.4 13.5 14.6 12.3

␧GC experimental (%)

8.0 27.0 27.4 7.3 13.2 22.5 22.3 22.6 19.0 16.6 14.9 20.6 14.9 14.0 50.9 56.1 52.1 54.2 28.0 12.6 16.7 11.7

␧GC predicted (%)

71 Deionized water Deionized water water–air (1 MPa) Deionized water–air (1.5 MPa) Deionized water–air (0.1 MPa)

Water–air 540 kPa Water–air 345 kPa Water–air 140 kPa Water–air 21 kPa 1.56% CMC 1% CMC Water–nitrogen Turpentine– nitrogen NaCl–Ar/He Water Water–He Water Water Water 4.3 4.5 5.9 2.1 3.1 3.2

250 278 375 120 220 250

3.0

2.0

173

6.2 5.4

233

400 300

1.0 1.7 4.0 1.7 2.1 1.8

2.8

200

125 166 231 166 198 171

1.7

167

3.05

3.25

4.32.9 2.9

3.45 3.2 2.8 2.2 2.75 3.05

2.95 3.75 1.10 2.70

2.6

2.6

3.05

4.35

Note: Air at atmospheric pressure was used as gas phase unless stated otherwise.

Wilkinson et al. (1992)

Yamashita and Inoue (1975) Wilkinson et al. (1992)

Krishna et al. (1991) Koetsier et al. (1976)

Buccholz et al. (1983)

Tarmy et al. (1984)

63.2

56.7

62.1 61

44.8 32.9 34.7 29.6 24.6 40.0

16.4 10.5 46.5 58.2

57.1

60.0

52.5

47.8

36.5

24.3

15.5 20.3

35.8 19.8 15.0 17.8 12.4 23.4

4.37 8.75 21.2 23.2

20.6

24.0

26.3

28.6

35.9

18.3

13.7 26.9

32.6 29.5 13.4 29.5 14.5 29.3

3.2 11.2 22.6 9.222

24.1

17.9

23.8

36.1

72

J. B. JOSHI ET AL. TABLE VI BACK CALCULATION OF PROPORTIONALITY CONSTANT FOR DISPERSION FROM EXPERIMENTAL DATA: GAS –LIQUID BUBBLE COLUMNS Cv

m ⫽ 1.0

m ⫽ 1.4

m ⫽ 1.9

m ⫽ 2.4

0.5 f (␧L) 1.0

2.41 3.56 3.65

2.07 2.86 3.11

1.75 2.21 2.61

1.50 1.91 2.22

treatment is strictly for axial direction and the axial nonuniformities are the source of transition. Analyses such as these have the limitation that the real fluidized beds are not of infinite extent and that some account must be taken of the boundaries at upper and lower surface of a bed of finite length, and also of the walls that bound the bed laterally. Therefore, an unbounded case is a simplification of the real systems and most of the criteria based on this approach have their limitations.

FIG. 30. Comparison of experimental and predicted critical hold-ups: bubble columns.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

73

Therefore, a more elaborate model taking into account the presence of the lateral walls and the sparger design will now be presented.

A. MATHEMATICAL MODEL In this section a criterion will be developed for the case of a twodimensional bubble column. A schematic diagram of the column with the coordinate system is shown in Fig. 31. The criterion will be developed for the continuous mode of operation. The criterion for batch operation will be a special case of the criterion for the continuous mode of operation, in the limiting case of superficial liquid velocity becoming zero. 1. Basic Equations The basic equations are similar to the equations for the unbounded case. However, for the bounded case, Z as well as X components of the equation are needed. The following additional assumptions are made:

FIG. 31. Schematic diagram of a rectangular bubble column with coordinate system.

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J. B. JOSHI ET AL.

1. The gas phase convection terms and gravity forces are negligible. 2. The gas phase stress terms are negligible. (However, a more generalized criterion that includes the effect of gas phase stress terms will be presented later without the detailed derivation.) 3. There is no variation of gas hold-up in the smallest dimension (y). However, variation of the velocities in the y direction is considered. 4. The virtual mass terms are not considered. In bounded case, the disturbances (destabilizing effects) due to the sparger are present. The equations of continuity and motion in their dimensionless form are as follows: Continuity equation for the liquid phase:

冉 冊 冉 冊

⭸␧ ⭸␧ ⭸␧L ⭸ ⭸ ⭸ ⭸ (␧LUx) ⫹ (␧LUz) ⫺ ⫹ DL L ⫺ DL L ⫽ 0. (94) ⭸␶ ⭸X ⭸Z ⭸X ⭸X ⭸Z ⭸Z Continuity equation for the gas phase:

冉

冊 冉 冊

⭸ ⭸ ⭸ ⭸ ⭸␧G ⭸␧ ⭸␧ (␧GVx) ⫹ (␧GVz) ⫺ ⫹ DG G ⫺ DG G ⫽ 0. (95) ⭸␶ ⭸X ⭸Z ⭸X ⭸X ⭸Z ⭸Z Equation of motion for the liquid phase, X component: (1 ⫺ ␧G)

再

⭸Ux ⭸U ⭸U ⫹ Ux x ⫹ U z x ⭸␶ ⭸X ⭸Z

⫽ ⫺(1 ⫺ ␧G) ⫹

冎

冎

再

(96)

再

(97)

⭸P (1 ⫺ ␧G) ⭸2Ux ⭸2Ux ⫹ B(Vx ⫺ UX) ⫹ ⫹ ⭸X ReL ⭸X 2 ⭸Y 2

再

冎

⭸ 2 Ux 2 ⭸Ux ⭸␧L ⭸Ux ⭸␧L ⫹ . ⫹ ⭸Z2 ReL ⭸X ⭸X ⭸Z ⭸Z

Equation of motion for the liquid, Z component: (1 ⫺ ␧G)

再

⭸Uz ⭸U ⭸U ⫹ Ux z ⫹ U z z ⭸␶ ⭸X ⭸Z

⫽ ⫺(1 ⫺ ␧G) ⫹

冎

冎

⭸P (1 ⫺ ␧G) ⭸2Uz ⭸2Uz ⫹ ⫹ B(Vz ⫺ Uz) ⫹ ⭸Z ReL ⭸X2 ⭸Y2

再

冎

⭸ 2 Uz 2 ⭸Uz ⭸␧L ⭸Uz ⭸␧L ⫹ ⫺ gˆ(1 ⫺ ␧G). ⫹ ⭸Z2 ReL ⭸X ⭸X ⭸Z ⭸Z

Equation of motion for the gas phase, X component: 0 ⫽ ⫺␧G

⭸P ⫺ B(Vx ⫺ UX). ⭸X

(98)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

75

Equation of motion for the gas phase, Z component: 0 ⫽ ⫺␧G

⭸P ⫺ B(Vz ⫺ Uz), ⭸Z

(99)

The dimensionless variables are defined by Ux ⫽

ux u v v ; Uz ⫽ z ; V s ⫽ s ; V x ⫽ x ; VB앝 VB앝 VB앝 VB 앝

Vz ⫽

x y H vz z ; Z⫽ ; X⫽ ; Y⫽ ; h⫽ ; VB 앝 L L L L

␶⫽

tVB앝 LVB앝␳L LVB앝␳L ; ReL ⫽ ; ReG ⫽ ; L 애L 애G

P⫽

p gL ˆ DL gˆ ⫽ 2 D ; L⫽ 2 ; ␳LV B앝 V B앝 LVB앝

(100)

ˆ G ⫽ D G ; B ⫽ ␭L ; ␳ ⫽ ␳ G ; D LVB앝 ␳ L V B앝 ␳L where 애L , 애G are the turbulent viscosities of the liquid phase and gas phase, respectively and VB앝 ⫽ ␷s(⑀G ⫽ 0). Thus, Eqs. (94)–(99) describe the dynamics of gas–liquid dispersion in a two-dimensional bubble column. 2. Boundary Conditions The pressure drop across the sparger for gas phase is due to viscous as well as turbulent resistance to the flow. It is expressed as pG ⫺ p(x, 0) ⫽ kVvsup ⫹ kTv2sup ,

(101)

where pG is the pressure below the sparger plate and p(x, 0) is the pressure just above the sparger plate, at the bottom of the column. kv and kT are the proportionality constants, and vsup is the superficial gas velocity. For a bubble column operated under net liquid flow conditions, both the gas and the liquid are introduced and removed continuously. The liquid flow may be cocurrent or countercurrent. The boundary condition at the bottom depends on the way in which the liquid is introduced or taken away. Consider the bottom design as shown in Fig. 32. The liquid phase pressure drop consists of three parts, unlike that for gas, which consisted of only two parts. The liquid-phase pressure drop is due to the viscous and turbulent modes of friction across the sparger and

76

J. B. JOSHI ET AL.

FIG. 32. Details of sparger.

due to the acceleration of the liquid as it passes through the sparger region. The liquid-phase pressure drop is given as pL ⫺ p(x, 0) ⫽ kVLusup ⫹ kTLu2sup ⫹

␳Du2z ⫺ ␳Lu2sup , 2

(102)

where kVL and kTL are the proportionality constants, usup is the superficial liquid velocity, and eD ⫽ ␳L (1 ⫺ ␧G) is the dispersion density. Let us introduce the dimensionless variables as follows: vsup kV k u k ; KV ⫽ ; KT ⫽ T; Usup ⫽ sup ; KVL ⫽ VL VB 앝 ␳LVB앝 ␳L VB앝 ␳LV B 앝 (103) ␳ kTL ⫺ L 2 pG p(x, 0) pL KTL ⫽ ; PG ⫽ ; P(X, 0) ⫽ ; PL ⫽ . ␳LV2B앝 ␳LV2B앝 ␳LV 2B앝 ␳L

Vsup ⫽

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

77

Thus, Eq. (101) and (102) in dimensionless form are, respectively, PG ⫺ P(X, 0) ⫽ KV Vsup ⫹ KTV 2sup

(104)

冋

册

1 PL ⫺ P(X, 0) ⫽ KVLUZ(1 ⫺ ␧G) ⫹ U 2Z KTL(1 ⫺ ␧G)2 ⫹ (1 ⫺ ␧G) . (105) 2 The boundary conditions are X ⫽ 0,

Ux ⫽ 0

(106a)

X ⫽ 1,

Ux ⫽ 0

(106b)

Z ⫽ 0,

U z ⫽ U0

(106c)

Z ⫽ h,

U z ⫽ U0

(106d)

y ⫽ ⫾d/2, all x and z, Ux ⫽ 0,

Uz ⫽ 0

(106e)

Vx ⫽ 0,

Vz ⫽ 0.

(106f )

The initial steady state of flow is the homogeneous flow regime (i.e., the uniform bubbly flow regime) and is represented by the following equations: At time t ⫽ 0: Ux ⫽ 0

(107a)

Uz ⫽ U0 ⫽ constant

(107b)

VX ⫽ 0

(107c)

Vz ⫽ V0 ⫽ constant

(107d)

␧G(X, Z) ⫽ ␧G0 ⫽ constant P(X, Z) ⫽

patm ⫹ gˆ(h ⫺ Z)(1 ⫺ ␧G0) ⫽ P0(X, Z). ␳LV 2B앝

(107e) (107f )

Substitution of the steady-state conditions in the equations of motion gives the following expression for the drag coefficient 웁: 웁⫽

␧G(␳L ⫺ ␳G) g . (v0 ⫺ u0)

(108)

3. Linear Stability Analysis According to the theory of hydrodynamic stability analysis, infinitesimally small perturbations are superimposed on the steady-state values of

78

J. B. JOSHI ET AL.

variables and their transient behavior is studied. The variables in the perturbed state are represented by subscript 1 and steady-state values of the variables are represented by subscript 0. Using Eq. (107), we get the variables in their perturbed state as follows: ␧G ⫽ ␧G0 ⫹ ␧G1

(109a)

Ux ⫽ Ux1

(109b)

Uz ⫽ U0 ⫹ Uz1

(109c)

Vx ⫽ Vx1

(109d)

Vz ⫽ V0 ⫹ Vz1

(109e)

P ⫽ P 0 ⫹ P1 .

(109f )

The interaction term 웁 or B is also linearized as follows: B(␧G) ⫽ B0(␧G0) ⫹ B⬘0␧G1 ,

(110)

冉 冊

(111)

where B⬘0 ⫽

⭸B ⭸␧G

␧G⫽␧G0

.

Before introducing perturbations, some simplification of equations can be done by following an averaging procedure, given next. a. Averaging in the Y Direction. Since L Ⰷ d and H Ⰷ d, it is assumed that the velocity profiles in the Y direction can be represented as

冉 冉

冊 冊

U(X, Y, Z, ␶) ⫽ Umax(X, Z, ␶) 1 ⫺

4Y 2L2 d2

(112a)

V(X, Y, Z, ␶) ⫽ Vmax(X, Z, ␶) 1 ⫺

4Y 2L2 , d2

(112b)

where Umax and vmax are the velocities in the central plane, Y ⫽ 0. (U includes both the components Ux and Uz , and similarly for V.) The average of these velocities in the Y direction can be obtained as

冕 ¯ (X, Z, ␶) ⫽ 冕 V

¯ (X, Z, ␶) ⫽ U

d/2L

⫺d/2L

d/2L

⫺d/2L

U(X, Y, Z, ␶) dY

(113a)

V(X, Y, Z, ␶) dY.

(113b)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

79

Substituting Eq. (112) in Eq. (113),

冉 冉

冊 冊

U(X, Y, Z, ␶) ⫽

3L ¯ 4Y 2L2 U(X, Z, ␶ ) 1 ⫺ 2d d2

(114a)

V (X, Y, Z, ␶) ⫽

3L ¯ 4Y 2L2 V(X, Z, ␶ ) 1 ⫺ . 2d d2

(114b)

It is assumed that the averaging holds for the perturbation variables as well, and that the hold-up and pressure are independent of Y. With the preceding assumptions, substituting Eq. (114) in Eq. (96)–(99), integrating each term w.r.t. Y from ⫺d/2L to ⫹d/2L, and dividing throughout by d/L, the Y-direction averaged equations can be obtained. After Ydirection averaging and substituting Eq. (109) in Eq. (95)–(99), and retaining only those terms that are linear in the perturbed variables, we can get the linearized continuity equations and equations of motion for both the phases. Proceeding in the usual manner, we assume perturbations of the form ¯ x(X, Z)es␶ ¯ x1(X, Z, ␶) ⫽ U U

(115a)

¯ z1(X, Z, ␶) ⫽ U ¯ z(X, Z)es␶ U

(115b)

¯ x(X, Z)es␶ ¯ x1(X, Z, ␶) ⫽ V V

(115c)

¯ z1(X, Z, ␶) ⫽ V ¯ z(X, Z)e V

s␶

(115d)

␧G1(X, Z, ␶) ⫽ ␧G(X, Z)es␶

(115e)

P1(X, Z, ␶) ⫽ P(X, Z)e .

(115f )

s␶

The term’s temporal growth rate in es␶ is complex. When the real part of s equals zero, there can be two types of marginal states, depending on whether the imaginary part of s is also zero or nonzero. It is known that when the imaginary part of s is also equal to zero, the marginal state is charecterized by a stationary pattern of motion, in the form of cellular convection or secondary flow. If the imaginary part of s is nonzero in the marginal state, the instability sets in as oscillations of growing amplitude, also known overstability or oscillatory mode of instability. In the case of multiphase reactors, instability occurs in the form of cellular convection. In bubble columns, the transition from the homogeneous regime to the heterogeneous regime is marked by intense liquid recirculation in form of a single cell or multiple cells. This indicates that the imaginary part of s is equal to zero. In contrast, in the bounded bed analysis the imaginary part of s was not zero. Jackson and co-workers, Homsy and co-workers, and

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J. B. JOSHI ET AL.

Batchelor have assumed that the instability in unbounded beds sets in as oscillations of growing amplitude. Using Eq. (115) and setting s ⫽ 0 gives the equations that determine the parameters at which the transition from homogeneous regime to the heterogeneous regime occurs. These neutral continuity equations and neutral equations of motion for both the phases after linearization and Ydirection averaging are as follows: Linearized neutral continuity equation, liquid phase: (1 ⫺ ␧G0)

冋

册

¯ x ⭸U ¯ ⭸U ⭸␧ ⭸2␧G ⭸2␧G ⫹ z ⫺ U0 G ⫹ D L ⫽ 0. (116) 2 ⫹ DL ⭸X ⭸Z ⭸Z ⭸X ⭸Z2

Linearized neutral continuity equation, gas phase: ␧G0

冋

册

¯ x ⭸V ¯ ⭸V ⭸␧ ⭸2␧G ⭸2␧G ⫽ 0. ⫹ z ⫹ V0 G ⫺ DG 2 ⫺ DG ⭸X ⭸Z ⭸Z ⭸X ⭸Z2

(117)

Linearized equation of motion for the liquid phase, X component: (1 ⫺ ␧G0) U0

¯x ⭸P ⭸U ⫽ ⫺(1 ⫺ ␧G0) ⭸Z ⭸X ⫹

␩ ¯ x ⫹ B0(V ¯x⫺U ¯ x). (1 ⫺ ␧G0)U ReL

(118)

Linearized equation of motion for the liquid phase, Z component: (1 ⫺ ␧G0)U0

¯Z ⭸P ⭸U ␩ ¯Z (1 ⫺ ␧G0)U ⫽ ⫺(1 ⫺ ␧G0) ⫹ gˆ␧G ⫹ ⭸Z ⭸Z ReL ¯z⫺U ¯ z) ⫹ BVs0␧G . ⫹ B0(V

(119)

Linearized equation of motion for the gas phase, X component: 0 ⫽ ⫺␧G0

⭸P ¯X⫺U ¯ X). ⫺ B0(V ⭸X

(120)

Linearized equation of motion for the gas phase, Z component: 0 ⫽ ⫺␧G0 Here,

⭸P ¯Z⫺U ¯ Z) ⫺ B 10 Vs0␧¯ G ⫺ B0(V ⭸Z

(121)

B 10 ⫽ (⭸B/⭸␧G )␧G ⫽ ␧G0 . Vs0 ⫽ V0 ⫺ U0 .

(122)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

81

The overbar indicates the velocities averaged in the Y direction, and ␩, the averaging parameter, is defined as

␩⫽

12L3 . d3

(123)

In writing equations (118) and (119), we have neglected the rate of momentum transferred in the X and Z directions as compared with that transferred in the smallest direction i.e., Y. As L Ⰷ d, H Ⰷ d, ␩ Ⰷ 1; therefore, we have ¯ x1 ⱖ ␩U

¯ x1 ⭸2U ¯ x1 ⭸ 2U . 2 ⫹ ⭸X ⭸Z2

(124)

b. Linearization of the Boundary Conditions. The linearized boundary conditions are as follows: X ⫽ 0,

¯x⫽0 U

(125a)

X ⫽ 1,

¯x⫽0 U

(125b)

Z ⫽ h,

P ⫽ 0.

(125c)

The gas sparger boundary condition is obtained by substituting Eq. (109) in Eq. (104) and linearizing. At Z ⫽ 0, ␧G ⫽ ⫺E1(P ⫹ F1Uz),

(126)

E1 ⫽ [(U0 ⫹ Vs ⫹ ␧G0V⬘s )(KV ⫹ 2KT␧G0(U0 ⫹ Vs))]⫺1

(127)

where

V ⬘s ⫽

冉 冊 ⭸Vs ⭸␧G

␧G⫽␧G0

F1 ⫽ KV ⫹ 2␧G0KT(U0 ⫹ Vs)␧G0 .

(128) (129)

To linearize the liquid sparger boundary condition, substituting Eq. (109) in Eq. (104) and rearranging, we have Uz ⫽ ⫺A1(P ⫺ C5␧G),

(130)

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J. B. JOSHI ET AL.

where

冋

冉

再

A1 ⫽ (1 ⫺ ␧G0) KVL ⫹ 2U0 KTL(1 ⫺ ␧G0) ⫹

冋

冉

C5 ⫽ U0 KVL ⫹ U0 2KTL(1 ⫺ ␧G0) ⫹

1 2

冊册

冎冊册

1 2

.

⫺1

(131)

(132)

From Eqs. (126) and (130), we get UZ ⫽ ⫺SP

(133a)

␧G ⫽ ⫺QP,

(133b)

where

S⫽

A1(1 ⫹ E1C5) , (1 ⫹ A1E1C5F1)

Q⫽

E1(1 ⫺ A1F1) . (1 ⫹ A1E1C5F1)

(134)

c. Solution of Continuity Equations. Let us introduce stream functions ␺ and ␺G as follows: Ux ⫽ ⫺

⭸␺ ⭸␺ , Uz ⫽ ⭸Z ⭸X

(135)

⭸ ␺G ⭸␺ , Vz ⫽ G . ⭸Z ⭸X

(136)

Vx ⫽ ⫺

Adding Eq. (116) and (117) using Eq. (135) and (136) and rearranging gives ⭸2␧G Vs ⭸␧G ⭸2␧G ⫹ ⫽ 0, ⫹ ⭸Z2 De ⭸X ⭸X 2

(137)

D e ⫽ DL ⫺ DG .

(138)

where

Let us use the method of separation of variables for solving Eq. (137). ␧G ⫽ f (X) g(Z)

(139)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

83

Using Eq. (139) and (137), we get d 2g Vs dg ⫺ ␭2g ⫽ 0 ⫹ dZ2 De dZ

(140a)

d 2f ⫹ ␭2f ⫽ 0, dZ2

(140b)

where ␭ is a separation constant. The general solution of Eq. (140b) is f (X) ⫽ C1 sin(␭X) ⫹ C2 cos(␭X).

(141)

At X ⫽ 0, ␧G ⫽ 0. This implies that C2 ⫽ 0. Therefore, f (X) ⫽ C1 sin(␭X ).

(142)

This admits periodic solutions for each value of ␭, and in general ␭ ⫽ n앟. This indicates multiple cells in the horizontal direction. We restrict ourselves to just one horizontal cell, i.e., n ⫽ 1. The value of n ⫽ 1 corresponds to one horizontal cell, and all the values of n that are greater than 1 correspond to multiple cells in the horizontal direction. In bubble column transitions, the multiple cells in horizontal direction have been observed only in shallow bubble columns in which the height-to-diameter ratio is less than 1. In the present case, we have considered H/D Ⰷ 1. Therefore, the transition to the heterogeneous regime is characterized by one cell in the horizontal direction. Thus, the first transition appears to correspond to a single cell in the horizontal direction:

␭1 ⫽ 앟.

(143)

g(Z) ⫽ C3eaZ ⫹ C4ebZ,

(144)

The solution to Eq. (140a) is

where a and b are given by

冋

a, b ⫽ 0.5 ⫺

Vs ⫾ De

冪冉 冊 册 Vs De

2

⫹ 4앟 .

(145)

Inspection of Eq. (145) reveals that the order of magnitude of vs /De is 50–100 under normal operating conditions, indicating that a is a small positive number and b is a large negative number. The disturbance is induced in a stable bubble column because of the sparger. Therefore, any small disturbance in the free region below the sparger will manifest itself very close to the sparger (as Z tends to zero), and this disturbance will decay off rapidly as we move up the column, at a rate of e bz, b being a

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J. B. JOSHI ET AL.

very large negative number. Thus, the term on the RHS of Eq. (144) represents the contribution of perturbations due to the sparger that die out within a very short distance from the sparger. These perturbations have little physical significance and hence can be neglected. Equation (144) can thus be approximated to g(Z) ⫽ C3eaZ.

(146)

Combining Eqs. (139), (142), (143), and (146), we get ␧G(X, Z) ⫽ C1C3 sin(앟X)eaZ.

(147)

␧G(X, 0) ⫽ C1C3 sin(앟X).

(148)

At Z ⫽ 0,

Combining Eqs. (147) and (148), ␧G(X, Z) ⫽ ␧G(X, 0)eaZ,

(149)

⭸␧G(X, Z) ⭸␧G(X, 0) aZ ⫽ e . ⭸X ⭸X

(150)

which then gives

d. Solution of the Momentum Equations. Adding Eqs. (118) and (120), we get the x direction linearized momentum equation for the gas–liquid dispersion as M4

¯x ⭸U ⭸P ¯ x, ⫽ ⫺M1 ⫹ M 3U ⭸Z ⭸X

(151)

M1 ⫽ 1

(152)

where

␩ M3 ⫽ (1 ⫺ ␧G0) ReL M4 ⫽ (1 ⫺ ␧G0)U0.

(153) (154)

Similarly, adding Eqs. (119) and (121) gives the combined z direction momentum equation for gas–liquid dispersion as M4

¯z ⭸U ⭸P ¯ z ⫹ M2gˆ, ⫽ ⫺M1 ⫹ M 3U ⭸Z ⭸Z

(155)

where M2 ⫽ 1.

(156)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

85

Eliminating P from Eqs. (151) and (155), and introducing stream functions from Eq. (135), we get M4

冉

冊 冉

冊

⭸␧ ⭸ ⭸2␺ ⭸ 2␺ ⭸ 2␺ ⭸ 2␺ ⫹ ⫹ M2gˆ G ⫽ 0. 2⫹ 2 ⫽ M3 ⭸Z ⭸X ⭸Z ⭸X 2 ⭸Z 2 ⭸X

(157)

Substitution of boundary condition (133b) in equation (150) gives ⭸P(X, 0) aZ ⭸␧G(X, Z) ⫽ ⫺Q e . ⭸X ⭸X

(158)

Equation (151) at the boundary (X, 0) takes the form M1

⭸␺(X, 0) ⭸2␺(X, 0) ⭸␺(X, 0) ⫺ M3 ⫽ M4 . ⭸X ⭸Z2 ⭸Z

From Eqs. (158) and (159), we get

冉

⭸2␺ ⭸␺ ⭸␧G(X, Z) Q ⫺M4 2 ⫹ M3 ⫽ ⭸X M1 ⭸Z ⭸Z Substitution of Eq. (160) in (157) gives M4

冊

Z⫽0

eaZ.

冋 冉 冊册 冉 冊 冉 冉 冊 ⭸ ⭸2␺ ⭸Z ⭸X 2

⫹

⭸2␺ ⭸2 ␺ ⭸ 2 ␺ ⫹ 2 ⫺ M3 ⭸Z ⭸X 2 ⭸Z 2

⭸2␺ ⭸␺ ⫹ ␴ M4 2 ⫺ M 3 ⭸Z ⭸Z

Z⫽0

e

aZ

冊

(159)

(160)

(161)

⫽ 0,

where

␴⫽

M2gˆQ . M1

(162)

The boundary conditions for ␺ are X ⫽ 0,

⭸␺ ⫽0 ⭸Z

(162a)

X ⫽ 1,

⭸␺ ⫽0 ⭸Z

(162b)

Z ⫽ h,

M4

Z ⫽ 0,

⭸ 2␺ ⭸2␺ ⭸␺ ⫽ ⫺SM ⫺ M3 4 ⭸X 2 ⭸Z2 ⭸Z

(162d)

Z ⫽ 0,

⭸␺ ⫽ 0. ⭸Z

(162e)

⭸2␺ ⭸␺ ⫽0 2 ⫺ M3 ⭸Z ⭸Z

(162c)

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J. B. JOSHI ET AL.

Let us define

␾ ⫽ M4

⭸2␺ ⭸␺ ⫺ M3 . ⭸Z 2 ⭸Z

(163)

Equation (162) suggests the following form for ␺:

␺ ⫽ exp

冉 冊冋冕 M3Z M4

Z

0

冉

冊

册

1 MZ exp ⫺ 3 ␾dZ ⫹ C . M4 M4

(164)

Equation (161) along with its boundary conditions represents an eigenvalue problem. The lowest eigenvalue is the value at which the transition from the homogeneous to the heterogeneous regime occurs. Using the method of separation for solving Eq. (164),

␺(X, Z) ⫽ X(X) ⭈ Z(Z).

(165)

Substitution of (165) in Eq. (161) gives d 2X ⫽ ⫺k2X dX2 d 2␾ ⫺ k2Z ⫽ ⫺M5eaZ, dZ2

(166a) (166b)

where k is a separation constant and M5 ⫽ ␴

dZ(0) . dZ

(167)

The boundary conditions for Eqs. (166) are X ⫽ 0, X ⫽ 0

(168a)

X ⫽ 0, X ⫽ 1

(168b)

Z ⫽ 0, ␾ ⫽ 0

(168c)

Z ⫽ h, ␾ ⫽ 0.

(168d)

The general solution of Eq. (166a) is X ⫽ A1 sin(kX) ⫹ A2 cos(kX ).

(169)

To satisfy the boundary condition (168a), A2 ⫽ 0. Further, to satisfy boundary condition (168b), k ⫽ n앟(n ⫽ 1, 2, 3, 4, . . .).

(170)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

87

The general solution of Eq. (166b) is

␾(Z) ⫽ A3 sinh(kZ) ⫹ A4 cosh(kZ) ⫹

M5 eaZ . k2 ⫺ a 2

(171)

Using the boundary conditions (168c) and (168d), we get the transition criterion at the lowest value of k, (i.e., n ⫽ 1): 앟 2 ⫺ a2 ((M2gˆQ/M1) ⫺ M3S) ⫽ coth(앟h). (172) 2 2 1 ⫺ ((M2gˆQ/M1) ⫺ M3S)(a/(앟 ⫺ a )) 앟 The homogeneous regime will prevail as long as the LHS of this equation is less than the RHS, and the transition to the heterogeneous regime will occur as the LHS becomes greater than the RHS. This criterion involves the assumption that the gas phase stress terms are negligible. This assumption may not be valid in case of solid–liquid fluidized beds or liquid–liquid dispersions. In this case, the criterion is of the same form as Eq. (172), with different definitions of the parameters M1 , M2 , and M3 , which are given in Table VII. Table VII also gives the parameters of the criterion when the dispersion terms are not included in the continuity equations of both the phases. Two special cases can be considered here. Case 1. The pressure drop across the liquid sparger is very much less than that caused by its acceleration over the gas sparger region. This case TABLE VII PARAMETERS APPEARING ON THE STABILITY CRITERION (BOUNDED ANALYSIS) FOR RECTANGULAR BUBBLE COLUMNS: CARTESIAN COORDINATES Parameter

␩␧G0(1 ⫺ ␧G0) ReGB0

M1

1⫺

M2

1⫺␳⫹

M3

M4

ˆG ⫽ 0 D

Generalized criterion 1

␩␧G0 ( gˆ ⫹ B⬘0VS0) ReGB0 gˆ ␩(1 ⫺ ␧G0) ⫹ ReL ␩␧G0 ␩(1 ⫺ ␧G0) 1⫺ ReG ReLB0

冉

冉

U0(1 ⫺ ␧G0) 1 ⫺

冊

冊

␩␧G0 ReGB0

1⫺␳

␩(1 ⫺ ␧G0) ReL

U0(1 ⫺ ␧G0)

Note: These parameter definitions hold for the semibatch operations with U0 ⫽ 0 and Vs ⫽ V0 .

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J. B. JOSHI ET AL.

corresponds to a situation when the column is connected at its bottom to a huge reservoir of liquid. Mathematically, this can be written as kVLvsup ⫹ kTLv2sup Ⰶ

␳uz ␳Lvsup ⫺ . 2 2

(173)

Under limiting conditions this means that kVL 씮 0, and kTL 씮 0 parameters A1 and C5 are defined as A1 ⫽

U 20 1 , C5 ⫽ . U0(1 ⫺ ␧G0) 2

(174)

Parameters E1 and F1 are defined as in Eqs. (127) and (129). The stability criterion is given by Eq. (172). Case 2. Pressure drop across the liquid sparger is sufficiently large to ensure uniform liquid flow into the column irrespective of the pressure perturbations in the column. Mathematically this means that kVLvsup ⫹ kTLv2L Ⰷ

␳uz ␳Lvsup ⫺ . 2 2

(175)

Under these limiting conditions, we have kVL 씮 앝 and kTL 씮 앝.

(176a)

Uz1(X, 0) ⫽ 0; hence, S ⫽ 0.

(176b)

Also,

Using these conditions in (131), (132), and (134), we get A1 씮 0, C5 씮 앝; therefore, Q ⫽ E1 .

(177)

Under these conditions, the stability criterion becomes (M2 /M1)( gˆ웁⬘1 /움⬘) 1 ⫺ (M2 /M1)( gˆ웁⬘1 /움⬘)(a/앟2 ⫺ a2)

⫽

앟 2 ⫺ a2 coth(앟h), 앟

(178)

where 움⬘ and 웁⬘1 are the modified parameters corresponding to the continuous mode of operation defined as follows: 움⬘ ⫽ (U0 ⫹ Vs ⫹ ␧G0V⬘s )

(179)

웁⬘1 ⫽ [Kv ⫹ 2KT␧G0(U0 ⫹ Vs)]⫺1.

(180)

e. Criterion for Batch Operation. The criterion for batch operation is a special case of the criterion for continuous mode of operation, when U0 ⫽ 0.

89

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

4. Criterion for Cylindrical Bubble Columns Following a procedure analogous to that used in derivation of the stability criterion for two-dimensional bubble columns, the stability criterion for cylindrical bubble columns is given as

␭22 ⫺ a2 sinh(␭2h) N2 Q , ⫽ 1 ⫺ N2Q(a/(␭22 ⫺ a2)) ␭2 cosh(␭2h) ⫺ eah

(181)

where ␭2 is the first nonzero root of the equation:

冉

J1 ␭2

冪NNGh冊 ⫽ 0, i.e., ␭ ⫽ 3.8693 冪NNGh . 1

1

2

3

(182)

3

The parameters appearing in Eqs. (181) and (182) are defined in Table VIII. The assumption of negligible gas phase stresses leads to some simplifications. The modified definitions of the parameters for these special cases are also listed in Table VIII. Other special cases depending upon the limiting values of pressure drop across the sparger are similar to those given for a rectangular column. The criteria for the special cases are given next. Case 1. kVL ⫽ 0 and kTL ⫽ 0, A1 and C5 are defined by Eq. (174), and the criterion is given by Eq. (181). TABLE VIII PARAMETERS APPEARING IN THE STABILITY CRITERION (BOUNDED ANALYSIS) FOR CYLINDRICAL BUBBLE COLUMNS: CYLINDRICAL COORDINATES Parameter N1 N2 N3

웂G ⫽ 0

Generalized criterion Y1 ⫺ Y3(B0 ⫺ 2Y1)/B0 1 ⫺ 2Y3(1 ⫺ ␧G0)/B0

Y1 ⫹ V0␧G0(B0 ⫺ 2Y1)/2B0h 1 ⫹ V0␧G0(1 ⫺ ␧G0)/B0h

1 ⫺ Y4( gˆ ⫺ B⬘0VS0)/B0 1 ⫺ Y4(1 ⫺ ␧G0)/B0 Y2 ⫺ Y4(B0h ⫺ Y2)/B0h hgˆ(1 ⫺ Y4(1 ⫺ ␧G0)/B0h)

1 ⫹ V0( gˆ ⫺ B⬘0VS0)/B0␧G0 1 ⫹ V0(1 ⫺ ␧G0)/B0␧G0 Y2 ⫹ V0(B0h ⫺ Y2)/␧G0B0h gˆh(1 ⫹ V0(1 ⫺ ␧G0)/␧G0)/␧G0B0h)

冉 冉

冊 冊

8 ⫹ 0.5(1/h)2 U0 ⫺ ReL 2h

Y1

(1 ⫺ ␧G0)

Y2

h(1 ⫺ ␧G0)

Y3

␧G0(1 ⫺ ␧G0)

Y4

h␧G0(1 ⫺ ␧G0)

8 ⫹ 2.5(1/h)2 U0 ⫺ ReL h

冉 冉

Remains same Remains same

冊 冊

8 ⫹ 0.5(1/h)2 V0 V␧ ⫺ ⫺ 0 G0 ReG 2h 2h 8 ⫹ 2.5(1/h)2 V0 V ⫺ ⫺ 0 ReG h ␧G0

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J. B. JOSHI ET AL.

Case 2. kVL ⫽ 앝 and kTL ⫽ 앝, and the criterion becomes N2(웁⬘1 /움) 1 ⫺ N2(웁⬘1 /움⬘)(a/(␭22 ⫺ a2))

⫽

␭22 ⫺ a2 sinh(␭2h) , ␭2 cosh(␭2h) ⫺ eah

(183)

where 웁⬘1 and 움⬘ are given by Eqs. (180) and (179).

B. ESTIMATION OF MODEL PARAMETERS In order to predict fractional gas hold-up and superficial gas velocity at the point of transition using the stability criteria developed, it is necessary to estimate the following parameters: 1. Slip velocity: vs(VB앝 , ␧G) 2. Dispersion coefficients 3. Pressure drop across the gas sparger: ⌬p(kV , kT , vsup) The estimation procedure for slip velocity and dispersion coefficient was given earlier in Section V. For the bounded bed analysis, the pressure boundary condition at the sparger becomes important. The procedure for predicting pressure drop at the sparger is given next. 1. Estimation of Sparger Pressure Drop Pressure drop across the gas sparger is expressed as follows: ⌬P ⫽ kVvsup ⫹ kTv2sup ⫽ kVVG ⫹ kTV 2G .

(184a)

The two terms on the right-hand side represent the contributions due to viscous and turbulent flow resistances. It was assumed that the pressure drop due to viscous resistance is negligible (kv ⫽ 0). However, the parametric sensitivity of kv on the critical superficial gas velocity was also studied. The turbulent pressure drop was estimated using the orifice equation ⌬P ⫽ kTV 2G ,

(184b)

where kT ⫽

冉 冊

␳G R 2 . 2 Chole

(184c)

R is the ratio of column cross-sectional area to orifice area, and Chole is the orifice coefficient and was taken as 0.61. In the case of fluidized beds, since all reported data on transition are

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

91

for very small particles of the order of a few hundred micrometers, the sparger resistance calculated by Eq. (184) is very high, and therefore, in the further analysis, it was assumed to be a very high quantity tending to infinity.

C. RESULTS AND DISCUSSION 1. Fluidized Beds The stability criterion is given by Eq. (183). The homogeneous flow regime can be maintained as long as the left-hand side of this equation is less than the right-hand side. For the aspect ratio (h ⫽ H/D) ⬎ 1, the righthand side is a constant quantity. The left-hand side depends on the design of the sparger and the physical properties of the fluid. As mentioned earlier, data on the design of a sparger for fluidized beds has not been reported. Therefore, the role of sparger design in transition cannot be studied. However, under the limiting case of infinite sparger resistance, the criterion can be simplified. Under these conditions kv and kT 씮 앝 and hence 웁⬘1 씮 0. In order to keep the RHS of Eq. (183) finite, 움⬘ should tend to zero. With this condition, we get ␧SC ⫽

1 , m

(185)

where m is the Richardson–Zaki index and is a function of the particle Reynolds number. 2. Bubble Columns The superficial gas velocity (vsup , VG), at which the transition from the homogeneous to the heterogeneous flow regime occurs depends mainly on the design of the gas sparger and the physical properties of the system. In order to understand the effect of these parameters on vsup , plots were constructed on the basis of the stability criterion [Eq. (183)]. The effects of the number of orifices, orifice diameter, bubble rise velocity, and gas hold-up index on VGC were studied. In the case of continuous operation, the effect of superficial liquid velocity on ␧GC and vsup was studied. While studying the effect of orifice diameter and the contribution of viscous flow resistance on ␧GC and vsup , the value of m was taken as 2.4 in the following equation: vsup ⫺1 ⫽ VB앝␧m . L ␧G

(186)

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J. B. JOSHI ET AL.

The homogeneous flow regime can be maintained as long as the LHS of Eq. (183) is less than the RHS. For the aspect ratio (h ⫽ H/D) ⬎ 1 the RHS is a constant quantity. The LHS depends on the design of the gas sparger and the physical properties of the liquid. The role of these parameters will now be discussed systematically. a. Semibatch Operation (i) Effect of Sparger Resistance. The sparger resistance is represented by Eq. (184). The sparger resistance increases with an increase in the area ratio (R). The effect of area ratio on ␧GC is shown in Fig. 33. It can be seen that the value of ␧GC increases with an increase in R up to about 800, and then ␧GC levels off. The limiting value of ␧gc can be obtained as R tends to infinity. Under these conditions kT 씮 앝 and hence 웁⬘1 씮 0. In order to keep the RHS of Eq. (183) finite, 움⬘ should tend to zero. With this condition, we get ␧GC ⫽

1 . m

FIG. 33. Effect of area ratio on critical gas hold-up.

(187)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

93

FIG. 34. Effect of area ratio on superficial gas velocity.

For a typical value of m equal to 1.42 (Richardson and Zaki, 1954), the value of ␧GC works out to be 0.42. In the published literature, the maximum value of hold-up (usup ⫽ 0) has been reported by Oels et al. (1978) and is in the range of 0.42 to 0.48. The experimental value agrees with the limiting value predicted by Eq. (186). The effect of area ratio on the transition gas velocity, VGC, is shown in Fig. 34. Figure 35 shows the parametric sensitivity of VGC on kv . The effect

FIG. 35. Parametric sensitivity of kV on critical superficial gas velocity.

94

J. B. JOSHI ET AL.

of hole size d0 on ␧GC and VGC is shown in Figs. 36 and 37, respectively. It can be seen that the values of ␧GC and VGC decrease with an increase in d0 . (ii) Effect of Bubble Rise Velocity. Liquid viscosity, surface tension, and liquid density govern the bubble rise velocity. It was thought desirable to study the effect of physical properties through the value of VB앝 . The relationship between physical properties and bubble size has been given by Kumar and Kuloor (1972) and the relationship between physical properties and the bubble rise velocity (for a given size) has been given by Clift et al. (1978). The effect of bubble rise velocity on VGC is shown in Fig. 38. It can be seen that the value of VGC increases with an increase in VB앝 . (iii) Effect of Gas Density. Effect of gas density was studied over a wide range of gas density. The turbulent contribution of sparger resistance, kT, depends proportionally on the gas density as given by Eq. (184c). An increase in the gas density therefore stabilizes the bed, as shown in Fig. 39.

FIG. 36. Effect of hole size of the sparger on critical gas hold-up.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

95

FIG. 37. Effect of hole size of the sparger on critical superficial gas velocity.

(iv) Effect of Hold-up Parameter. In a bubble column the bubble rises in the presence of many bubbles. Because of the presence of other bubbles, the actual bubble rise velocity is lower than the terminal value, and this hindrance effect increases with an increase in the gas hold-up. With the hindrance effect, the relationship between superficial gas velocity and fractional gas hold-up is given by Eq. (186). The hold-up parameter n depends upon the bubble Reynolds number. The effect of m on VGC and ␧GC is shown in Fig. 40 and 41 for hole diameters of 0.5 mm. It can be seen that at a constant R the value of VGC increases with a decrease in m. The asymptotic value of ␧GC given by Eq. (186) is shown in Fig. 41. (v) Effect of Dispersion. The dispersion coefficient has a stabilizing effect on the performance of bubble columns. This implies that the transition is delayed with the inclusion of dispersion. Figures 42 and 43 show the effect of dispersion on the superficial gas velocity and gas hold-up for a fixed column geometry over a wide range of dispersion coefficient. It can be seen that the effect of dispersion on VGC and ␧GC is nominal over a 100fold variation of the dispersion coefficient. Figure 44 shows the effect of relaxing most of the assumptions made by Shnip et al. (1992).

96

J. B. JOSHI ET AL.

FIG. 38. Effect of terminal bubble rise velocity on critical superficial gas velocity.

(vi) Effect of Column Diameter. Figure 45 shows the effect diameter for a fixed sparger design and dispersion height. As the plots, a decrease in the column diameter has a stabilizing the critical superficial gas velocity increases with a decrease column diameter.

of column seen from effect and in D, the

(vii) Effect of Dispersion Height. As noted earlier, once the aspect ratio h becomes equal to 1, the RHS of Eq. (183) becomes more or less a constant. Figure 46 shows the effect of dispersion height for a fixed diameter and sparger design. The curves are seen to become almost insensitive to h, above a value of h ⫽1–1.5. The liquid flow is either cocurrent or countercurrent to the gas flow. Liquid is introduced with the help of a distributor. Two limiting cases of the distributor resistance (kVL) are considered: kVL ⫽ 앝 and kVL ⫽ 0. For the first case, the stability criterion was given by Eq. (183). The results

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

97

FIG. 39. Effect of gas density on critical superficial gas velocity.

FIG. 40. Effect of m on critical superficial gas velocity for orifice diameter of the sparger ⫽ 0.5 mm.

98

J. B. JOSHI ET AL.

FIG. 41. Effect of m on critical gas hold-up for orifice diameter of the sparger ⫽ 0.5 mm.

are shown in Fig. 47 for cocurrent (uL ⫽ ⫹10 and ⫹20 mm/s) and countercurrent (⫺10 and ⫺20 mm/s) flows. It can be seen that the value of VGC increases with an increase in the cocurrent liquid velocity, whereas it decreases with an increase in the countercurrent liquid velocity. These predictions agree with the experimental observations of Oels et al. (1978). The effect of liquid velocity can be qualitatively explained as follows. In the heterogeneous regime, liquid circulation is developed that is upward in the central region and downward in the annular region. With cocurrent liquid upflow, the downward annular flow is restricted, resulting into a reduction in the liquid circulation. Therefore, the cocurrent liquid flow delays the transition. The countercurrent liquid flow has the opposite effect. When the liquid distributor resistance is zero (kVL ⫽ 0), the results are shown in Fig. 48. In this case also, the transition delays with an increase in the cocurrent vsup . However, the increase in VGC is less for the case of kVL ⫽ 0 as compared to the case of kVL ⫽ 앝. When kVL ⫽ 0, the disturbances at the bottom grow because of the absence of the possible dampening effect of the liquid distributor. For this case, the homogeneous regime is not possible when the liquid flow is countercurrent.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

99

FIG. 42. Effect of dispersion coefficient on critical superficial gas velocity.

D. COMPARISON WITH EXPERIMENTAL DATA 1. Fluidized Beds Table IX comparies model predictions with experimental values for solid–liquid fluidized beds. As mentioned earlier, in fluidized beds of very fine heavy particles, transition occurs because of radial nonuniformity. Further, almost all reported data on fluidized beds have mentioned that the sparger resistance was very large. Therefore, the comparison is made for the limiting case of the model, i.e., Eq. (183). Also, to bring out the limitation of the unbounded analysis, the same system data are compared. The comparison is favorable using the model of the present work. Table X shows the comparison for gas–solid fluidized beds. Again, fairly good agreement can be seen between the model predictions and the experimental observations. 2. Bubble Columns Yamashita and Inoue (1975), Maruyama et al. (1981), and Chisti (1989) have measured the values of critical superficial gas velocity for transition.

100

J. B. JOSHI ET AL.

FIG. 43. Effect of dispersion coefficient on critical gas hold-up.

The details pertaining to the experiments along with the comparison between model predictions and experimental observations is shown in Table XI. It can be seen that the agreement is favorable over a wide range of column widths, hole diameters, and numbers of holes. It has been pointed out earlier that the maximum possible hold-up in the homogeneous regime is 42%. This prediction compares favorably with the experimental observations of Maruyama et al. (1981) and Koetsier et al. (1976).

VII. Comparison of Bounded and Unbounded Analysis In Sections II and VI, the stability of multiphase systems was analyzed for unbounded and bounded beds, respectively. In the unbounded case, it was dispersion that was the most important parameter in deciding the stability of the system. An increase in the dispersion coefficient led to a

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

101

FIG. 44. Comparison with the predictions of Shnip et al. (1992).

more stable system. This has been depicted in Fig. 17, 18, 19, and 20 for gas–solid and solid–liquid fluidized beds. In the bounded case as well, the dispersion figures in the final criterion. However, the dependence of stability on the dispersion is very weak. Figure 42 shows, the dependence of critical superficial gas velocity, VGC , on the dispersion coefficient. The plot is almost insensitive to the value of the dispersion coefficient over a wide range. This apparent discrepancy which arises because of the fundamental difference between the bounded and unbounded analysis, is explained next. We define more clearly what is meant by an unbounded bed. In an unbounded bed, the bed dimensions are infinite, there are no lateral walls, and the presence of a sparger is not considered. Thus, unbounded analysis is valid in a situation when the sparger resistance is relatively small and the column dimensions are large. Once any perturbation or disturbance starts, dispersion is quick (or large) enough to nullify the gradients and make the bed stable. Thus, the dispersion always levels off all the gradients and is always a stabilizing mechanism. At this point, the question arises: If dispersion is always stabilizing, what is its upper limit? Unbounded bed analysis predicts that a sufficiently high value of dispersion coefficient can make the bed totally stable for a given particle diameter and physical properties. However, in reality, the destabilizing mechanisms associated

102

J. B. JOSHI ET AL.

FIG. 45. Effect of column diameter on critical superficial gas velocity.

with the sparger and/or lateral walls may become important. When this happens; the stabilizing effect of dispersion is no longer unlimited, the unbounded analysis breaks down, and we have to resort to the bounded bed analysis. Thus, the comparison of experimental transition hold-up using the unbounded bed criterion in Eq. (25), and the values of proportionality constant for dispersion equal to 3, shows that the unbounded analysis cannot predict the transitions where ␧L ⬎ 60%, i.e., the unbounded bed analysis predicts the fully particulate regime when the experimental transitions are occuring above the hold-up value of 60%. As the sparger resistance increases, the transition occurs at higher values of hold-up. However, the increase in sparger resistance also reduces the applicability of unbounded analysis. It must be emphasized that the applicability of unbounded analysis is limited to low values of sparger resistance. In the case of high sparger resistance, bounded bed analysis should be used to get realistic predictions. From this discussion, we propose the following procedure to predict transitions in two-phase systems in a quantitative manner: 1. Check the stability using the unbounded bed criterion, neglecting the sparger and column dimensions. If the bed is unstable, then the bed will be unstable throughout, no matter how well the sparger is de-

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

103

FIG. 46. Effect of dispersion height on superficial gas velocity.

signed or how the column dimensions are altered. However, if the unbounded criterion predicts a stable bed, it just means that the bed is stable against the axial disturbances. It may or may not be stable against the radial disturbances due to sparger or wall. Therefore: 2. Check the stability using the bounded bed criterion. If the bounded criterion also predicts a stable bed, the bed is fully stable and will exhibit all the characteristics of the homogeneous regime. However, if the bounded criterion predicts an unstable bed, then the bed will exhibit all the characteristics of the heterogeneous regime even though the unbounded criterion predicted a stable bed. Stability can, however, be achieved in this case by increasing the sparger resistance or by reducing the column dimensions. VIII. Three-Phase Fluidization A. INTRODUCTION Three-phase fluidization refers to operation in which an upward flow of gas and cocurrent or countercurrent flow of liquid supports a bed of

104

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FIG. 47. Effect of liquid superficial velocity for cocurrent and countercurrent operation [kVL ⫽ 앝 ].

solid particles. The gas flows as bubbles. The three-phase fluidization is characterized by the unique phenomenon that an increase in the gas velocity may cause an increase in the height (expansion) or a decrease in the bed height (contraction), whereas an increase in the liquid velocity always causes an increase in the bed height. This phenomenon was first observed by Turner (1964) and subsequently by Stewart and Davidson (1964), Ostergaard (1965), Ostergaard and Thiesen (1966), Adlington and Thompson (1965), and Epstein and co-workers. All these investigators agree that (a) the bed expansion occurs with an increase in the liquid velocity, and (b) an increase in the gas velocity can cause either contraction or expansion. Bed expansion and contraction are important phenomena in three-phase fluidization, since these affect the bed volume and the residence time(s) of all the phase(s). The phenomenon of bed contraction/expansion is also very closely linked to the problem of transition. The relationship between the expansion/contraction behavior and the transition will be clearly brought out later.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

105

FIG. 48. Effect of liquid superficial velocity for cocurrent and countercurrent operation [kVL ⫽ 0].

B. HEURISTIC MODELS Stewart and Davidson (1964) and Ostergaard (1965) proposed similar mechanisms to explain the contraction phenomenon. They offered qualitative and semiquantitative explanations based on the assumption that gas bubbles in the bed are followed by wakes. The wakes travel at velocities equal to the bubble velocities and thus considerably higher than the average superficial liquid velocity in the bed. Therefore, it follows from the continuity equation that the velocity in the bed outside the bubble wakes is lower than the average superficial velocity, and thus the expansion of this part of the bed must be correspondingly reduced. Epstein and co-workers (Bhatia and Epstein, 1974; Epstein, 1976; Epstein and Nicks, 1976; El-Temtamy and Epstein, 1978, 1979) have made very valuable contributions to the understanding of expansion/contraction characteristics of three-phase fluidized beds. Bhatia and Epstein (1974) proposed a wake model in which the solid contents of the liquid wakes behind the gas bubbles is negligible compared to the solid contents of the remaining

b

a

655 655 275 275 165 165 82.5 82.5

Unbounded bed analysis. Bounded bed analysis.

Gibilaro et al. (1986)

Investigator

dP (mm) 8710 8710 8710 8710 8710 8710 8710 8710

␳S (kg/m3) 1000 1000 1000 1000 1000 1000 1000 1000

␳L (kg/m3) 1.25 0.60 1.05 0.60 1.25 0.75 1.25 0.75

애L ⫻ 103 Pas

0.48 0.45 0.60 0.51 0.74 0.66 0.75 0.72

␧LC experimental (%)

0.55 0.49 0.67 0.59 0.81 0.73 0.91 0.86

␧LC predicteda (%)

0.478 0.452 0.595 0.520 0.726 0.676 0.742 0.7345

␧LC predictedb (%)

TABLE IX COMPARISON OF EXPERIMENTAL DATA WITH BOUNDED BED ANALYSIS PREDICTIONS FOR SOLID –LIQUID FLUIDIZED BEDS

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

107

TABLE X COMPARISON OF EXPERIMENTAL DATA WITH PREDICTIONS OF BOUNDED-BED ANALYSIS: GAS –SOLID FLUIDIZED BEDS

Investigator Jacob and Weimer (1987)

Musters and Rietema (1977)

a b

dP (mm)

␳S (kg/m3)

␧GC experimental (%)

␧GC predicteda (%)

␧GC predictedb (%)

0.044 0.044 0.044 0.112 0.112 0.112 0.0397 0.078 0.103

850 850 850 850 850 850 920 920 920

76.5 76.7 79.8 79.8 79.8 79.8 77.6 74.2 70.1

79.0 80.0 81.0 64.0 62.0 60.0 76.0 68.0 65.0

75.2 76.1 78.2 77.1 78.3 79.3 78.2 76.5 76.5

Unbounded bed analysis. Bounded bed analysis.

liquid. More direct experimental evidence for wakes that are nearly solidsfree was cited in the published discussion of this paper. The equations governing the solids-free wake model, assuming steady state, are as follows: ␧L ⫹ ␧G ⫹ ␧S ⫽ 1.

(188)

Since the liquid divides itself between the solids-free wake and the particulate region of the liquid-fluidized solids, ␧L ⫽ ␧k ⫹ (1 ⫺ ␧G ⫺ ␧k)␧lf .

(189)

TABLE XI COMPARISON OF EXPERIMENTAL DATA WITH PREDICTIONS OF BOUNDED-BED ANALYSIS: TWO-DIMENSIONAL GAS –LIQUID BUBBLE COLUMNS

Investigator Chisti (1989) Maruyama et al. (1981) Yamashita and Inoue (1975)

Column Column length width (mm) (mm)

Orifice diameter (mm)

Area ratio

Sparger thickness (mm)

VGC experimental (mm/s)

VGC predicted (mm/s)

460 300

155 10

1.0 0.2

50 29

2.0 20.0

63.41 39.13

42.45 46.05

300 300

10 10

0.3 0.5

29 29

0.7 0.7

50.00 44.10

46.08 43.40

Note: Air–water system was used by all investigators.

108

J. B. JOSHI ET AL.

The ratio of wake volume to gas bubble volume is given by ␧k ⫽ kA(␧G , ␧S). ␧G

(190)

vsup ⫽ v␧G ,

(191)

usup ⫽ ␧kv ⫹ (1 ⫺ ␧G ⫺ ␧k)␧lfvlf .

(192)

A gas balance yields

and a liquid balance gives

The velocity of the gas bubbles relative to the velocity of liquid in the particulate region is given by vs ⫽ v ⫺ vlf .

(193)

The interstitial velocity of the liquid in particulate region can be represented by a Richardson–Zaki type equation, assuming uniform-sized solids: vlf␧lf ⫽ aVS앝␧lfm .

(194)

Combining Eqs. (188)–(193), we get the bed voidage, ⑀, ␧ ⫽ 1 ⫺ ␧S ⫽

usup ⫺ vsupkA vsup(1 ⫹ kA) ⫹ . vlf vlf ⫹ vs

(195)

Combining equations (190)–(194), we get

冉 冊

aVS앝 (usup ⫺ vsupkA) vlf

1 m⫺1

⫽ vlf ⫺

VGvlf(kA ⫹ 1) vlf ⫹ vs

(196)

Equation (196) can be arranged to give vsup explicitly: vsup ⫽ vsup(vlf , vs , kA , aVS앝 , m, usup).

(197)

Substitution of Eq. (197) in Eq. (195) yields ␧ ⫽ ␧(vlf , vs , kA , aVS앝 , m, usup).

(198)

It was assumed that the wake volume to bubble volume ratio, kA , is a known constant and vs can be determined if the bubble size is specified. It was further assumed that vsup , a, m, and vs앝 are fixed. (This assumption is tantamount to the assumption of starting with liquid-fluidized bed at t ⫽ 0, or with a bed of specified properties and degree of expansion.) With these assumptions, Eqs. (197) and (198) were simplified to vsup ⫽ f (vlf)

(199)

109

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

and ␧ ⫽ h(vlf).

(200)

Bhatia and Epstein (1974) proposed that bed contraction/expansion behavior can be predicted by evaluating ⭸⑀/⭸vsup at vsup ⫽ 0. This derivative was obtained by ⭸␧ ⭸vsup

冏

vsup⫽0

⫽

⭸␧/⭸vlf ⭸vsup /⭸vlf

冏

vsup⫽0

,

(201)

which gave ⭸␧ ⭸vsup

冏

vsup⫽0

⫽

(m/(m ⫺ 1) ⫹ kA)u ⫺ (1 ⫹ kA)usup ⫺ (kAvs /(m ⫺ 1)) , (202) (m/(m ⫺ 1))u(u ⫺ vs)

where, for zero gas flow (i.e., ␧lf ⫽ ␧L and vlf ⫽ usup), Eq. (194) gives u⫽

usup (m⫺1)/m ⫽ (aVs앝)1/mu sup . ␧L

(203)

The magnitude and the sign of the bed expansion on first introducing gas can be predicted by using LHS of Eq. (202), with the negative sign indicating contraction. If it is desired to predict qualitatively whether the expansion or contraction will occur, only the numerator in the RHS of Eq. (202), ␾A, needs to be considered, as the denominator is always positive: k v m 1 ␾A ⫽ ⫺ (1 ⫹ kA) usup ⫺ A s . ⫹ kA (204) m⫺1 ␧L m⫺1

冋冉

冊

册

The criterion of Eq. (202) is a simplified fluid dynamic description of three-phase fluidized beds capable of application over a wide range of conditions. They have also compared the predictions of Eq. (202) with their experimental data, showing a good agreement. In Eq. (202) a value of kA ⫽ 1 was assumed as a first approximation. However, a better estimate of kA for the case of negligible gas hold-up was obtained from kA ⫽ 3.5(1 ⫺ ␧s)3.

(205)

For the case of kA ⫽ 0, the numerator of Eq. (204) becomes

␾A ⫽

m 1 ⫺ 1, m ⫺ 1 ␧L

(206)

which is always positive for m ⬎ 1. In other words, Eq. (202) correctly predicts that, in the absence of bubble wakes, bed expansion will always occur when gas is introduced to a liquid fluidized bed.

110

J. B. JOSHI ET AL.

1. Effect of Various Parameters. It is instructive to analyze the qualitative criterion in terms of the numerator in Eq. (202), ␾A , to understand the effect of various parameters. Rearranging Eq. (204), we get,

冋

␾A ⫽ kA usup

册

冋

册

vs 1 ⫺ ␧L m ⫺ ⫹ usup ⫺1 . ␧L m⫺1 (m ⫺ 1)␧L

(207)

1. Equation (207) reveals that a high ratio of wake volume to bubble volume favors bed contraction. 2. Increase in the bubble velocity vs favors bed contraction. At vsup ⫽ 0 ⫽ ␧G , vs represents the rise velocity of bubble in a stagnant liquid. Since the rise velocity of a single bubble increases with gas–liquid interfacial tension, it follows that any decrease in surface tension favors expansion. This effect of surface tension has been observed by, among others, Dakshinamurty et al. (1971) and Kim et al. (1975). 3. Increasing the liquid viscosity favors contraction. This follows from the fact that in Eq. (204), the coefficient of usup is always positive, and since the value of usup required to produce a fixed value of ⑀L decreases with an increase in liquid viscosity, the value of ␾A decreases as the viscosity is increased. The contraction favoring behaviour of liquid viscosity has been observed by Bhatia and Epstein (1974) and Kim et al. (1975). 4. Increasing the solid particle size or density favors bed expansion. This effect was observed by Bhatia and Epstein (1974), Kim et al. (1975), and Nicklin (1962). Darton and Harrison (1975) derived a criterion for the point of transition to predict whether a solid–liquid fluidized bed will expand or contract when the gas is first introduced. The definition of kA used by Darton and Harrison was the ratio of upper clear (particle-free) wake volume to the bubble volume. But since they did not consider the circulation of solids associated with the lower nonclear portion of the wake, their kA was effectively the same as that of Bhatia and Epstein (1974). The use of the Wallis drift flux approach by Darton and Harrison (1975) also represents no real difference from the relative velocity approach taken by Bhatia and Epstein (1974), since the two methods are rigorously interrelated. It is therefore not surprising that the final criteria of Bhatia and Epstein (1974) and Darton and Harrison (1975) are identical. El-Temtamy and Epstein (1979) later found that the assumption of solids-free liquid wake behind the bubbles is tenable only for relatively coarse and/or heavy particles and becomes increasingly untenable for solids smaller than 1–2 mm, especially if the particle densities do not exceed 3000

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

111

kg/m3 . They rederived the criterion without the assumption of a solidsfree wake. The rederived expression for ␾A together with the correction suggested by Jean and Fan (1987) is given as

␾A ⫽

冋

册 冋

k v m ⫹ kA u ⫺ (1 ⫹ kA)usup ⫹ A s m⫺1 m⫺1

⫹

␹kA(u ⫹ vs) m␹(usup ⫺ u)kA ⫹ m⫺1 m⫺1

册

(208)

where, ␹ denotes the ratio of solid hold-up in bubble wakes to the solid hold-up in the liquid fluidized region.

C. NEW CRITERION FOR THE PREDICTION OF CONTRACTION /EXPANSION It was thought desirable to develop an alternative model for the prediction of contraction/expansion. The proposed model is based on the behavior of the particle settling velocity in solid–liquid and three-phase fluidization. In the case of solid–liquid fluidized beds, the settling velocity of the particle is less than the terminal settling velocity. The hindered settling velocity is given by the well-known Richardson–Zaki equation, usup ⫽ VS앝␧Lm ,

(209)

where m is a parameter that depends on the particle Reynolds number. When the gas is introduced in the solid–liquid fluidized bed, the situation becomes more complex. The settling velocity in the presence of gas can either be higher or lower than the terminal settling velocity. Imafuku et al. (1968) estimated the effective settling velocity of a particle in the presence of gas. They found that the settling velocity of the particles in the presence of gas was always larger than the hindered settling velocity. They attributed the increase in settling velocity in the presence of gas to the formation of aggregates. They proposed the correlation (for 1.3 ⬍ VS앝 ⬍ 27.2mm/s) VGLS ⫽ 1.45V S⫺앝0.35 , VSL

(210)

where VSL represents the settling velocity in the absence of a gas phase and VGLS represents the settling velocity in the presence of gas. Kato et al. (1972) have reported the following correlation: 0.25 2.5 VGLS ⫽ 1.33V 0.75 S앝 V G ␧L .

(211)

Smith and Reuther (1985) also have given a correlation with a form similar to the correlation of Kato et al. (1972). However, the values of

112

J. B. JOSHI ET AL.

exponents and constants are different. A careful analysis of the settling data reported by Kato et al. (1972) and Smith and Reuther (1985) indicates that VGLS may be higher or lower than the terminal settling velocity. However, it was always higher than that in the absence of gas (only solid–liquid fluidized bed). We will try to use this information for prediction of contraction/expansion behavior. Consider a solid–liquid fluidized bed in which the gas phase is introduced. From a stability viewpoint, there are three distinct possibilities regarding the stability of the bed before and after introducing the gas phase. 1. Homogeneous–heterogeneous. If the solid–liquid fluidized bed becomes heterogeneous after the introduction of gas, the settling velocity of the solids increases, and the bed tends to contract. The contracting tendency due to an increase in the settling velocity of solids is higher than the expanding tendency due to the additional gas hold-up, and contraction is observed. We have tested data of Epstein (1976) on three-phase fluidization, and the results are given in Table XII. In all the cases when the bed contraction was experimentally observed, the stability criterion for the solid–liquid fluidized beds given by Eq. (224) predicts that the initial solid– liquid bed was indeed in the homogeneous regime. 2. Heterogeneous–heterogeneous. On the other hand, if the bed is heterogeneous before as well as after the introduction of the gas phase, the settling velocity of solids does not change much and the additional hold-up of the gas phase causes bed expansion. This inference was also tested using the data of Epstein (1976). The stability criterion for the solid–liquid fluidized bed given by Eq. (20) does predict the initial heterogeneous regime, when the bed expansion was observed experimentally as shown in Table XII.

TABLE XII PREDICTION OF CONTRACTION /EXPANSION BEHAVIOR IN THREE-PHASE FLUIDIZED BEDS

No.

System

1 2

Water–glass beads Aqueous glycerol– glass beads Water–sieved sand Water–glass beads Water–glass beads Water–glass beads Water–glass beads Water–lead shots Aqueous PEG–glass balls Water–glass beads

3 4 5 6 7 8 9 10

␳S (kg/m3)

애L ⫻ 105 (Pas)

Observed behavior

Predicted behavior

1.08 1.08

2824 2824

80 210

Contraction Contraction

Contraction Contraction

0.458 0.273 0.456 1.08 1.08 2.18 3.18

2578 2938 2935 2824 2949 11030 7756

80 80 80 80 80 80 6300

Contraction Contraction Contraction Contraction Contraction Expansion Contraction

Contraction Contraction Contraction Contraction Contraction Expansion Contraction

1.08

2824

6300

Expansion

Expansion

dP (mm)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

113

3. Homogeneous–homogeneous. If the solid–liquid bed remains homogeneous after introduction of the gas phase, the settling velocity of the solids does not change much. However, the additional gas phase hold-up should cause the expansion of the bed. This case can be observed if the experiments are performed with sufficiently high sparger resistance that the sparger-related disturbances themselves do not induce heterogeneity. In order to check the validity of this case, experimental data are needed. Joshi (1983) used a simpler form of this model for the prediction of contraction/expansion in three-phase fluidized beds.

IX. Conclusions 1. Jackson and co-workers, Wallis, Homsy, and co-workers, Gibilaro, Foscolo and co-workers, Rietema and co-workers, and Batchelor have made pioneering contributions to the analysis of transition in fluidized beds. 2. A unified approach has been developed for the prediction of transition in multiphase reactors such as gas–liquid bubble columns, liquid– liquid spray columns, solid–liquid fluidized beds, gas–solid fluidized beds, and three-phase fluidized beds. 3. The role of sparger (distributor) design on the transition has been brought out. When the sparger resistance is small, the bed can be considered as unbounded. The bed is considered bounded at high sparger resistance. For both these cases, transition criteria have been developed using the theory of linear stability. Clear physical significance has been attributed to all the terms in the transition criteria. A comparison between the predictions for unbounded and bounded beds has been presented. 4. All the published literature has been critically reviewed. The reported criteria have been classified into (a) fundamental and (b) heuristic approaches. An attempt has been made to establish a relationship between the fundamental approach and the heuristic approach. It has been shown that the criterion based on the heuristic approach can be considered as a special case of the generalized criterion based on the fundamental approach. 5. Parametric sensitivity of the criteria has been presented in the form of stability maps. The particle phase dispersion coefficient has been shown to be the most important parameter governing the stability. It has also been shown that the stability maps can be conveniently and advantageously drawn in terms of particle Reynolds number. 6. All the published data on transition for solid–liquid fluidized beds, gas–solid fluidized beds, three-phase fluidized beds, and gas–liquid

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bubble columns have been compared with the predictions of the criteria. Reasonably good agreement has been shown in all the cases. A stepwise procedure has been presented for the use of criteria based on unbounded and bounded beds. 7. The phenomenon of expansion/contraction in three-phase fluidized beds was analyzed. A brief review of literature has been presented. Epstein and co-workers have made outstanding contributions in this area. An alternative method has been presented. A favorable comparison between the model predictions and the experimental observations was observed.

X. Suggestions for Future Work 1. The most important parameter governing stability is the dispersion coefficient of the dispersed phase such as bubbles, drops, and particles. The published information is not sufficient. A comprehensive research program is needed for the measurement of dispersion in all multiphase reactors over a wide range of terminal velocities, column diameters, column heights, sparger designs, phase velocities, and continuousphase physical properties. 2. It will be useful to develop an understanding of the relationship between turbulent flow field and dispersion. Measurement techniques need to be developed for the measurement of turbulent flows in multiphase systems. The relationship between eddy diffusion and the dispersion coefficient needs to be brought out over a wide range of particle sizes, settling velocities, column diameters, column heights, phase velocities, and physical properties. 3. The debate pertaining to the definition of buoyancy force is given in Appendix B. A systematic experimental program is still needed in this area. 4. Homsy and co-workers and Gibilaro, Foscolo, and co-workers have proposed neat methods for the experimental measurement of transition. The techniques of measurement of dynamic pressure and dynamic measurement of bed heights will prove to be extremely useful in understanding the hydrodynamics of multiphase systems. The measurements will also be useful for understanding other transport phenomena. 5. Additional experimental information is needed from liquid–liquid spray columns, liquid–gas spray columns, solid–gas transport reactors, and three-phase fluidized beds.

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APPENDIX A: MODELING OF CORRELATION OF FLUCTUATING PRESSURE AND HOLD-UP The equation of motion for a single phase is written as 1 ⭸p 애 2 ⭸ui ⭸u ⫹ uj i ⫽ ⫺ ⫹ ⵜ u ⫹ gh. ⭸t ⭸xj ␳ ⭸xi ␳

(212)

To derive the pressure interaction correlation, we neglect viscous effects and write the time-averaged momentum equation as ⭸u⬘i u⬘j 1 ⭸p¯ ⭸u¯i ⭸u¯ ⫹ gh ⫺ . ⫹ u¯j i ⫽ ⫺ ⭸t ⭸xj ␳ ⭸xi ⭸xj

(213)

Now subtracting Eq. (212) from Eq. (213) we get the equation of motion in terms of fluctuating components: u⬘j

⭸u⬘i ⭸ 1 ⭸p⬘ ⭸u¯i ⫹ u¯j ⫺ (u⬘i u⬘j ⫺ u⬘i u⬘j ) ⫽ ⫺ . ⭸xj ⭸xj ⭸xj ␳ ⭸xi

(214)

We can get the correlation of fluctuating pressure and hold-up by taking the product of Eq. (214) with ␧⬘ and then time averaging the equation so obtained: ␧⬘u⬘j

⭸␧⬘u⬘i ⭸␧⬘u⬘i u⬘j 1 ⭸␧⬘p⬘ ⭸u¯i ⫹ u¯j ⫺ ⫽⫺ . ⭸xj ⭸xj ⭸xj ␳ ⭸xi

(215)

Therefore, the pressure correlation term can be obtained by neglecting the triple product of the fluctuations:

冋

册

⭸␧⬘u⬘i ⭸u ⭸␧⬘p⬘ ⫽ ⫺␳ ␧⬘u⬘j i ⫹ u¯j . ⭸xi ⭸xj ⭸xj

(216)

Using this equation, we can write

冋

册

⭸␧⬘Sp⬘ ⭸␧⬘Sv⬘z ⭸␧⬘Sv⬘z ⭸v¯ ⭸v¯ ⫽ ⫺␳ ␧⬘Sv⬘z z ⫹ ␧⬘Sv⬘x z ⫹ v¯z ⫹ v¯x . ⭸z ⭸z ⭸x ⭸z ⭸x

(217)

APPENDIX B: FORCES ACTING ON A PARTICLE IN A FLUIDIZED BED For a single particle in an infinite fluid medium, the buoyancy force was defined by Archimedes and is given by FB ⫽ vP␳g,

(218)

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where vp is the particle volume and ␳ is the fluid density. This definition of buoyancy force for a single particle is well established. However, for multiparticle systems (such as fluidized beds), there has been a vigorous debate regarding the formulation of buoyancy force. Epstein (1984), Joshi (1983, 1984), Clift et al. (1987), Fan et al. (1987), Grbavcic et al. (1992), and Clift (1993) have defined the buoyancy force similarly to Eq. (218). It is given by FB ⫽ vP␳g.

(219)

In contrast, Richardson and Meikle (1961), Barnea and Mizrahi (1973), Rietema (1982), Rowe (1984), Foscolo et al. (1984), and Gibilaro et al. (1987a, 1987b) have defined the buoyancy force on the basis of average suspension density (␳¯ ) and FB ⫽ vP␳g,

(220)

␳ ⫽ ␧␳ ⫹ (1 ⫺ ␧)␳S .

(221)

where

␳¯ is the average density of the suspension and ␧ is the volume fraction of the continuous phase fluid in the bed. Though there has been a difference of opinion regarding the definition of buoyancy force, there is no disagreement on the definition of gravitational force on a bed particle in a fluidized bed, which is given as vp␳sg, where ␳s is the particle density. However, when this force is equated with the sum of the buoyancy force and fluid dynamic drag force, the buoyancy question affects the magnitude of the drag portion of the gravitational force, since vP␳Sg ⫽ vP␳g ⫹ (␳S ⫺ ␳)vPg,

(222)

vP␳Sg ⫽ vP␳g ⫹ (␳S ⫺ ␳)vPg.

(223)

or For an isolated particle i.e., ␧ 씮 1, Eq. (223) reduces to Eq. (222). The buoyancy term represents the net force arising from the pressure distribution over the surface of the particle with the fluid at rest. The drag term represents the additional force exerted on the particle by a flowing fluid. Drag gives rise to mechanical energy dissipation. ENERGY DISSIPATION Consider the case of a particle settling in an infinite liquid medium with a terminal velocity vS앝 . The rate of decrease of potential energy is given by et ⫽ (␳S ⫺ ␳L)vP g

dh dt

(224)

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

⫽ (␳S ⫺ ␳L)vPgVS앝 .

117 (225)

The rate of energy dissipation at the solid–liquid interface is also given by this equation. As the particle settles, the equivalent volume of the liquid rises so that the potential energy of the particle at any height h is equal to (␳s ⫺␳L)vp gh. Since the medium is infinite, the rising superficial liquid velocity is negligible as compared with vS앝 . Now consider the case of particulate fluidized bed. Though the particles in the bed are moving randomly, there is no net displacement and the particles can be considered stationary with respect to the column wall. The superficial velocity is usup and the interstitial velocity is vs ⫽ usup /␧.

(226)

It is important to know whether the energy dissipation at the solid–liquid interface is FDusup or FDvs . Let us write the energy balance for the two situations of a falling bed and a stationary bed, respectively. These are shown in Fig. 49A and 49B, respectively. In the first case, the liquid phase is stationary, the support plate is absent, and the bed falls. The second case is the usual solid–liquid fluidized bed where the particle phase is stationary and the liquid moves upward and supports the solid phase. In the case of stationary fluid (Fig. 49A), let the bed level be at an elevation h from a certain reference level. At time t, let the falling bed reach the reference level. The change in potential energy of the bed is EP ⫽ VP␳S gh,

(227)

where Vp is the total volume of solid phase. As the bed falls, the equivalent volume of the liquid rises and the gain in potential energy of liquid is EL ⫽ VP␳L gh.

(228)

The net energy dissipated during the fall of the bed is EP ⫺ EL ⫽ VP(␳S ⫺ ␳L) gh,

(229)

and the energy dissipation rate is E D ⫽ V P (␳ S ⫺ ␳ L ) g

dh . dt

(230)

The bed falls at the rate dh/dt. Now, consider the usual case of Fig. 49B. The energy input rate to the fluidized bed is given by 앟 Ei ⫽ D2usupH (␧S␳S ⫹ ␧L␳L) g, 4

(231)

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(c) FIG. 49. (a) Falling bed (liquid phase is stationary). (b) Stationary bed (solid–liquid fluidized bed). (c) Particle in a flowing fluid.

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

119

where D and H are the diameter and height of the fluidized bed, respectively. At height H, the liquid phase gains potential energy at a rate given by 앟 EL ⫽ D2usup H␳Lg. 4

(232)

The net energy dissipation rate is 앟 ED ⫽ D2usupH␧S(␳S ⫺ ␳L) g 4

(233)

⫽ VPusup(␳S ⫺ ␳L) g.

If the two situations of stationary fluid and rising fluid are to be equivalent, dh/dt must equal the superficial liquid velocity. Therefore, if FD is the drag force on the solid phase, the energy dissipation rate is given by ED ⫽ FDusup .

(234)

Substitution of Eqs. (234) in (233) gives FD ⫽ VP(␳S ⫺ ␳L) g.

(235)

e ⫽ vP(␳S ⫺ ␳L) gusup

(236)

For a single particle, ⫽ Fdusup .

(237)

From Eqs. (236) and (237), Fd ⫽ vP(␳S ⫺ ␳L) g.

(238)

Equations (236) and (238) represent the energy and the force balance for a single particle. The following additional discussion may be useful. Consider the case of Fig. 49A. The bed descends with a velocity dh/dt equal to usup . Because of the fall of the bed, the equivalent amount of liquid rises and the generated liquid velocity on the superficial basis is given by 앟 U * ⫽ D2(1 ⫺ ␧L)usup 4

冫앟4 D ␧ 2

L

(239)

1 ⫺ ␧L ⫽ usup . ␧L The relative velocity between the particle and fluid is usup ⫹ U* ⫽ usup ⫹ ⫽

usup . ␧L

1 ⫺ ␧L usup ␧L (240)

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It may be pointed out that the rising fluid gains potential energy at a rate given by 앟 2 D U*gh␳L , 4

(241)

where h is the position of the bed at any time. Thus, the velocity U* is responsible for increasing the potential energy of liquid phase and is not available for doing work on the solid particles, and the energy dissipation rate is governed by usup and not vs . However, Gibilaro et al. (1987) defined the energy balance on the basis of vs : ED ⫽ VP(␳S ⫺ ␳L) gvs .

(242)

Therefore, FD ⫽ VP[(␳S ⫺ (␧S␳S ⫹ ␧L␳L)] g ⫽ VP(␳S ⫺ ␳) g.

(243)

From the foregoing discussion, it can be said that the buoyancy force is decided by the fluid density and not the suspension density. In order to understand the formulation of buoyancy force and hence the drag force, Grbavcic et al. (1992) did ingenious experiments. They started with the following formulation:

␳s ⫽ ␳* ⫹

3 ␳LCD(␧L) 2 vs . 4 dS g

(244)

Grbavcic et al. (1992) used equation (244) for finding the value of ␳*. The experimental plan consisted of the measurement of slip velocity (vs) over a wide range of ␳s . A plot of ␳s vs v2s gave the value of ␳*. For this purpose, ␧L , CD(␧L), and dS were held constant. In a typical experiment (say, particle size 1.2 mm), the desired value of ␧L was obtained by adjusting the superficial liquid velocity. In such a bed, the slip velocities of 10- and 19.5- mm particles were measured over a wide range of solid phase density (324–8320 kg/m3), and a plot of ␳s vs. v2s was constructed. Several such plots were obtained by varying ␧L . They found that ␳* equals ␳L when the diameter of sedimenting (or rising) particle was less than 3.36 times the particle diameter in a fluidized bed. Above this particle size, ␳* was found to be equal to ␳¯ . Therefore, it may be concluded that the buoyancy force for any particle in the bed (the sizes of all the particles are within a range) is decided by the fluid density and not the suspension density. PRESSURE GRADIENTS Gibilaro et al. (1987a, 1987b) proposed that the buoyancy force can be obtained by integrating the pressure gradients, irrespective of the source

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

121

and direction of pressure gradient. Thus, FB ⫽ v P and

冉

⭸p ⭸z

Fd ⫽ vP ␳S g ⫺

(245)

冊

⭸p . ⭸z

(246)

Joshi (1987) and Clift (1993) have addressed this point in detail. Clift (1993) has provided the following systematic derivation. Consider a particle of arbitrary shape that is stationary in a fluid approaching with velocity usup , where the fluid velocity need not be in the vertical direction (Fig. 49C). At any point on the surface of the particle, the fluid exerts a normal stress ␴ and shear stress ␶s . The force obtained by integrating ␶ over the surface of the particle is the skin friction, and the component of this force parallel to usup is the skin friction drag. There is no argument over the formulation of ␶ and hence skin friction. The buoyancy argument centers on ␴. Integrating ␴ over the surface gives another force including both buoyancy and form drag, and the argument concerns the division between these two. Taking the simple case of an incompressible Newtonian fluid, the fluid motion is described by the Navier–Stokes equation:

␳

Du ⫽ ␳g ⫺ ⵜp ⫹ 애ⵜ2u. Dt

(247)

Equation (247) has a directionality introduced by the body force. To remove this directionality, it is common to introduce the modified or reduced pressure, p, where p ⫽ po ⫹ ␳gx ⫹ Pr

(248)

where x is the position vector and po a reference pressure. The difference between Pr and p is the hydrostatic pressure. Integrating the hydrostatic component of ␴ over the surface of the particle always gives the simple Archimedean buoyancy, which always acts in the vertical direction. In terms of the reduced pressure, Eq. (248) becomes

␳

Du ⫽ ⫺ⵜPr ⫹ 애ⵜ2u. Dt

(249)

Integrating the component of ␴ arising from the modified pressure, p, gives a force, additional to skin friction, arising from the fluid motion. The component of this force parallel to u is called the form drag. From the foregoing discussion, it can be seen that, for a fluidized bed, some part of ⭸p/⭸z or ␳¯ g contributes to form drag. When the form drag is subtracted from ␳¯ g, we get ␳Lg and the buoyancy force is given by vp␳Lg.

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Nomenclature a A A1 AS B B0 B⬘0 C C1 C2 C3 C4 C5 Chole CD 앝 CD C ⬘D CV d dB dˆB d0 dP dPL dPU D ˆ D De e et E E1 E* ED Ei EL Eo EP Es f1

Wall effect factor (194) Constant defined by (22a) Parameter defined by (131) Area, m2 Parameter of the stability criterion Dimensionless interaction parameter at steady state [ Eq. (110)] Parameter defined in (111) Parameter of the stability criterion Constant defined in Eq. (141) Constant defined in Eq. (141) Constant defined in Eq. (144) Constant defined in Eq. (144) Parameter defined by (132) Orifice coefficient Drag coefficient under terminal rise conditions Drag coefficient in presence of other bubbles Dimensional drag coefficient, kg/m 3s or N · s/m 4 Virtual mass coefficient Width of two dimensional column, m Dimensionless bubble diameter defined by (70) Bubble diameter, m Orifice diameter Particle diameter, m Lower limit of critical particle diameter, m Upper limit of critical particle diameter, m Coefficient of dispersion, m2 /s Dimensionless coefficient of dispersion Difference in dimensionless dispersion coefficients of the gas and liquid phases Energy dissipation rate for a single particle Rate of decrease of potential energy of a single particle, J/s Parameter of the stability criterion Parameter defined by (127) Mean elasticity modulus of the bed Energy dissipation rate, J/s Energy input rate, J/s Potential energy for the liquid, J Energy output rate, J/s Potential energy for a swarm of particles, J Elasticity modulus of the bed material Interaction force per unit volume term defined by (10), N/m3

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

fx fz fzd fzv F F1 FB Fd Fd앝 FD Fr g gˆ G Ga h H I J k kA kT kTL kV kVL K0 KT KV KTL KVL l L m mP M1 M2 M3 M4 N1 N2 N3 Nm Ns

123

x component of interaction force per unit volume, N/m3 function defined by equation (25) which determines stability z component of interaction force per unit volume, N/m3 Drag part of interaction force per unit volume, N/m3 Virtual mass part of interaction force per unit volume, N/m3 Parameter of the stability criterion Parameter defined by equation (129) Buoyancy force, N/m2 Drag force on a particle in fluidized bed, N/m2 Drag force on a particle in an infinite medium, N/m2 Drag force on the solid phase (all the particles put together) Froude number Acceleration due to gravity, m/s2 Dimensionless acceleration due to gravity [equation (100)] Parameter of the stability criterion Galileo number defined in Eq. (82) Height of the column, m Dimensionless height of bubble column Parameter of the stability criterion The drift velocity, m/s Separation constant in (166) Ratio of wake volume to bubble volume Constant in pressure drop–velocity relationship, Eq. (101) Constant in pressure drop–velocity relationship, Eq. (102) Constant in pressure drop–velocity relationship, Eq. (101) Constant in pressure drop–velocity relationship, Eq. (102) Proportionality constant in Eq. (31) Dimensionless constant defined in Eq. (103) Dimensionless constant defined in Eq. (103) Dimensionless constant defined in Eq. (103) Dimensionless constant defined in Eq. (103) Length scale of turbulence, m Width of two-dimensional bubble column, m Richardson–Zaki index Poisson ratio of the bed material Parameter defined in (152) Parameter defined in (156) Parameter defined in (153) Parameter defined in (154) Parameter defined in Table VIII Parameter defined in Table VIII Parameter defined in Table VIII LHS of the stability criterion Number of particles

124 p p¯ p⬘ p⬘s Patm P P0 Pr Q r0 rc R Re ReB Rep s S t ˆt u u0 u1 ue u␧ usup ux u¯x u⬘x uz u¯z u⬘z U ¯ U UD Ux Uz v

J. B. JOSHI ET AL.

Instantaneous pressure, N/m2 Time-averaged pressure, N/m2 Fluctuating pressure, N/m2 Particle phase pressure in (26), N/m2 Atmospheric pressure, N/m2 Dimensionless pressure Dimensionless pressure at steady state Reduced pressure, Pa Term defined in (134) Characteristic dimension of bubble or particle, m Radius of curvature, m Ratio of the orifice area to the column area Reynolds number Reynolds number of a bubble Particle Reynolds number Growth rate constant in time, s Term defined in (134) Time, s Dimensionless time Liquid phase (or continuous phase) velocity, m/s Steady-state continuous phase velocity, m/s Perturbation in steady-state continuous phase velocity, m/s Elastic wave velocity, m/s Voidage propagation velocity, m/s Superficial velocity of the continuous phase, m/s Instantaneous value of X component of the continuous phase velocity, m/s Time-averaged value of X component of the velocity of the continuous phase, m/s Fluctuating value of X component of the continuous phase velocity, m/s Instantaneous value of Z component of the continuous phase velocity, m/s Time-averaged value of Z component of the velocity of the continuous phase, m/s Fluctuating value of Z component of the continuous phase velocity, m/s Dimensionless liquid phase (or continuous phase) velocity Dimensionless liquid phase velocity averaged in y-direction Downward liquid velocity, m/s Dimensionless liquid phase velocity in x-direction Dimensionless liquid phase velocity in z-direction Gas (dispersed phase) velocity, m/s

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

v0 v1 vP vs vlf vsup vx v¯x v⬘x vz v¯z v⬘z V ¯ V VB 앝 VG VGC VGLS Vmf Vp VS VS0 V ⬘S VS 앝 VSL Vx Vz We W w wG wL x X y Y z Z

125

Steady-state dispersed phase velocity, m/s Perturbation in steady-state dispersed phase velocity, m/s Particle volume, m3 Slip velocity, m/s Velocity in the wake region, m/s Superficial velocity of the dispersed phase, m/s Instantaneous value of X component of the dispersed phase velocity, m/s Time-averaged value of X component of the velocity of the dispersed phase, m/s Fluctuating value of X component of the dispersed phase velocity, m/s Instantaneous value of Z component of the dispersed phase velocity, m/s Time-averaged value of Z component of the velocity of the dispersed phase, m/s Fluctuating value of Z component of the dispersed phase velocity, m/s Dimensionless velocity Dimensionless gas phase velocity averaged in y-direction Terminal bubble rise velocity, m/s Superficial gas velocity, m/s Critical superficial gas velocity at transition Settling velocity in presence of the gas phase, m/s Minimum fluidization velocity, m/s Volume of the solid phase, m3 Dimensionless slip velocity, vs /VB00 or vs /VS00 Value of VS at steady state ⭸VS /⭸␧G Terminal settling velocity, m/s Settling velocity in absence of the gas phase, m/s Dimensionless gas phase velocity in x-direction Dimensionless gas phase velocity in z-direction Effective weight, kg Dimensionless component of axial liquid velocity, w/VB00 Component of true axial liquid velocity, m/s Average linear velocity of the gas phase, m/s Average linear velocity of the liquid phase, m/s Coordinate Dimensionless x coordinate Coordinate Dimensionless y coordinate Axial coordinate Parameter of the stability criterion

126 Greek 움 움⬘ 웁 웁⬘1 웃 웃␶ ⌬h ⌬t ⌬usup ␴ ␴0 웂 웂G ⌫ ␧ ␧c ␧G ␧¯ G ␧⬘G ␧GA ␧GB ␧GC ␧K ␧L ␧¯ L ␧⬘L ␧lf ␧L0 ␧L1 ␧LC ␧S ␧¯ S ␧⬘S ␧SC ␾ ␾A ␩ ␺ ␭ 애 애C ␯L ␯t

J. B. JOSHI ET AL.

Symbols Proportionality constant for dispersion A parameter defined in (179) Two-phase interaction term, kg/m 3 · s A parameter defined in (180) Characteristic dimension of pore space, m Ratio of distance between particles to particle diameter Change in the height Change in the time Change in superficial velocity of continuous phase Surface tension Defined in Eq. (72) Dimensionless gas throughput Dimensionless gas throughput Interfacial energy Phase hold-up, dimensionless Continuous phase hold-up, dimensionless Instantaneous gas phase hold-up Time-averaged gas phase hold-up Fluctuating gas phase hold-up Gas hold-up after suddenly reducing the gas flowrate Gas hold-up before suddenly reducing the gas flowrate Critical gas hold-up at which transition occurs Hold-up in the wake region Instanataneous liquid phase hold-up Time-averaged liquid phase hold-up Fluctuating liquid phase hold-up Average liquid hold-up in the wake region Steady-state liquid hold-up Perturbation in steady-state liquid hold-up Critical liquid hold-up at which transition occurs Instantaneous solid phase hold-up Time-averaged solid phase hold-up Fluctuating solid phase hold-up Critical solid hold-up Defined in Eq (163) Defined in Eq. (204) Averaging parameter defined in Eq. (123) Stream function Separation constant, Eq. (140) Viscosity, kg/m s Viscosity of the continuous phase, kg/m s Molecular kinematic viscosity of the liquid, m2 /s Turbulent kinematic viscosity, m2 /s

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

127

␯0 ␳ ␳¯ ␳C ␳D ␳f ␶ ␶S ␶S,xx ␶S,zz ␹

Mobility in Eq. (31) Density, kg/m3 Average density of two-phase dispersion, kg/m3 Density of continuous phase,kg/m3 Density of dispersed phase, kg/m3 Density of the fluid, kg/m3 Dimensionless time Shear stress Normal stress in solid phase, N/m2 Shear stress in solid phase, N/m2 Ratio of solid hold-up in bubble wakes to that in liquid fluidization, in equation (208) ␨ Parameter defined in Eq. (68) ␰ Proportionality constant in Eq. (12) Subscripts 0 Initial steady state 1 Perturbation G Gas phase L Liquid phase max Maximum sup Superficial S Solid phase

ACKNOWLEDGMENT The research work was supported by a grant under the Indo-U.S. Collaborative Research Program (DST/INT/USHRF/336/92). Authors express sincere thanks to Professor M. C. Jones for several valuable suggestions.

REFERENCES Abdel-Ghani, M., Petrie, J. G., Seville, J. P. K., Clift, R., and Adams, J. Powder Tech. 65, 113 (1991). Adlington, D., and Thompson, E. in ‘‘Proc. Third European Symp. Chem. React. Eng.,’’ 203 (1965). Agarwal, G. P., Hudson, J. L., and Jackson, R. Ind. Eng. Chem. Fundamentals 19, 59 (1980). Anderson, T. B., and Jackson, R. Ind. Eng. Chem. Fundamentals 6, 478 (1967a). Anderson, T. B., and Jackson, R. Ind. Eng. Chem. Fundamentals 6, 527 (1967b). Anderson, T. B., and Jackson, R. Ind. Eng. Chem. Fundamentals 7, 12 (1968). Anderson, T. B., and Jackson, R. Ind. Eng. Chem. Fundamentals 8, 137 (1969). Astarita, G. Chem. Eng. Sci. 48, 3438 (1993). Bach, H. F., and Pilhofer, T. Ger. Chem. Eng. 1, 278 (1978).

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Barnea, E., and Mizrahi, J. Chem. Eng. J. 5, 171 (1973). Batchelor, G. K. J. Fluid Mech. 193, 75 (1988). Bhatia, V. K., and Epstein, N. in ‘‘Proc. Int. Conf. on Fluidization and Its Applications,’’ 380, Toulouse (1974). Biesheuvel, A., and Gorrisen, W. C. M. Int. J. Multiphase Flow 16, 211 (1990). Buchholz, R., Tsepetonides, J., Steinmann J., and Onken, U. Ger. Chem. Eng. 6, 105 (1983). Chisti, M. Y. ‘‘Airlift Bioreactors.’’ Elsevier, New York, 1989. Clift, R. A. I. Ch. E. Symp. Ser. 89, 1 (1993). Clift, R., Grace, J. R., and Weber, M. E. ‘‘Bubbles, Drops, and Particles.’’ Academic Press, New York, 1978. Clift, R., Seville, J. P. K., Moore, S. C., and Chavarie, C. Chem. Eng. Sci. 42, 191 (1987). Cook, T. L., and Harlow, F. H., Int. J. Multiphase Flow, 12, 35 (1986). Dakshinamurty, P., Subramhanyam, V., and Rao, J. N. Ind. Eng. Chem. Proc. Des. Dev. 11, 151 (1971). Darton, R. C., and Harrison, D. Chem. Eng. Sci. 30, 581 (1975). Davidson, J. F., and Harrison, D. Chem. Eng. Sci. 21, 731 (1966). Davies, R. M., and Taylor, Sir G. I. Proc. Roy. Soc. A200, 375 (1950). Deckwer, W. D., Allenbach, U., and Bretschneider, H. Chem. Eng. Sci. 32, 43 (1977). Didwania, A. K., and Homsy, G. M. Int. J. Multiphase Flow 7, 563 (1981a). Didwania, A. K., and Homsy, G. M. Ind. Eng. Chem. Fundamentals 20, 318 (1981b). Didwania, A. K., and Homsy, G. M. J. Fluid Mech. 122, 433 (1982). Dorgelo, E. A. H., Van Der Meer, A. P., and Wesseslingh, J. A. Chem. Eng. Sci. 40, 2150 (1985). Drew, D., Cheng, L., and Lahey, R. T. Int. J. Multiphase Flow 5, 233 (1979). Elgobashi, S. E., and Abou-Arab, T. W. Phys. Fluids 26, 931 (1983). El-Kaissy, M. M., and Homsy, G. M. Int. J. Multiphase Flow 2, 379 (1976). El-Temtamy S. A., and Epstein, N. Int. J. Multiphase Flow 4, 19 (1978). El-Temtamy, S. A., and Epstein, N. Can. J. Chem. Eng. 57, 520 (1979). Epstein, N. Can. J. Chem. Eng. 54, 259 (1976). Epstein, N. Chem. Eng. Sci. 39, 1533 (1984). Epstein, N. and Nicks, D. in ‘‘Fluidization Technology,’’ Vol. I (D. L. Keairns, Ed.). Hemisphere, Washington, D. C., 1976. Ergun, S. Chem. Eng. Progr. 48, 89 (1952). Fan, L. S., Han, L. S. and Brodkey, R. S. Chem. Eng. Sci. 42, 1269 (1987). Foscolo, P. U., and Gibilaro, L. G. Chem. Eng. Sci. 39, 1667 (1984). Gaudin, A. M. in ‘‘Flotation,’’ 2nd ed. McGraw-Hill, New York, 1977. Gibilaro, L. G., Waldram, S. P., and Foscolo, P. U. Chem. Eng. Sci. 39, 607 (1984). Gibilaro, L. G., Hossain, I., and Foscolo, P. U. Can J. Chem. Eng. 64, 93 (1986). Gibilaro, L. G., Di Felice, R., Waldram, S. P., and Foscolo, P. U. Chem. Eng. Sci. 42, 194 (1987a). Gibilaro, L. G., Di Felice, R., Foscolo, P. U., and Waldram, S. P. Chem. Eng. Sci. 42, 1272 (1987b). Gibilaro, L. G., Di Felice, R., Hossain, I., and Foscolo, P. U. Chem. Eng. Sci. 44, 101 (1989). Gibilaro, L. G., Di Felice, R., and Foscolo, P. U. Chem. Eng. Sci. 45, 1561 (1990). Grbavcic, Z. B., Vukovic, D. V., Jovanovic, S. D. J., and Littman, H. Chem. Eng. Sci. 47, 2120 (1992). Green, D., and Homsy, G. M. Int. J. Multiphase Flow 13, 447 (1987a). Green, D., and Homsy, G. M. Int. J. Multiphase Flow 13, 459 (1987b). Ham, J. M., Thomas, S., Guazzelli, E., Homsy, G. M., and Anselmet, M. C. Int. J. Multiphase Flow 16, 171 (1990). Handley, D., Doraiswami, A., Butcher, K. L., and Franklin, N. L. Trans. Inst. Chem. Engr. 44, T266 (1966). Homsy, G. M., and El-Kaissy, M. M. Int. J. Multiphase Flow 6, 305 (1980).

HYDRODYNAMIC STABILITY OF MULTIPHASE REACTORS

129

Imafuku, K., Wang, T. Y., Koide, K., and Kuboto, H. J. Chem. Eng. Japan 1, 153 (1968). Jackson, R. Trans. Inst. Chem. Engr. 41, 13 (1963a). Jackson, R. Trans. Inst. Chem. Engr. 41, 21 (1963b). Jackson, R. Chem. Eng. Sci. 19, 509 (1964). Jackson, R. in ‘‘Fluidization’’ (Davidson, J. F., Clift, R., and Harrison, D., Eds.). Academic Press, London, 1985. Jacob, K. V., and Weimer, A. W. A.I.Ch.E.J. 33, 1698 (1987). Jean, R. H., and Fan, L. S. Can. J. Chem. Eng. 65, 351 (1987). Joshi, J. B. Trans. Instn. Chem. Engr. 58, 155 (1981). Joshi, J. B. Chem. Eng. Res. Des. 61, 143 (1983). Joshi, J. B. Chem. Eng. Res. Des. 62, 271 (1984). Joshi, J. B. unpublished (1987). Joshi, J. B., and Lali, A. M. in ‘‘Frontiers in Chemical Reaction Engineering’’ (L. K. Doraiswamy and R. A. Mashelkar, Eds.). Wiley Eastern Ltd., 1984, p. 314. Kastanek, F., Nyvlt, V., and Rylek, M. Collect. Czech. Chem. Commun. 39, 3246 (1974). Kataoka, I. Int. J. Multiphase Flow 12, 745 (1986). Kataoka, I., and Serizawa, A. Int. J. Multiphase Flow 15, 843 (1989). Kato, Y., Nishiwaki, A., Fukuda, T., and Tanaka, S. J. Chem. Eng. Japan 5, 112 (1972). Kelkar, B. G., Phulgaonkar S. P., and Shah, Y. T. Chem. Eng. J. 27, 125 (1983). Khare, A. S., and Joshi, J. B. Chem. Eng. J. 44, 11 (1990). Kim, S. D., Baker, C. G .J., and Bergougnou, M.A. Can. J. Chem. Eng. 53, 134 (1975). King, D. F., and Harrison, D. Trans. Instn. Chem. Engrs. 60, 26 (1982). Koetsier, W. T., Van Swaij, W. P. M., and Van der Most, M. J. J. Chem. Eng. Japan 9, 332 (1976). Krishna, R., Wilkinson, P. M., and Van Dierendonck, L. L. Chem. Eng. Sci. 46, 2491 (1991). Kumar, R., and Kuloor, N. R. Adv. Chem. Eng. 8, 55 (1972). Lali, A. M., Khare, A. S., Joshi, J. B., and Nigam, K. D. P. Powder Technol. 57, 39 (1989). Lisseter, P. E., and Fowler, A. C. Int. J. Multiphase Flow 18, 195 (1992). Liu, J. T. C. Proc. Royal Soc. London A380, 228 (1982). Lockett, M. J., and Kirkpatrick, R. D. Trans. Instn. Chem. Engrs. 53, 267 (1975). Marucci, G. Ind. Eng. Chem. Fundamentals 4, 224 (1965). Maruyama, T., Yoshida, S., and Mizushina, T. J. Chem. Eng. Japan 14, 352 (1981). Medlin, J., and Jackson, R. Ind. Eng. Chem. Fundamentals 14, 315 (1975). Medlin, J., Wang, H. W., and Jackson, R. Ind. Eng. Chem. Fundamentals 13, 247 (1974). Mendelsen, H. D. A.I.Ch.E.J. 13, 250 (1967). Mersmann, A. Ger. Chem. Eng. 1, 1 (1978). Molerus, O. in ‘‘Principles of Flow in Disperse Systems.’’ Chapman and Hall, London, 1993. Musters, S. M. P., and Rietema, K. Powder Technol. 18, 239 (1977). Nicklin, D. J. Chem. Eng. Sci. 17, 693 (1962). Oels, U., Lucke, J., Buchholz, R., and Schugerl, K. Ger. Chem. Eng. 1, 115 (1978). Ostergaard, K. Chem. Eng. Sci. 20, 165 (1965). Ostergaard, K., and Thiesen, P.I. Chem. Eng. Sci. 21, 413 (1966). Pal, R., and Masliyah, J. Can J. Chem. Eng. 67, 916 (1989). Pandit, A. B., and Joshi, J. B. Reviews in Chem. Eng. 2, 1 (1984). Pandit, A. B., and Joshi, J. B. Chem. Eng. Res. Des. 64, 125 (1986). Pauchon, C., and Banerjee, S. Int. J. Multiphase Flow 14, 253 (1988). Ranade, V. V., and Joshi, J. B. in ‘‘Proceedings of International Symposium on Transfer Processes in Multiphase Systems’’ (Upadhyay, S. N., Ed.). Banaras Hindu University Press, Varanasi (India), 1987, p. 113. Richardson, J. F. in ‘‘Fluidization’’ (Davidson, J.F., and Harrison, D., Eds.). Academic Press, New York, 1971, p. 25. Richardson, J. F., and Meikle, R. A. Trans. Instn. Chem. Engrs. 39, 348 (1961).

130

J. B. JOSHI ET AL.

Richardson, J. F., and Zaki, W. N. Trans. Inst. Chem. Engr. 32, 35 (1954). Rietema, K. Chem. Eng. Sci. 28, 1493 (1973). Rietema, K. Chem. Eng. Sci. 37, 1125 (1982). Rowe, P. N. Chem. Eng. Res. Des. 62, 190 (1984). Rowe, P. N., Foscolo, P. U., Hoffman, A. C., and Yayes, J. G. Chem. Eng. Sci. 37, 355 (1982). Sangani, A. S., and Didwania, A. K. J. Fluid Mech. 248, 27 (1993). Schugerl, K., Lucke, J., and Oels, U. Adv. Biochem. Eng. 7, 1 (1977). Shnip, A. I., Kolhatkar, R. V., Dinakar, S., and Joshi, J. B. Int. J. Multiphase Flow 18, 705 (1992). Slis, P. L., Willemse, Th. W., and Kramers, H. Appl. Sci. Research A8, 209 (1959). Smith, D. N., and Reuther, J. A. Chem. Eng. Sci. 40, 741 (1985). Stewart, P. S. B., and Davidson, J. F. Chem. Eng. Sci. 21, 165 (1964). Taitel, Y., Barnea, D., and Dukler, A. E. A.I.Ch.E.J. 26, 345 (1980). Tarmy, B. L., Chang, M., Coulaloglou, C. A. and Ponzi, P. R., Inst. Chem. Eng. Symp. Ser. 87, 303 (1984). Tsinontides, S. C., and Jackson, R. J. Fluid Mech. 255, 237 (1993). Turner, R. in ‘‘Fluidization.’’ Soc. Chem. Eng., London, 1964, p. 42. Turner, R. Chem. Eng. Sci. 21, 971 (1966). Verloop, J., and Heertjes, P. M. Chem. Eng. Sci. 25, 825 (1970). Wallis, G. B. in ‘‘One-Dimensional Two-Phase Flow.’’ McGraw-Hill, New York, 1969. Whithman, G. B., ‘‘Linear and Nonlinear Waves.’’ Wiley, New York, 1974. Wilkinson, P. M., Spek A. P., and Van Dierendonck, L. L. A.I.Ch.E.J. 38, 544 (1992). Willhelm, R. H., and Kwauk, M. Chem. Eng. Prog. 44, 201 (1948). Yamashita, F., and Inoue, H. J. Chem. Eng. Japan 8, 334 (1975). Zhou, Z.A., and Egiebor, N.O. Minerals Eng. 6, 199 (1993). Zuber, N., and Hench, J., Rept. No. 62GL100, General Electric Company, Schenectady, NY (1962).

MODEL PREDICTIVE CONTROLLERS: A CRITICAL SYNTHESIS OF THEORY AND INDUSTRIAL NEEDS Michael Nikolaou Department of Chemical Engineering, University of Houston, Houston, Texas 77204 I. Introduction II. What Is MPC? A. A Traditional MPC Formulation B. Expanding the Traditional MPC Formulation C. MPC without Inequality Constraints III. Stability A. What Is Stability? B. Is Stability Important? IV. The Behavior of MPC Systems A. Feasibility of On-Line Optimization B. Nonminimum Phase and Short Horizons C. Integrators and Unstable Units D. Nonlinearity E. Model Uncertainty F. Fragility G. Constraints V. A Theory for MPC with Predictable Properties A. Stability B. Robust Stability and Fragility of Constrained MPC C. Performance and Robust Performance VI. How Can Theory Help Develop Better MPC Systems? A. Conceptual Unification and Clarification B. Improving MPC VII. Future Needs A. Is Better MPC Needed? B. Is More MPC Theory Needed? References

132 133 137 139 142 145 145 155 156 156 157 159 162 165 165 169 170 170 176 185 187 187 188 198 198 198 199

After several years of effort, constrained model predictive control (MPC), the de facto standard algorithm for advanced control in process industries, has finally succumbed to rigorous analysis. Yet successful practical implementations of MPC were already in place almost two decades before a rigorous stability proof for constrained 131 ADVANCES IN CHEMICAL ENGINEERING, VOL. 26

Copyright  2001 by Academic Press. All rights of reproduction in any form reserved. 0065-2377/01 $35.00

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MPC was published. What then, is the importance of recent theoretical results for practical MPC applications? In this publication we present a pedagogical overview of some of the most important recent developments in MPC theory and discuss their implications for the future of MPC theory and practice.  2001 Academic Press. There is nothing more practical than a good theory. —Boltzmann

I. Introduction The past couple of decades have witnessed a steady growth in the use of computers for advanced control of process plants. Rapid improvements in computer hardware, combined with stiff foreign and domestic competition and government regulations, have been largely responsible for this development. With more than 2000 industrial installations, model predictive control (MPC) is currently the most widely implemented advanced process control technology for process plants (Qin and Badgwell, 1997). As is frequently the case, the idea of MPC appears to have been proposed long before MPC came to the forefront (Propoi, 1963; Rafal and Stevens, 1968; Nour-Eldin, 1971). Not unlike many technical inventions, MPC was first implemented in industry—under various guises and names—long before a thorough understanding of its theoretical properties was available. Academic interest in MPC started growing in the mid-eighties, particularly after two workshops organized by Shell (Prett and Morari, 1987; Prett et al., 1990). The understanding of MPC properties generated by pivotal academic investigations (Garcı´a and Morari, 1982; Rawlings and Muske, 1993) has now built a strong conceptual and practical framework for both practitioners and theoreticians. Although several issues in that framework are still open, there is now a strong foundation. The purpose of this paper is to examine some of the recent developments in the theory of MPC, discuss their theoretical and practical implications, and propose directions for future development and research on MPC. We hope that both practitioners and theoreticians will find the discussion useful. We would like to stress that this work does not purport to be an exhaustive discussion on MPC to any extent beyond what the title of the work implies. In particular, important practical issues such as the efficiency and effectiveness of various numerical algorithms used to solve the on-line optimization problems, human factors, fault tolerance, detection and diagnosis, and programming environments for MPC implementation are hardly touched upon in any

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way other than what pertains to their implications for theoretically expected MPC properties.

II. What Is MPC? Although the MPC paradigm encompasses several different variants, each one with its own special features, all MPC systems rely on the idea of generating values for process inputs as solutions of an on-line (real-time) optimization problem. That problem is constructed on the basis of a process model and process measurements. Process measurements provide the feedback (and, optionally, feedforward) element in the MPC structure. Figure 1 shows the structure of a typical MPC system. It makes it clear that a number of possibilities exist for the following: • Input–output model • Disturbance prediction

FIG. 1. Model predictive control scheme.

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Objective Measurement Constraints Sampling period (how frequently the on-line optimization problem is solved)

Regardless of the particular choice made for these elements, on-line optimization is the common thread tying them together. Indeed, the possibilities for on-line optimization (Marlin and Hrymak, 1997) are numerous, as discussed in Section VI,B,2,a. Figure 1 also makes it clear that the behavior of an MPC system can be quite complicated, because the control action is determined as the result of the on-line optimization problem. Although engineering intuition may frequently be used in the analysis of the behavior or in the design of MPC systems, theory can provide valuable help. Theory can augment human judgement and intuition in the development and implementation of better MPC systems that can realize their full potential as ‘‘advanced’’ control systems. Some of the benefits of improved MPC systems are better control performance, less down time, reduced maintenance requirements, and improved flexibility and agility. The origins of MPC applications in industry are quite interesting. Although the author is more familiar with U.S. industry, developments overseas seem to have followed a similar path. The first use of computers to calculate an on-line economic optimal operating point for a process unit ˚ stro¨m and Wittenmark appears to have taken place in the late 1950s. A (1984, p. 3) cite March 12, 1959, as the first day when a computer control system went on-line at a Texaco refinery in Port Arthur, Texas. The computer control system, designed by Ramo-Wooldridge (later TRW), relied on an RW-300 computer. Baxley and Bradshaw (1998) mention that around the same time (1959) Union Carbide, in collaboration with Ramo-Wooldridge, implemented an on-line computer control and optimization system, based also on the RW300, at the Seadrift, Texas, plant’s ethylene oxide unit. The implementation was not a classical mathematical programming type optimization. It was an implied ‘‘maximize production’’ optimization with a feed allocation algorithm for multiple parallel reactors followed by a serial train of reactors to convert all the remaining ethylene before exhausting to the air. However, there was no open publication reporting this venture. Baxley and Bradshaw (1998) believe that the first open report related to a similar computer control venture was by Monsanto. It appears that computer control and on-line optimization were ideas whose time had come. It also appears that on-line optimization was performed every few hours at the supervisory level, using steady-state models. Certainly, the

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speed and storage capacity of computers available at the time must have played a role. As the capability of computers increased, so did the size and sophistication of on-line optimization. Early projects usually included ethylene units and major oil refinery processes such as crude distillation units and fluid catalytic cracking (FCC) units (Darby and White, 1988). ‘‘The objective function was generally an economic one but we had the flexibility to select alternative ones if the operating and/or business environment suggested another, e.g., maximize ethylene production, minimize ethylene costs, etc. We were getting the tools to be more sophisticated and we took advantage of them where it made economic sense’’ (Baxley and Bradshaw, 1998). In the early seventies, practitioners of process control in the chemical industry capitalized on the increasing speed and storage capacity of computers by expanding on-line optimization to process regulation through more frequent optimization. This necessitated the use of dynamic models in the formulation of on-line optimization problems that would be solved every few minutes. What we today call MPC was conceived as a control algorithm that met a key requirement, not explicitly handled by other control algorithms: The handling of inequality constraints. Efficient use of energy, of paramount importance after the 1973 energy crisis, must have been a major factor that forced oil refineries and petrochemical plants to operate close to constraints, thus making constrained control a necessity. At the time, the connection of MPC to classical control theory was, at best, fuzzy, as manifested by the title of perhaps the first journal publication reporting the successful application of the MPC algorithm in the chemical process industry: ‘‘Model Predictive Heuristic Control: Applications to Industrial Processes’’ (Richalet et al., 1978). As is often the case, ingenuity and engineering insight arrived at the same result that others had reached after taking a different route, whether theoretical or heuristic. Where the MPC idea first appeared as a concept is difficult to trace. Prett and Garcia (1988) cite Propoi (1963) as the first who published essentially an MPC algorithm. Rafal and Stevens (1968) presented essentially an MPC algorithm with quadratic cost, linear constraints, and moving horizon of length 1. They controlled a distillation column for which they used a first-principles nonlinear model that they linearized at each time step. In many ways, and quite amazingly, that publication contained several of the elements that today’s MPC systems include. It is evident that the authors were fully aware of the limitations of the horizon of length 1, but were limited by the computational power available at the time: . . . the step-by-step optimal control need not be overall optimal. . . . In the present work, the one-step approach is taken because it is amenable to

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practical solution of the problem and is well suited to nonlinear situations where updating linearization is useful. (Rafal and Stevens, 1968, p. 85).

Mayne et al. (1998) provide a quote from Lee and Markus (1967, p. 423) which essentially describes the MPC algorithm. Nour-Eldin (1971, p. 41), among others, explicitly describes the on-line constrained optimization idea, starting from the principle of optimality in dynamic programming: 1

Zusammenfassend: Beim Zeitpunkt tk⫺1 wird das Optimum von Zk gesucht. Der resultierende Steuerungsvektor U*(k) ha¨ngt von x(k ⫺ 1) ab und entha¨lt sa¨mtliche Steuervektoren u*k , u*k⫹1 , . . . , u*N welche den Prozess wa¨hrend dem Intervall [tk⫺1,T] optimal steuern. Von diesem Steuervektoren verwendet man den Vektor u*k (welcher von x(k ⫺ 1) abha¨ngt) als Steuervektor fu¨r das na¨chste Intervall [tk⫺1, tk]. Beim na¨chsten Zeitpunkt tk wird ein neuer Steuervektor u*k⫹1 bestimmt. Dieser wird aus der Zielfunktion Zk⫹1 berechnet und ist von x(k) abha¨ngig. Damit wird der Vektor uk , welcher im Intervall ␶k verwendet wird, vom Zustandsvektor x(k ⫺ 1) abha¨ngig. Das gesuchte Ru¨ckfuhrungsgesetz besteht somit aus der Lo¨sung einer convexen Optimierungsaufgabe bei jedem Zeitpunkt tk⫺1 (k ⫽ 1,2, . . ., N). [Underlining in the original text.]

Although the value of on-line constrained optimization is explicitly identified in this passage, the concept of moving horizon is missing. That is, perhaps, due to the fact that the author was concerned with the design of autopilots for airplane landing, a task that has a finite duration T. The algorithm described above is, essentially, the mathematical equivalent of MPC for a batch chemical process. In the sixties and seventies, in contrast to literature references to the constrained on-line optimization performed by MPC, which were only sporadic, there was an already vast and growing literature on a related problem, the linear-quadratic regulator (LQR) either in deterministic or stochastic settings. Simply stated, the LQR problem is

冘 x[i] Qx[i] ⫹ u[i] Ru[i], p

min x,u

1

T

T

(1)

i⫽0

Summarizing: At the time point tk⫺1 the optimum of the [quadratic objective function] Zk is sought. The resulting control [input] vector U*(k) depends on x(k⫺1) and contains all control [input] vectors u*k , u*k⫹1 , . . . , u*N which control the process optimally over the interval [tk⫺1 , T]. Of these control [input] vectors, one implements the vector u*k (which depends on x(k ⫺ 1)) as input vector for the next interval [tk⫺1, tk]. At the next time point tk a new input vector u*k⫹1 is determined. This is calculated from the objective function Zk⫹1 and is dependent on x(k). Therefore, the vector uk , which is implemented in the interval ␶k , is dependent on the state vector x(k ⫺ 1). Hence, the sought feedback law consists of the solution of a convex optimization problem at each time point tk⫺1 (k ⫽ 1, 2, . . . , N). (Translation by the author.)

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x[i ⫹ 1] ⫽ Ax[i] ⫹ Bu[i], x[0] ⫽ x0 ,

(2)

where

and the optimization horizon length p could be finite or infinite. A celebrated result of LQR theory was uopt[i] ⫽ K[i]xopt[i], i ⫽ 1, . . . , p,

(3)

known as the ‘‘feedback form’’ of the optimal solution to the optimization problem posed by Eqs. (1) and (2). The state feedback gain K[i] is not fixed and is computed from the corresponding Riccati equation. Yet, for finite p, this viewpoint of feedback referred to a set of shrinking horizons ending at the same time point, thus corresponding to a control task that would end at time p. Of course, p could be equal to infinity (in which case K would be fixed), but then that formulation would not lend itself to optimization subject to inequality constraints (for which no explicit solution similar or Eq. (3) could be obtained) because it would involve an infinite number of decision variables. The ideas of a finite moving horizon and online optimization somehow did not occupy much of the research community, although it appears that they were known. In fact, a lot of effort was expended to avoid on-line optimization, given the limited capabilities of computers of that era and the fast sampling of systems for which LQR was developed (e.g., aerospace). Over the years, the heuristics of the early works on MPC were complemented by rigorous analysis that elucidated the essential features, properties, advantages, and limitations of MPC. In the next section, we will start with a smooth introduction to a simple (if not limiting) MPC formulation that was popular in the early days of MPC. We will then identify some of its many variants.

A. A TRADITIONAL MPC FORMULATION Consider a stable single-input–single-output (SISO) process with input u and output y. A formulation of the MPC on-line optimization problem can be as follows: At time k find

冘 w ( y [k ⫹ i兩k] ⫺ y p

min

u[k兩k],⭈⭈⭈,u[k⫹p⫺1兩k] i⫽1

i

) ⫹

SP 2

冘 r ⌬u[k ⫹ i ⫺ 1兩k] , m

i

2

(4)

j⫽1

subject to umax ⱖ u[k ⫹ i ⫺ 1兩k] ⱖ umin , i ⫽ 1, ⭈ ⭈ ⭈ , m

(5)

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⌬umax ⱖ ⌬u[k ⫹ i ⫺ 1兩k] ⱖ ⫺ ⌬umax , i ⫽ 1, ⭈ ⭈ ⭈ , m

(6)

ymax ⱖ y[k ⫹ i兩k] ⱖ ymin , i ⫽ 1, ⭈ ⭈ ⭈ , p,

(7)

where p and m ⬍ p are the lengths of the process output prediction and manipulated process input horizons, respectively; u[k ⫹ i ⫺ 1兩k], i ⫽ 1, . . . , p, is the set of future process input values with respect to which the optimization will be performed, where u[k ⫹ i兩k] ⫽ u[k ⫹ m ⫺ 1兩k], i ⫽ m, ⭈ ⭈ ⭈ , p ⫺ 1;

(8)

ySP is the set-point; and ⌬ is the backward difference operator, i.e., ⌬u[k ⫹ i ⫺ 1兩k] ⫽ ˆ u[k ⫹ i ⫺ 1兩k] ⫺ u[k ⫹ i ⫺ 2兩k].

(9)

In typical MPC fashion (Prett and Garcia, 1988), this optimization problem is solved at time k, and the optimal input u[k] ⫽ uopt [k兩k] is applied to the process. This procedure is repeated at subsequent times k ⫹ 1, k ⫹2兩, etc. It is clear that the foregoing problem formulation necessitates the prediction of future outputs y[k ⫹ i兩k]. This, in turn, makes necessary the use of a model for the process and external disturbances. To start the discussion on process models, assume that the following finite-impulse-response (FIR) model describes the dynamics of the controlled process: y[k] ⫽

冘 h u[k ⫺ j ] ⫹ d [k], n

i

(10)

j⫽1

where hi are the model coefficients (convolution kernel) and d is a disturbance. Then y[k ⫹ i兩k] ⫽

冘 h u[k ⫹ i ⫺ j兩k] ⫹ d [k ⫹ i兩k], n

j

(11)

j⫽1

where u [k ⫹ i ⫺ j兩k] ⫽ u [k ⫹ i ⫺ j ], i ⫺ j ⬍ 0.

(12)

The prediction of the future disturbance d[k ⫹ i兩k] clearly can be neither certain nor exact. An approximation or simplification has to be employed, such as d [k ⫹ i兩k] ⫽ d [k兩k] ⫽ y [k] ⫺

冘 h u[k ⫺ j ], n

j

j⫽1

(13)

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MODEL PREDICTIVE CONTROLLERS

where y[k] is the measured value of the process output y at sampling point k and u[k ⫺ j] are past values of the process input u. Substitution of Eqs. (11)–(13) into Eqs. (4)–(7) yields

冘 w 冉冘 h u[k ⫹ i ⫺ j兩k] ⫺ 冘 h u[k ⫺ j ] ⫹ y[k] ⫺ y 冊 p

min

n

u[k兩k],⭈⭈⭈,u[k⫹p⫺1兩k] i⫽1

⫹

2

n

i

j

SP

j

j⫽1

j⫽1

冘 r ⌬u [k ⫹ i ⫺ 1兩k] ,

(14)

m

2

i

i⫽1

subject to umax ⱖ u [k ⫹ i ⫺ 1兩k] ⱖ umin , ⌬umax ⱖ ⌬u [k ⫹ i ⫺ 1兩k] ⱖ ⫺⌬umax , ymax ⱖ

i ⫽ 1, 2, ⭈ ⭈ ⭈ , m i ⫽ 1, 2, ⭈ ⭈ ⭈ , m

冘 h u[k ⫹ i ⫺ j 兩k] ⫺ 冘 h u[k ⫺ j ] ⫹ y[k] ⱖ y n

(15) (16)

n

j

j⫽1

j

j⫽1

min ,

(17)

i ⫽ 1, ⭈ ⭈ ⭈ , p . The preceding optimization problem is a quadratic programming problem, which can be easily solved at each time k.

B. EXPANDING THE TRADITIONAL MPC FORMULATION The foregoing formulation of MPC was typical in the first industrial implementations that dealt with stable processes modeled by finite-impulseresponse models. FIR models, although not essential for characterizing a model-based algorithm such as MPC, have certain advantages from a practical implementation viewpoint: Time delays and complex dynamics can be represented with equal ease. Mistakes in the characterization of colored additive noise as white in open-loop experiments introduce no bias in parameter estimates. No advanced knowledge of modeling and identification techniques is necessary if simple step-response experiments are used for process identification. Instead of the observer or state estimator of classic optimal control theory, a model of the process is employed directly in the algorithm to predict future process outputs (Morari, 1988). Their main disadvantage is the use of too many parameters (overparametrization), which becomes even more pronounced in the multivariable case. Although a large class of processes can be treated by that formulation, more general classes can be handled by more general MPC formulations concentrating on the following characteristics:

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• Unstable process model. In that case, an FIR process model cannot be used. A state-space model such as x[k ⫹ 1] ⫽ Ax[k] ⫹ Bu[k] ⫹ Ed[k] y [k] ⫽ Cx[k] ⫹ Du[k] ⫹ Fe[k]

(18)

or a deterministic autoregressive-moving-average with exogenous input (DARMAX) model,

冘 h u [k ⫺ i] ⫹ 冘 g y [k ⫺ i] ⫹ 冘 f d [k ⫺ i], nu

y [k] ⫽

ny

i

i⫽1

nd

i

i⫽1

i

(19)

i⫽1

can be used. As MPC systems grow in size, the probability of including an unstable or marginally stable (integrating) unit increases. For such units, models like in eqs. (18) or (19) are necessary. • Nonlinear process model. The nonlinearity of chemical processes is well documented (Shinskey, pp. 55–56, 1967; Foss, 1973; Buckley, 1981; Garcı´a and Prett, 1986; Morari, 1986; NRC Committee Report, p. 148, 1988; Fleming, 1988; Prett and Garcı´a, p. 18, 1988; Edgar, 1989; Longwell, 1991; Bequette, 1991; Kane, 1993; Allgo¨wer and Doyle, 1997; Ogunnaike and Wright, 1997). Typical examples are distillation columns and reactors. Because nonlinear models are defined by what they are not (namely, linear), there exist a number of possibilities for representing nonlinear systems. First-principles, empirical, or hybrid models in state-space or input–output frameworks are all possible. In addition, the development and adaptation of such models is a central issue in MPC. • Stochastic disturbance model. There are various possibilities for using stochastic disturbance models other than the zero-order model shown in Eq. (13) (Ljung, 1987). • Stochastic objective function. The preceding MPC formulation assumes that future process outputs are deterministic over the finite optimization horizon. For a more realistic representation of future process outputs, one may consider a probabilistic (stochastic) prediction for y[k ⫹ i兩 k] and formulate an objective function that contains the expectation of appropriate functionals. For example, if y[k ⫹ i兩 k] is probabilistic, then the expectation of the functional in Eq. (4) could be used. This formulation, known as open-loop optimal feedback, does not take into account the fact that additional information would be available at future time points k ⫹ i and assumes that the system will essentially run in open-loop fashion over the optimization horizon. An alternative, producing a closed-loop optimal feedback law relies

141

MODEL PREDICTIVE CONTROLLERS

on the dynamic programming formulation of an objective function such as min (w1( y [k ⫹ 1兩k] ⫺ ySP)2 ⫹ ⌬u[k兩k]2

u[k兩k]

⫹

min

u[k⫹1兩k⫹1]

(20) (w2( y[k ⫹ 2兩k ⫹ 1] ⫺ y ) ⫹ ⌬u[k ⫹ 1兩k ⫹ 1] ⫹ ⭈ ⭈ ⭈)). SP 2

2

Although the open-loop optimal feedback law does not result in unwileldy computational requirements, the closed-loop optimal feedback law is considerably more complicated. For several practical problems the open-loop optimal feedback law produces results that are close to those produced by the closed-loop optimal feedback law. However, there are cases for which the open-loop optimal feedback law may be far inferior to the closed-loop optimal feedback law. Rawlings et al. (1994) present a related example on a generic staged system. Lee and Yu (1997) show that open-loop optimal feedback is, in general, inferior to closed-loop optimal feedback for nonlinear processes and linear processes with uncertain coefficients. They also develop a number of explicit closed-loop optimal feedback laws for a number of unconstrained MPC cases. • Available measurements. For controlled variables that are not directly measurable, measurements have to be inferred by measurements of secondary variables and/or laboratory analysis of samples. Good inference relies on reliable models. In addition, the results of laboratory analysis, usually produced much less frequently than inferential estimates, have to be fused with the inferential estimates produced by secondary measurements. For MPC systems that use state-space models, usually not all states are measurable, thus making state estimators necessary (Lee et al., 1994). • Constraints. Although constraints placing bounds on process inputs are trivial to formulate, constraints on process outputs are more elusive, because future process outputs y[k ⫹ i兩 k] are predicted in terms of a model. If the probability density function of y[k ⫹ i兩 k] is known, then deterministic constraints on y[k ⫹ i兩k] can be replaced by probabilistic constraints of the form Pr兵y[k ⫹ i兩k] ⱕ ymax其 ⱖ 움

(21)

(Schwarm and Nikolaou, 1997). • Sampling period. The selection of the time points tk at which on-line optimization is performed (Fig. 1) is an important task, albeit not as widely studied as other MPC design tasks. Things become more interesting when measurements or decisions take place at different

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time intervals for different variables (Lee et al., 1992). The multitude of different time scales is usually handled through decomposition of the overall on-line optimization problem into independently solved subproblems, each at a different time scale.

C. MPC

WITHOUT

INEQUALITY CONSTRAINTS

When there are no inequality constraints [Eqs. (15)–(17)], the minimization of the quadratic objective function in Eq. (4) has a simple closed-form solution, which can be expressed as follows. Equations (11) and (13) yield ykk⫹⫹p1兩兩kk ⫽ Hukk⫹兩klp⫺1兩k ⫹ Gukk⫺⫺n1 ⫹ y[k],

(22)

where, assuming that p ⬎ n, ykk⫹⫹p1兩兩kk ⫽ ˆ [ y[k ⫹ 1兩k] ⭈ ⭈ ⭈ y[k ⫹ p兩k]T ukk⫹兩kp⫺1兩k

⫽ ˆ [u[k兩k] ⭈ ⭈ ⭈ u[k ⫹ p ⫺ 1兩k]]

(23) T

(24)

(26)



y[k] ⫽ ˆ [ y[k] ⭈ ⭈ ⭈ y[k]]T



(25)



ukk⫺⫺n1 ⫽ ˆ [u[k ⫺ n] ⭈ ⭈ ⭈ u[k ⫺ 1]]T

p

h1

0 .. .

.. . 0

.. . .. .

0 .. . .. . 0

⭈ ⭈ ⭈ h1

hn



0 .. .

⭈⭈⭈ ⭈⭈⭈

冥冧

p

(27)



H⫽ ˆ hn

0



冤

h1 .. .

p

冤 冥冧 冤 冥冧 n

0

hn ⭈ ⭈ ⭈ h 1 n







0 ⭈⭈⭈ ⭈⭈⭈

0 .. .

hn ⭈ ⭈ ⭈ h 1 .. .. . . .. .. . . .. p ⫺ .. . . .. .. . .



⭈ ⭈ ⭈ h2 .. .. . . .. . hn



hn .. .



0 .. . .. . G⫽ ˆ .. . .. .

p,

(28)

MODEL PREDICTIVE CONTROLLERS

143

and Eq. (4) becomes T

min (ykk⫹⫹p1兩兩kk ⫺ ySP)TW(ykk⫹⫹p1兩兩kk ⫺ ySP) ⫹ ⌬ukk兩⫹km⫺1兩k R⌬ukk⫹兩km⫺1兩k

uk⫹p⫺1兩k k兩k

⫽ u min (HJukk⫹兩km⫺1兩k ⫹ Gukk⫺⫺n1 ⫹ y[k] ⫹ ySP)TW(HJukk⫹兩km⫺1兩k k⫹m⫺1兩k k兩k

(29)

⫹ Gukk⫺⫺n1 ⫹ y[k] ⫺ ySP) ⫹((I ⫺ P)ukk⫹兩km⫺1兩k ⫺ Qukk⫺⫺n1 )TR((I ⫺ P)ukk⫹兩km⫺1兩k ⫺ Qukk⫺⫺n1 ) where (30)







ySP ⫽ [ y SP ⭈ ⭈ ⭈ y SP]T p ukk⫹兩km⫺1兩k ⫽ ˆ [u[k兩k] ⭈ ⭈ ⭈ u[k ⫹ m ⫺ 1兩k]]T

(31)

⌬ukk⫹兩km⫺1兩k ⫽ ˆ [⌬u[k兩k] ⭈ ⭈ ⭈ ⌬u[k ⫹ m ⫺ 1兩k]]T

(32)

W⫽ ˆ Diag(w1 ⭈ ⭈ ⭈ wp)

(33)

R⫽ ˆ Diag(r1 ⭈ ⭈ ⭈ rm)

(34)

m







冤

0 ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ 0 .. .. 1 . . .. .. .. P⫽ . . . ˆ 0 .. .. .. .. . . . . . .. 0 ⭈⭈⭈

0

1

0

冥冧

m

(35)

n







冤

冥冧

0 ⭈⭈⭈ ⭈⭈⭈ 0 .. .. .. . . . Q⫽ ˆ .. . 0 0 0 ⭈⭈⭈

0

1

m

(36)

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MICHAEL NIKOLAOU

and

J⫽ ˆ

冤 冥 1

0

0 .. . .. .

1

⭈⭈⭈

0

.. .

1 0

0 ⭈⭈⭈ ⭈⭈⭈ 0 1

⭈ ⭈⭈ ⭈ ⭈ ⭈ ⭈ ⭈⭈ ⭈ ⭈ ⭈ ⭈ ⭈⭈ ⭈ ⭈ ⭈

0 ⭈⭈⭈ ⭈⭈⭈ 0 1 .. .. .. . . .

冧

.

(37)

p⫺m







0 ⭈⭈⭈ ⭈⭈⭈ 0 1

冧

m

m The straightforward solution of the optimization problem in the preceding equation is uoptkk⫹兩kp⫺1兩k ⫽ Juoptkk⫹兩km⫺1兩k ⫽ J[(I ⫺ P)TR(I ⫺ P) ⫹ JTHTWHJ]⫺1[JTHTWe[k]

(38)

⫺ ((I ⫺ P)TRQ ⫹ JTHTWG)ukk⫺⫺n1 ], where e[k] ⫽ ˆ ySP [k] ⫺ y[k], and the input u[k] that will eventually be implemented will be







u[k] ⫽ [1 0 ⭈ ⭈ ⭈ 0]uoptkk⫹兩kp⫺1兩k . p Therefore, the controller is a linear time-invariant controller, and no online optimization is needed. Linear control theory, for which there is a vast literature, can equivalently be used in the analysis or design of unconstrained MPC (Garcı´a and Morari, 1982). A similar result can be obtained for several MPC variants, as long as the objective function in Eq. (4). remains a quadratic function of uoptkk⫹兩kp⫺1兩k and the process model in Eq. (22) remains linear in uoptkk⫹兩kp⫺1兩k . Incidentally, notice that the appearance of the measured process output y[k] in Eq. (22) introduces the measurement information needed for MPC to be a feedback controller. This is in the spirit of classical linear optimal control theory, in which the controlled

MODEL PREDICTIVE CONTROLLERS

145

process state x[k] contains the feedback information needed by the controller. Whether one performs the analysis and design using directly a closed form of MPC such as in Eq. (4) or its equivalent on-line optimization form, Eq. (38), is a matter of convenience in the translation of engineering requirements into equations. For example, Eq. (38) can be used to determine the poles of the controller, and consequently, the closed-loop behavior (e.g., stability, zero offset). On the other hand, Eq. (4) can be directly used to help tune the MPC system. For example, it intuitively makes it clear that the ‘‘smaller’’ the matrix R, the faster the closed loop will be, at the risk of approaching instability. Similarly, the process output y will track step changes in the setpoint ySP if the prediction horizon length p is long enough. An overview of MPC within a linear control framework can be found in Mosca (1995). Clarke and co-workers have used the term generalized predictive control (GPC) to describe an essentially unconstrained MPC algorithm (Clarke et al., 1987a, 1987b; Bitmead et al., 1990). The situation is quite different when inequality constraints are included in the MPC on-line optimization problem. In the sequel, we will refer to ‘‘inequality constrained MPC’’ simply as ‘‘constrained MPC.’’ For constrained MPC, no closed-form (explicit) solution can be written. Because different inequality constraints may be active at each time, a constrained MPC controller is not linear, making the entire closed loop nonlinear. To analyze and design constrained MPC systems requires an approach that is not based on linear control theory. We will present the basic ideas in Section III. We will then present some examples that show the interesting behavior that MPC may demonstrate, and we will subsequently explain how MPC theory can conceptually simplify and practically improve MPC.

III. Stability A. WHAT IS STABILITY? The concept of stability is central in the study of dynamical systems. Loosely speaking, stability is a dynamical system’s property related to ‘‘good’’ long-run behavior of that system. Although stability by itself may not necessarily guarantee satisfactory performance of a dynamical system, it is not conceivable that a dynamical system may perform well without being stable. Stability can be quantified in several different ways, each providing insight into particular aspects of a dynamical system’s behavior. Mathemati-

146

MICHAEL NIKOLAOU

cal descriptions of the system and its surroundings are necessary for quantitative results. Two broad classes of stability definitions are associated with (a) stability with respect to initial conditions and (b) input–output stability, respectively. The two classes are complementary to each other and can also be combined. For linear systems the two classes are, in general, equivalent. However, they are different (although interrelated) for nonlinear dynamical systems. Next, we make these ideas precise and illustrate their implications through a number of examples. The discussion encompasses both discreteand continuous-time systems. Full details of pertinent mathematical underpinnings can be found in standard textbooks on nonlinear systems, such as Vidyasagar (1993). 1. Stability with Respect to Initial Conditions Consider a system described by the vector difference equation x[k ⫹ 1] ⫽ f(k, x[k]), k ⱖ 0,

(39)

where x : Z⫹ 씮 ᑬn and f : Z⫹ ⫻ ᑬn 씮 ᑬn. Without loss of generality, the vector 0 is assumed to be an equilibrium point of the system (39). The system is said to be stable around the equilibrium point 0 if the state x eventually returns to 0 when its initial value is anywhere within a small neighborhood around 0. The preceding statement is accurately captured in the following definitions. Definition 1—Stability. The equilibrium point 0 at time k0 of Eq. (39) is said to be stable at time k0 if, for any ␧ ⬎ 0, there exists a 웃(k0 , ␧) ⬎ 0 such that 储x[k0]储 ⬍ 웃(k0, ␧) ⇒ 储x[k]储 ⬍ ␧, ᭙k ⱖ k0 .

(40)

Definition 2—Uniform Stability. The equilibrium point 0 at time k0 of Eq. (39) is said to be uniformly stable over [k0 , 앝) if, for any ␧ ⬎ 0, there exists a 웃(␧) ⬎ 0 such that

冎

储x[ᐉ]储 ⬍ 웃(␧) ⇒ 储x[k]储 ⬍ ␧, ᭙k ⱖ ᐉ. ᐉ ⱖ k0

(41)

Definition 3—Asymptotic Stability. The equilibrium point 0 at time k0 of Eq. (39) is said to be asymptotically stable at time k0 if (a) it is stable at time k0 and (b) there exists a 웃(k0) ⬎ 0 such that 储x[k0]储 ⬍ 웃(k0) ⇒ lim 储x[k]储 ⫽ 0. k씮앝

(42)

MODEL PREDICTIVE CONTROLLERS

147

Definition 4—Uniform Asymptotic Stability. The equilibrium point 0 at time k0 of Eq. (39) is said to be uniformly asymptotically stable over [k0 , 앝) if (a) it is uniformly stable over [k0 , 앝) and (b) there exists a 웃 ⬎ 0 such that

冎

储x[ᐉ]储 ⬍ 웃 ⇒ lim 储x[k]储 ⫽ 0. ᐉ ⬎ k0 k씮앝

(43)

Definition 5—Global Asymptotic Stability. The equilibrium point 0 at time k0 of Eq. (39) is said to be globally asymptotically stable if lim 储x[k]储 ⫽ 0

k씮앝

(44)

for any x[k0]. Remarks • Although there are no requirements on the magnitude of 웃 that appears in the above definitions, in practice 웃 is desired to be as large as possible, to ensure the largest possible range of initial states that eventually go to 0. • It is not important what particular norm 储*储 in ᑬn is used in Eqs. (40) and (41), because any two norms 储*储a and 储*储b in ᑬn are equivalent, i.e., there exist positive constants k1 and k2 such that k1储x储a ⱕ 储x储b ⱕ k2储x储a for any x 僆 ᑬn. We will see that the choice of particular norm is important in input–output stability. • As the preceding definitions imply, for a system not to be stable around 0, it is not necessary for the system to produce signals that grow without bounds. The following example illustrates the case. Example 1—Unstable System with Bounded Output. For the feedback system described by the equation x[k ⫹ 1] ⫽ 0.5x[k] ⫹ Sat(2x[k]), x[0] ⫽ 0.01,

(45)

where the saturation function is defined as

Sat( y) ⫽

冦

1

if y ⬎ 1

y

if ⫺1 ⱕ y ⱕ 1,

⫺1 if y ⬍ ⫺1

(46)

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MICHAEL NIKOLAOU

FIG. 2. A bounded-output system that is unstable with respect to initial conditions (x[0] ⫽ 0.01).

the point 0 is an unstable equilibrium point, because x moves away from 0 for any x[0] ⬆ 0, but it does not ever grow without bound, as Fig. 2 shows. Example 2—Unstable CSTR with bounded output. Consider the reaction R 씮 P occurring in a nonisothermal jacket-cooled continuous stirred tank reactor (CSTR) with three steady states, A, B, C, corresponding to the intersection points of the two lines shown in Fig. 3 (Stephanopoulos,

FIG. 3. The three steady states of the nonisothermal CSTR in Example 2.

MODEL PREDICTIVE CONTROLLERS

149

1984, p. 8). The steady state B corresponding to the temperature T2 is unstable with respect to initial conditions. Indeed, if the CSTR is initially at temperature T2 ⫹ ␧, then it will eventually reach either of the finite temperatures T1 or T3 , according to whether ␧ is negative or positive.

2. Input-Output Stability Input–output stability refers to the effect of system inputs to system outputs. Therefore, the emphasis is on how the magnitudes of inputs and outputs are related. First, we make these ideas of magnitude, system, and stability more precise. Definition 6—p-norms. The magnitude of a signal x : Z⫹ 씮 ᑬn is quantified by its p-norm, defined as 储x储p ⫽ ˆ

冉冘 앝

k⫽1

储x[k]储p

冊

1/p

,

(47)

where 1 ⱕ p ⱕ 앝 and 储x[k]储 can be any Euclidean norm in ᑬn. Based on this definition, we can provide a first definition of stability. Definition 7—Bounded-input–bounded-output (BIBO) stability. A system S, mapping an input signal u to an output signal x with S(0) ⫽ 0, is stable if bounded inputs produce bounded outputs, i.e., 储u储p ⬍ 앝 ⇒ 储x储q ⬍ 앝.

(48)

An alternative statement of Eq. (48) is u 僆 l pm ⇒ x 僆 l qm ,

(49)

l pm ⫽ ˆ 兵z: Z 씮 ᑬm兩 储z储p ⬍ 앝其.

(50)

where

A usual convention is to chose p ⫽ q in Definition 7, although different values for p and q may be selected. For example, the option p ⫽ 앝, q ⫽ 2 may be selected so that step inputs (for which 储u储2 ⫽ 앝 but 储u储앝 ⬍ 앝) can be included. Although Definition 7 is useful in characterizing instability in a meaningful way, it is not always as useful in characterizing stability in a meaningful way as well, as the following example shows.

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MICHAEL NIKOLAOU

Example 3—BIBO stability. Consider the system of Fig. 4. A pulse disturbance d of any amplitude drives the output y of the system to the final steady-state value 2, while the input u reaches a steady state value of 1, and the variable v reaches a steady-state value of 4. Figure 5 shows the response of the system for pulses d of amplitude 1 (diamond), 0.1 (square), and 0.01 (triangle). It is clear that the system output y does not return to the desired value of 0, so the system should not be characterized as stable. Yet the BIBO stability criterion with the 앝-norm would characterize this system as stable, because 储y储앝 ⬍ 앝 for any input d. Of course, a different norm 储y储p with 1 ⱕ p ⬍ 앝 might be used, in which case the system would be characterized as BIBO unstable. The compromise would be that systems generating signals such as y[k] ⫽ 1/(k ⫹ 1)1/p would also be BIBO unstable, because, 储y储pp ⫽

冘 冉(k ⫹11) 冊 ⫽ 앝, 앝

p

1/p

k⫽0

although lim y[k] ⫽ 0. A better definition of stability would require not k씮앝

only that bounded inputs produce bounded outputs, but also that the amplification of bounded inputs by the system be finite. More precisely, we have the following definition of finite-gain stability. Definition 8—Finite-gain (FG) stability. A system S : lpm 씮 lqn : u 씮 x ⫽ ˆ Su is finite-gain stable if the gain (induced norm) of S is finite, i.e., 储x储q ⬍ 앝. u僆l pm⫺兵0其 储u储p

储S储i,pq ⫽ ˆ sup

(51)

The advantage of FG stability over BIBO stability is that systems such as in Example 3 no longer have to be characterized as stable, a conclusion that agrees with intuition. Indeed, for Example 3 we have that pulses of infinitesimally small amplitude drive the output y to 2; consequently, 储S储i,앝앝 ⫽ ˆ sup

d僆l1앝⫺兵0其

储y储앝 2 ⱖ lim ⫽ 앝. 储d储앝 储d储앝씮0 储d储앝

FIG. 4. The feedback system of Example 3.

(52)

MODEL PREDICTIVE CONTROLLERS

151

FIG. 5. A system that is input–output BIBO stable.

The shortcoming of Definition 8 is that the input signal u can vary over the entire space lpm . This creates two problems: (a) The entire space lpm may contain physically meaningless signals (b) The stability characteristics of S may be different over different subsets of lpm .

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MICHAEL NIKOLAOU

The following two examples clarify the above statements. Example 4—Selecting physically meaningful inputs to characterize stability. Consider a continuous stirred-tank heater modeled by the following equations, in continuous time:

冋

册

UA(Tc ⫺ T (t)) dT 1 (FS ⫹ u(t))(Ti ⫺ T (t)) ⫹ ⫽ dt V ␳c p y(t) ⫽ T (t) ⫺ Ti ⫺

(53)

UA/ ␳cp (Tc ⫺ Ti), FS ⫹ UA/ ␳cp

where V ⫽ heater volume Fs ⫽ volumetric feed flowrate at steady state Ti ⫽ feed temperature T ⫽ heater temperature Tc ⫽ heating coil temperature U ⫽ heat transfer coefficient A ⫽ heat exchange area

␳ ⫽ density of liquid in heater cp ⫽ specific heat of liquid in heater. The above equation defines an operator S : u 哫 y, where u refers to the feed flowrate and y to temperature, both in deviation form. It can be shown (Nikolaou and Manousiouthakis, 1989) that 储 y储앝 ⫽ 앝, u僆l앝⫺兵0其 储u储앝

储S储i,앝앝 ⫽ ˆ sup

(54)

1

where the supremum is attained for u(t) ⫽ ⫺FS ⫺

UA UA ⇒ F (t) ⫽ ⫺ , t ⱖ 0. ␳cp ␳cp

(55)

This suggests that a negative flowrate F(t) would result in instability. However, since the flowrate is always nonnegative, this instability warning is of

153

MODEL PREDICTIVE CONTROLLERS

little value. In fact, computation of the gain of S over the set W ⫽ ˆ 兵u兩umin ⱕ u(t) ⬍ 앝, where ⫺Fs ⱕ umin, yields UA (Tc ⫺ Ti) ␳cp 储y储앝 ⫽ 储S储i,앝앝,W ⫽ ˆ sup u僆W⫺兵0其 储u储앝 UA ⫹ FS ⫹ umin ␳cp

冉

冊

2

⬍ 앝,

(56)

implying that the system is indeed FG stable for all physically meaningful inputs, as one would intuitively expect. Example 5—Stability dependence on the set of inputs. Consider a continuous stirred-tank reactor modeled by the following equations, in continuous time: dcA F (t) ⫽ (CAi ⫺ CA(t)) ⫺ CA(t)움e⫺E/RT(t) dt V dT F (t) ⌬ HR Q ⫽ (Ti ⫺ T (t)) ⫺ C (t)움e⫺E/RT(t) ⫺ dt V ␳c p A V␳ c p u(t) ⫽ F (t) ⫺ FS y(t) ⫽ T (t) ⫺ TS , where F ⫽ volumetric feed flowrate V ⫽ CSTR volume ⫽ 1.36 m3 CA ⫽ concentration of A in CSTR CAi ⫽ concentration of A in inlet stream ⫽ 8008 mol/m3 움 ⫽ kinetic constant ⫽ 7.08 ⫻ 107 1/hr E/R ⫽ activation energy/gas constant ⫽ 8375 K T ⫽ temperature in CSTR Ti ⫽ inlet temperature ⫽ 373.3 K ⌬HR ⫽ heat of reaction ⫽ ⫺69,775 j/mol

␳ ⫽ density of liquid in CSTR ⫽ 800.8 kg/m3 cp ⫽ specific heat of liquid in CSTR ⫽ 3140 j/kg ⫺ K Q ⫽ heat removal rate ⫽ 1.055 ⫻ 108 j/hr

(57)

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MICHAEL NIKOLAOU

冦

Fs, t ⬍ 0

FIG. 6. Response of CSTR to the flowrate pulse F(t) ⫽ 1.5Fs, 0 ⱕ t ⬍ 25. Fs, t ⱖ 25

The reactor has three steady states. Eigenvalue analysis of the linearized system around the steady state corresponding to FS ⫽ 1.133 m3 /hr TS ⫽ 547.6 K CAs ⫽ 393.2 mol/m

(58) 3

reveals that the preceding steady state is locally stable with respect to initial conditions. Consequently, it is input–output stable for inputs of small enough magnitude. To determine what ‘‘small enough’’ means in the previous sentence requires careful analysis. For example, single pulse changes in u of magnitude ⫺0.5FS or ⫹0.5FS do not reveal any instabilities (the CSTR returns to the original steady state, Figs. 6 and 7), but successive

冦

Fs, t ⬍ 0

FIG. 7. Response of CSTR to the flowrate pulse F(t) ⫽ 0.5Fs, 0 ⱕ t ⬍ 25. Fs, t ⱖ 25

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MODEL PREDICTIVE CONTROLLERS

pulse changes of magnitudes ⫺0.5FS and then ⫹0.5FS drive the CSTR to instability (储S储i,22 ⫽ 앝), as Fig. 8 demonstrates. Therefore, this CSTR is not input–output stable for u bounded in the interval [⫺0.5FS, 0.5FS ]. Choi and Manousiouthakis (1997) have introduced the concept of finite-gain/ initial conditions stability, to combine the insights provided by each of the above two kinds of stability: Definition 9—Finite-gain/initial conditions stability. A system S : l pm 씮 lqn : u 哫 x ⫽ ˆ Su is finite-gain stable over the set U for initial conditions s[0] in the set S if the following inequality holds: sup

u僆U s[0]僆S

储x储q ⬍ 앝. 储u储p

(59)

The advantage of this definition is that it gives a complete characterization of the stability behavior of a system. Its disadvantage is that the computation of the left-hand side in Eq. (59) is not trivial.

B. IS STABILITY IMPORTANT? As Section III,A emphasized, stability is a fundamental property of a dynamic system that summarizes the long-term behavior of that system. There are two important implications of this statement: 1. MPC controllers should result in closed loops that are stable. Therefore, if an optimal MPC system is to be designed, only candidates from a set of stabilizing MPC controllers should be considered in the design. Ideally, the set of all stabilizing MPC controllers should be known. That set

冦

Fs, t ⬍ 0

FIG. 8. Response of CSTR to the flowrate pulses F(t) ⫽ 0.5Fs, 0 ⱕ t ⬍ 25. 1.5Fs, t ⱖ 25

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MICHAEL NIKOLAOU

is difficult to determine for constrained MPC, given the richness allowed in the structure of constrained MPC. However, one can find subsets of the set of all stabilizing MPC controllers with constraints. Selecting MPC controllers from such a subset can have significant implications on closedloop performance, as Section V,C demonstrates. For unconstrained MPC controllers with linear models, which are equivalent to linear, time-invariant controllers as shown in Section II,C, the set of all controllers that can stabilize a given linear plant can be explicitly parametrized in terms of a single stable transfer function through the celebrated Youla parametrization (Vidyasagar, 1985). For stable plants, the Youla parametrization is the same as the internal model control (IMC) structure (Morari and Zafiriou, 1989). 2. The discussion in Section III,A is most relevant for continuous processes, for which operating time can theoretically extend to infinity. For batch processes, operating time is finite; consequently, stability should be examined in a different framework. For example, instability that would generate signals that grow without bounds might not be detrimental, provided that the rate of growth is very small with respect to the batch cycle time.

IV. The Behavior of MPC Systems The examples that follow attempt to expose some MPC pitfalls. All examples are intentionally kept as simple as possible, to easily expose the theoretical issues associated with each one of them. Because of that simplicity, the issues highlighted in each example could be easily addressed by intuitive design (tuning) improvements of the corresponding MPC systems. Our purpose, however, is not to use these examples to test the ultimate capabilities of MPC. Large multivariable systems, for example, would be better suited for that purpose. Our intention is to help the reader understand the theoretical issues associated with these examples by easily following some of the associated calculations.

A. FEASIBILITY OF ON-LINE OPTIMIZATION Because MPC requires the solution of an optimization problem at each time step, the feasibility of that problem must be ensured. For example, the optimization problem posed in Eqs. (14)–(17) may be infeasible. If the on-line optimization problem is not feasible, then some constraints would

MODEL PREDICTIVE CONTROLLERS

157

have to be relaxed. Finding what constraints to relax in order to get a feasible problem with optimal deterioration of the objective function is extremely difficult, since it is an np-hard problem. A possible (and partial) remedy to the problem is to consider constraint softening variables ␧ on process output constraints, e.g., ymax ⫹ ␧ ⱖ y[k ⫹ i兩k] ⱖ ymin ⫺ ␧, i ⫽ 1, ⭈ ⭈ ⭈ , p,

(60)

and include a penalty term such as ␧2 in the objective function. Feasibility, in addition to being a practical consideration, is also important for closedloop stability of MPC. In fact, algorithms have been developed by Mayne and co-workers that merely require the existence of a feasible instead of optimal solution of the on-line optimization problem, to guarantee closedloop stability of an MPC system.

B. NONMINIMUM PHASE AND SHORT HORIZONS Example 6—Closed-loop stability for a nonminimum-phase process. Consider a nonminimum-phase process described by the equation y [k] ⫽ h1u[k ⫺ 1] ⫹ h2u[k ⫺ 2] ⫹ h3u[k ⫺ 3] ⫹ h4u[k ⫺ 4] ⫹ d [k], (61) with h1 ⫽ 0 (dead time), h2 ⫽ ⫺1 (inverse response), h3 ⫽ 2, h4 ⫽ 0 (Genceli and Nikolaou, 1993; Genceli, 1993). Assume no modeling uncertainty. For that process, consider the constrained MPC on-line optimization

冘 兩 y [k ⫹ i兩k] ⫺ y 兩 ⫹ 冘 r 兩⌬u[k ⫹ i兩k]兩, p

min

u[k兩k],⭈⭈⭈,u[k⫹p⫺1兩l] i⫽1

m

SP

i

(62)

i⫽0

with p ⫽ 3 and m ⫽ 1, subject to the constraints umin ⫽ ⫺0.2 ⱕ u[k ⫹ i兩k] ⱕ 0.2 ⫽ umax . This optimization can be trivially transformed to linear programming. A step disturbance equal to ⫺0.05 and a step setpoint change equal to 0.05 enter the closed loop at time k ⫽ 0. For move suppression coefficient values r0 ⫽ r1 ⬍ 0.5, the resulting closed-loop response is shown in Fig. 9. The closed loop is clearly unstable. If the move suppression coefficients take values r0 ⫽ r1 ⱖ 0.5 (to penalize the move suppression coefficients even more) the closed loop remains unstable, as shown in the Fig. 10. The instability is due to the nonminimum phase characteristics of the process. Although a longer optimization horizon length, p, might easily solve the

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MICHAEL NIKOLAOU

FIG. 9. Closed-loop response for the process of Example 6, with r0 ⫽ r1 ⬍ 0.5.

problem, a simple remedy can also be obtained by considering the following end constraint on u in the on-line optimization: u[k ⫹ i兩k] ⫽

ySP ⫺ y[k] ⫹ 兺jN⫽1 gj u[k ⫺ j ] for all i ⱖ m. 兺jN⫽1 gj

(63)

Here, gj are model estimates of the coefficients hj. The meaning of Eq. (63) is that the value of the process input u at the end of its horizon should correspond to a steady-state value that would produce zero steady-state offset,





d [k ⫹ 앝兩k]



冉

y SP ⫺ u[k ⫹ i兩k]兺jN⫽1 gj ⫹

y[k] ⫺ 兺jN⫽1 gj u[k ⫺ j ]

冊

,

FIG. 10. Closed-loop response for the process of Example 6, with r0 ⫽ r1 ⱖ 0.5.

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159

for the process model output y[k ⫹ 앝兩k]. The closed-loop response for move suppression coefficient values r0 ⫽ r1 ⫽ 2.7 is shown in Fig. 11. It turns out that these values for the move suppression coefficients are sufficient for robust stability of the closed loop, when the modeling errors e1, e2, e3, e4 for the coefficients h1, h2, h3, h4 are bounded as 兩 e1 兩 ⫽ 兩 h1 ⫺ g1 兩ⱕ 0.12, 兩 e2 兩 ⫽ 兩 h2 ⫺ g2 兩ⱕ 0.10, 兩 e3 兩 ⫽ 兩 h3 ⫺ g3 兩ⱕ 0.08, 兩 e4 兩 ⫽ 兩 h4 ⫺ g4 兩ⱕ 0.05 (Genceli and Nikolaou, 1993). The key to achieving stability was the end constraint, Eq. (63). It turns out that inclusion of an end constraint of the type x[ p] ⫽ 0

(64)

in the on-line optimization performed by MPC is a convenient way to generate a controller structure for which stability can be easily shown (Section V,A).

C. INTEGRATORS AND UNSTABLE UNITS With the dimension of multivariable MPC systems ever increasing, the probability of dealing with a MIMO process that contains an integrator or an unstable unit also increases. For such units FIR models, as used by certain traditional commercial algorithms such as dynamic matrix control (DMC), is not feasible. Integrators or unstable units raise no problems if state-space or DARMAX model MPC formulations are used. As we will discuss later, theory developed for MPC with state-space or DARMAX models encompasses all linear, time-invariant, lumped-parameter systems and consequently has broader applicability.

FIG. 11. Closed-loop response for the process of Example 6, with r0 ⫽ r1 ⫽ 2.7, end condition enforced.

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MICHAEL NIKOLAOU

In contrast to constrained MPC of stable plants, constrained MPC of unstable plants has the complication that the tightness of constraints, the magnitude and pattern of external signals, and the initial conditions all affect the stability of the closed loop. The following simple example illustrates what may happen with a simple unstable system. Example 7—Stability regions. Consider the unstable plant P(s) ⫽ ⫺(s ⫹ 1)/(s ⫺ 3) controlled by the P controller C(s) ⫽ 2. The controller output is constrained between ⫺1 and 1. A disturbance, d, is added to the controller output, to create the final input to the plant. The accompanying figures show responses to three different disturbances. The response in Fig. 12, resulting from the step disturbance d ⫽ 0.5, clearly corresponds to unstable closed-loop behavior. The response in Fig. 13, resulting from a smaller step disturbance d ⫽ 0.26, shows a bounded plant output. However, one cannot say that disturbances of amplitude 0.26 or smaller result in stable closed-loop behavior. Indeed, as Fig. 15 shows, the plant response to the pulse disturbance of amplitude 0.26, shown in Fig. 14, is clearly unstable. The above simple example makes it clear that characterization of external inputs that do not destabilize a closed loop is by no means trivial, when the plant is unstable. Similar comments can be made about dependence of

FIG. 12. Plant response to step disturbance d ⫽ 0.5 at t ⫽ 1 (k ⫽ 10) for Example 7.

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161

FIG. 13. Plant response to step disturbance d ⫽ 0.26 at t ⫽ 1 (k ⫽ 10) for Example 7.

stability on the initial state of the system. For the case of a linear, unstable system with bounded inputs and without external disturbances, Zheng and Morari (1995) have developed an algorithm that can determine the domain of attraction for the initial state of the system.

FIG. 14. Pulse disturbance for Example 7.

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MICHAEL NIKOLAOU

FIG. 15. Output for pulse disturbance in Example 7.

D. NONLINEARITY MPC systems that employ nonlinear models may exhibit increased complexity due to two main factors: (a) Nonlinear programming, required for the solution of the MPC online optimization problem, does not produce exact solutions, but rather solutions that are optimal within a certain prespecified precision tolerance, or even locally optimal, if the optimization problem is nonconvex. (b) Even if the global optimum of the on-line optimization problem is assumed to be exactly reached, MPC behavior may show patterns that would not be intuitively expected. For instance, Rawlings et al. (1994) discuss two simple examples of MPC applied to nonlinear systems, where the state feedback law turns out to be a discontinuous function of the state, either because of stability requirements, or due to the structure of MPC. As a result, standard stability results that rely on continuity of the feedback law cannot be employed. Example 8—A nonlinear process that cannot be stabilized by a continuous feedback law. In the first example, the following two-state, one-input

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MODEL PREDICTIVE CONTROLLERS

system is considered: x1[k ⫹ 1] ⫽ x1[k] ⫹ u[k] x2[k ⫹ 1] ⫽ x2[k] ⫹ u[k]3

(65)

x1[0], x2[0] given Meadows et al. (1995) showed that the following MPC nonlinear program, corresponding to a moving horizon of length 3, results in a closed loop that is globally asymptotically stable around the equilibrium point xe ⫽ (0, 0).(Recall that an equilibrium point xe is globally asymptotically stable if x[k] 씮 xe as k 씮 앝 for any initial point x[0].)

冘 (x [k ⫹ i兩k] ⫹ x k ⫹ i兩k] ⫹ u[k ⫹ i兩k] ) 2

min

u[k兩k],u[k⫹1兩k],u[k⫹2兩k] i⫽0

1

2

2

2

2

(66)

subject to the terminal constraint x1[k ⫹ 3兩k] ⫽ x2[k ⫹ 3兩k] ⫽ 0.

(67)

Meadows et al. (1995) also showed that horizons of length less than 3 cannot globally asymptotically stabilize this system, while horizons of length larger than 3 will result in less aggressive control action. The control law u(x) for the horizon of length 3 turns out to be continuous at the origin, but has discontinuity points away from the origin. In fact, no continuous state feedback law can stabilize the system of Eqs. (65). To show that, following Meadows et al. (1995), first note that any stabilizing control law must allow both positive and negative input values for x. If the control is strictly positive, trajectories originating in the first quadrant move away from the origin under positive control action. If the control is strictly negative, trajectories originating in the third quadrant also move away from the origin. Yet u(x) cannot be identically zero for any nonzero x. If it were, then this x would be a fixed point of the dynamic system and trajectories containing this x would not converge to the origin. We have the situation in which the feedback control law must assume both negative and positive values away from the origin, yet must be zero nowhere away from the origin. Therefore, the control law must be discontinuous. Example 9—A finite prediction horizon may not be a good approximation of an infinite one for nonlinear processes. In the second example, consider the single-state, single-input system x[k ⫹ 1] ⫽ x[k]2 ⫹ u[k]2 ⫺ (x[k]2 ⫹ u[k]2)2

(68)

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MICHAEL NIKOLAOU

with the MPC controller

冘 (x [k ⫹ i兩k] ⫹ u[k ⫹ i兩k] ) 1

2

min

u[k兩k],u[k⫹1兩k] i⫽0

2

(69)

subject to the terminal constraint x[k ⫹ 2兩k] ⫽ 0,

(70)

which is feasible if the initial state is restricted such as 兩x[k]兩 ⱕ 1. The control law resulting from this optimization, Eqs. (69) and (70), is u(x) ⫽

再

0 ⫾兹1 ⫺ x2

x⫽0 , 0 ⬍ 兩x兩 ⱕ 1

(71)

resulting in an optimal cost Jopt(x) ⫽

再

0 1

x⫽0 . 0 ⬍ 兩x兩 ⱕ 1

(72)

Both u(x) and Jopt(x) are discontinuous at the origin. Therefore, stability theorems that rely on continuity cannot be used. Yet, it is simple to check by inspection that the feedback law of Eq. (71) (with either sign chosen) is asymptotically stabilizing. However, the continuous feedback control law u(x) ⫽ 0, resulting in the closed-loop system x[k ⫹ 1] ⫽ x[k]2 ⫺ x[k]4

(73)

is asymptotically stabilizing for initial conditions in [⫺1, 1]. The actual closed-loop cost incurred using this feedback control law is ⌺k앝⫽0 x[k]2. It turns out (Rawlings et al., 1994) that for initial conditions in [⫺1, 1], the closed-loop cost of the zero control action is always less than that for the optimal MPC controller with fixed horizon. Since the actual incurred cost is calculated over an infinite horizon, it is reasonable to ask whether the minimum cost of the finite horizon MPC problem would approach the cost incurred using the zero controller as the horizon length tends to infinity. The answer is negative. In the finite horizon MPC on-line optimization problem we require that the terminal constraint x[p] ⫽ 0 be satisfied, where p is the horizon length. Because of the structure of the problem, the closedloop MPC cost is 1 for all horizon lengths. This example demonstrates that the intuitive idea of using the terminal constraint x[p] ⫽ 0 and a large value of p in order to approximate the desired infinite horizon behavior, an idea that works for linear systems, does not work in general for nonlinear systems.

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MODEL PREDICTIVE CONTROLLERS

E. MODEL UNCERTAINTY Example 10—Ensuring robust stability of a heavy oil fractionator. Vuthandam et al. (1995) considered the top 2 ⫻ 2 subsystem of the heavy oil fractionator modeled in the Shell Standard Process Control Problem (Prett and Garcia, 1988) as

冤

4.05e⫺27s 50s ⫹ 1 P(s) ⫽ 5.39e⫺18s 50s ⫹ 1

冥

1.77e⫺28s 60s ⫹ 1 . 4.05e⫺27s 60s ⫹ 1

(74)

Discretization for a sampling period of 4 min yields a corresponding discrete-time model that is used in the MPC on-line objective function J [k] ⫽ ˆ

冘 冘 ( y [k ⫹ i兩k] ⫺ y ⫹ 冘 ␧ [k ⫹ i兩k] 2

7

j

SP 2 j )

j⫽1 i⫽1 7

1

⫹

冘 冘 r ⌬u [k ⫹ i兩k] 2

3

ji

j

j⫽1 i⫽0

2

2

(75)

i⫽1

with the constraints ⫺3 ⱕ ⌬u2[k] ⱕ 3 ⫺5 ⱕ u2[k] ⱕ 5 ⫺0.5 ⫺ ␧1[k ⫹ i兩k] ⱕ y1[k ⫹ i兩k] ⱕ 0.5 ⫹ ␧1[k ⫹ i兩k],

(76)

with setpoints SP y SP 1 ⫽ y2 ⫽ 0

(77)

and step disturbances d1 ⫽ 1.2 d2 ⫽ ⫺0.5.

(78)

In addition to these constraints, the end constraint u[k ⫹ m ⫹ i兩k] ⫽ G⫺1(ySP ⫺ d[k兩k]), i ⱖ 0,

(79)

is considered, where G is the steady-state gain of the process. Simulation of this system verified the robust stability analysis of the above authors, as shown in Table I and the corresponding figures (Figs. 16, 17, and 18). F. FRAGILITY Because MPC relies on the numerical solution of an on-line optimization problem, it may find a solution to that problem that is not exactly equal to the

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MICHAEL NIKOLAOU TABLE I

End condition, Eq. (79) Not used

Used

Used

Input move suppression coefficients r10 r11 r12 r13 r10 r11 r12 r13 r10 r11 r12 r13

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

r20 r21 r22 r23 r20 r21 r22 r23 r20 r21 r22 r23

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

0.10 0.12 0.12 0.12 0.06 0.07 0.07 0.07 10.82 11.15 11.46 11.86

Closed-loop behavior Unstable

Unstable

Robustly stable (rigorous proof)

FIG. 16. Closed-loop response for Example 10, Case 1.

FIG. 17. Closed-loop response for Example 10, Case 2.

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167

FIG. 18. Closed-loop response for Example 10, Case 3.

expected theoretical solution. Is closed-loop stability going to be adversely affected by that discrepancy between the theoretically expected MPC behavior and the actual (numerical) MPC behavior? An affirmative answer was given by Keel and Bhattacharya (1997), who demonstrated, by example, that there are linear time-invariant stabilizing controllers for which extremely small variations of their coefficients may render the controllers destabilizing, even though the controllers may nominally satisfy optimality criteria such as H2 ,H앝 ,l1 , or 애, as well as robustness criteria. Borrowing from these authors, consider the following example: Example 11—Sensitivity of closed-loop stability to small variations in controller parameters. For the stable transfer function ⫺s ⫹ 1 , s ⫹s⫹2

(80)

q6s6 ⫹q5s5 ⫹ q4s4 ⫹ q3s3 ⫹ q2s2 ⫹ q1s ⫹ q0 p 6 s6 ⫹ p 5 s5 ⫹ p 4 s4 ⫹ p 3 s3 ⫹ p 2 s2 ⫹ p 1 s

(81)

P(s) ⫽

2

the optimal controller C (s) ⫽

is designed by minimizing a weighted H2 norm of the closed-loop transfer function. The values of the controller parameters are given in Table II. The poles of the resulting closed loop are well in the left half-plane, as supported by the Nyquist plot of P(s)C(s) shown in Fig. 19. Therefore the nominal closed-loop is stable. Yet a small change ⌬p in the nominal controller parameters p, such that 储⌬p储2 ⫽ 3.74 ⫻ 10⫺6, 储p储2

(82)

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MICHAEL NIKOLAOU TABLE II Parameter

Value

Parameter

Value

q6 q5 q4 q3 q2 q1 q0

1.0002 3.0406 8.1210 13.2010 15.2004 12.08 4.0

p6 p5 p4 p3 p2 p1

0.0001 1.0205 2.1007 5.1403 6.06 2.0

can destabilize the closed loop. For example, if ⌬p ⫽ 10⫺4 ⫻ [⫺0.321 ⫺0.009 0.002 0.000 ⫺0.000 ⫺0.000 0.000 ⫺1.000 0.332 0.005 ⫺0.002 ⫺0.000 0.000]T, (83) then the closed loop has a pole at 앑 108. This problem of extreme sensitivity of closed-loop stability to small variations in controller parameters has been termed fragility. Given that unconstrained MPC with quadratic cost and linear model is equivalent to a linear time-invariant controller, as demonstrated in Section II,C it is clear that similar fragility problems may appear with MPC as well. Fragility might have a more realistic probability of being an instability threat in constrained MPC, where the results of on-line optimization may not be exact, such as in the case of nonlinear programming with multiple optima or with equality constraints. Fragility problems may even emerge in computer implementation of control algorithms where floating-point arithmetic introduces a truncation (round-off) error (Williamson, 1991). Of course, MPC controllers would have to be robust with respect to plant uncertainty, which is usually orders of magnitude larger than controller

FIG. 19. Nyquist plot of P(s)C(s) for Example 11.

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169

uncertainty. From that viewpoint, controller fragility would be an issue of practical significance if small controller uncertainty could cause instability for plants close to or at the boundary of the set of uncertain plants considered in controller design.

G. CONSTRAINTS Zafiriou (1991) used a number of examples to demonstrate that the presence of constraints can have a dramatic and often counterintuitive effect on MPC stability properties and can render tuning rules developed for stability or robustness of unconstrained MPC incorrect. The following examples show how the addition of constraints to a robustly stable unconstrained MPC system can lead to instabilities. Example 12. Consider the process p(s) ⫽

e⫺0.15s s⫹1

(84)

p˜(s) ⫽

1 . s⫹1

(85)

modeled by

A sampling period of 0.1 is used. The following MPC system is used to control the process: min y[k ⫹ 1兩k]2 ⫹ 0.16⌬u[k兩k]2

(86)

⫺1 ⱕ y[k ⫹ 1兩k] ⱕ 1.

(87)

⌬u[k]

Two points are to be made: (a) Although for step disturbances d ⱕ 1.70 the output y returns to the setpoint ySP ⫽ 0, for step disturbances d ⱖ 1.75 the output oscillates with amplitude that grows without bound. Therefore, unlike the case of linear systems, the stability characteristics of this constrained MPC system depend on the magnitude of external disturbances. (b) Perhaps counterintuitively, relaxing the controller by removing the output constraint, Eq. (87), can be shown to result in a linear timeinvariant controller that robustly stabilizes the closed loop for disturbances of any amplitude.

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MICHAEL NIKOLAOU

Example 13. Consider the process p(s) ⫽ p˜(s) ⫽

⫺s ⫹ 1 . (s ⫹ 1)(2s ⫹ 1)

(88)

A sampling period of 0.3 and an FIR model with n ⫽ 50 coefficients are used. The following MPC system is used to control the process:

冘 y[k ⫹ i兩k] p

min

2

⌬u[k兩k],⭈⭈⭈,⌬u[k⫹m⫺1兩k] i⫽1

⫺0.3 ⱕ y[k ⫹ 2兩k] ⱕ 0.3.

(89) (90)

If the output constraint, Eq. (4), were not present, then the choice m ⫽ 1 and a sufficiently large p ⱖ n ⫹ m would stabilize the closed loop, in the absence of process/model mismatch. However, the presence of the output constraint destabilizes the closed loop. As p 씮 앝, the closed loop largest root approaches 1.45, for m ⫽ 1, and 2.63, for m ⫽ 2. Again, the presence of output constraints destabilizes the closed loop instead of tightening control.

V. A Theory for MPC with Predictable Properties A. STABILITY Because no equivalent linear time-invariant controller exists for a constrained MPC controller, determining the stability of a constrained MPC closed-loop system cannot rely on methods suitable only for linear systems, such as determination of the poles of the closed-loop transfer matrix. Developments have shed light on the stability of constrained MPC. The seeds of proving stability for constrained MPC were already planted by Economou (1985), who postulated, before his life reached a tragic end, that closed-loop stability for constrained MPC could be established using the contraction mapping theorem. Mayne and Michalska (1990) and Rawlings and Muske (1993), through seminal papers, developed working frameworks for establishing closed-loop stability of continuous-time and discrete-time constrained MPC systems, respectively. These works provided the right tools for the study of a control problem that had long been considered difficult. Following is a discussion of pertinent ideas. 1. MPC with Linear Model: A Prototypical Stability Proof Consider the following state-space description of a linear, time-invariant (not necessarily stable) process:

MODEL PREDICTIVE CONTROLLERS

171

x[k ⫹ 1] ⫽ Ax[k] ⫹ Bu[k].

(91)

Here, x 僆 ᑬ x, u 僆 ᑬ u. We assume perfect knowledge of the matrices A and B, full state information x[k], and absence of disturbances. The MPC objective at time k is n

n

min

u[k兩k],⭈⭈⭈,u[k⫹p⫺1兩k]

J [k],

(92)

where J [k] ⫽ ˆ

冘 x[k ⫹ i兩k] Wx[k ⫹ i兩k] ⫹ 冘 u[k ⫹ i ⫺ 1兩k] Ru[k ⫹ i ⫺ 1兩k]. p

p

T

T

i⫽1

i⫽1

(93) W is a positive definite matrix and R is a positive semidefinite matrix. State and input constraints are G[k ⫹ i ⫺ 1]u[k ⫹ i ⫺ 1兩k] ⱕ g[k ⫹ i ⫺ 1], i ⫽ 1, ⭈ ⭈ ⭈ , p

(94)

H[k ⫹ i]x[k ⫹ i兩k] ⱕ h[k ⫹ i], i ⫽ 1, ⭈ ⭈ ⭈ , p.

(95)

These constraints are assumed to define nonempty (convex) regions containing the point (0, 0). Closed-loop MPC stability can be established using the following Lyapunov argument (Rawlings et al., 1994). Assume that G[k ⫹ i ⫺ 1], g[k ⫹ i ⫺ 1], H[k ⫹ i], and h[k ⫹ i] are independent of k and i. Consider a solution Uopt kk⫹兩kp⫺1兩k ⫽ 兵uopt[k兩k], ⭈ ⭈ ⭈ , uopt[k ⫹ p ⫺ 1兩k]其

(96)

to Eq. (92) at time k, and assume that p is large enough so that x[p] ⫽ 0. Consider the following candidate for control input at time k ⫹ 1: Ukk⫹⫹p1兩兩kk⫹⫹11 ⫽ 兵uopt[k ⫹ 1兩k], ⭈⭈⭈, uopt[k ⫹ p ⫺ 1兩k], 0其.

(97)

Ukk⫹⫹p1兩兩kk⫹⫹11 is feasible at time k ⫹ 1, because it contains inputs u that satisfied the same constraints at time k. This feasible input results in a value of the objective function J [k ⫹ 1] that satisfies J [k ⫹ 1] ⫽ Jopt[k] ⫺ x[k]TWx[k] ⫺ u[k]TRu[k].

(98)

Because of optimality, this equation yields Jopt[k ⫹ 1] ⱕ J [k ⫹ 1] ⫽ Jopt[k] ⫺ x[k]TWx[k] ⫺ u[k]TRu[k] ⱕ Jopt[k],

(99)

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MICHAEL NIKOLAOU

where the last inequality results from the positive semidefiniteness of W and positive definiteness of R. Therefore, the sequence 兵J[k]其k앝⫽k0 is nonincreasing. It is also bounded from below by 0. Consequently, the sequence 兵J[k]其k앝⫽k0 converges, i.e., limk씮앝J[k] ⫽ a. To show that a ⫽ 0, rearrange Eq. (99), to get x[k]TWx[k] ⫹ u[k]T Ru[k] ⱕ Jopt[k] ⫺ Jopt[k ⫹ 1] ⇒ lim (x[k]TWx[k] ⫹ u[k]T Ru[k]) ⱕ lim Jopt[k] ⫺ lim Jopt[k ⫹ 1] ⫽ 0 k씮앝

k씮앝

k씮앝

⇒ lim x[k] ⫽ 0, lim u[k] ⫽ 0, k씮앝

k씮앝

and closed-loop stability is proven.

2. MPC with Nonlinear Model The Lyapunov-like stability proof presented in Section V, A, 1 can be extended to nonlinear systems. Because nonlinear systems are defined by what they are not (namely, linear), the framework of the ensuing proof must be precisely defined. Borrowing from Rawlings et al. (1994) and Meadows et al. (1995), we consider dynamical systems of the form x[k ⫹ 1] ⫽ f(x[k], u[k]),

(100)

where f : ᑬn ⫻ ᑬm 씮 ᑬn is continuous and f(0,0) ⫽ 0. The state x is assumed to be measurable. The MPC online optimization problem becomes min

u[k兩k],⭈⭈⭈,u[k⫹p⫺1/k]

I [k],

(101)

where I[k] ⫽ ˆ

冘 L(x[k ⫹ i兩k], u[k ⫹ i兩k]),

p⫺1

(102)

i⫽0

subject to Eq. (101) and the state, input, and terminal constraints x[k ⫹ i兩k] 僆 X [k ⫹ i], i ⫽ 0, ⭈⭈⭈, p ⫺ 1,

(103)

u[k ⫹ i兩k] 僆 U [k ⫹ i], i ⫽ 0, ⭈⭈⭈ , p ⫺ 1.

(104)

The function L: ᑬn ⫻ ᑬm 씮 ᑬ that appears in Eq. (102) is assumed to satisfy the following properties: L(0, 0) ⫽ 0

(105)

MODEL PREDICTIVE CONTROLLERS

There exists a nondecreasing function 웂: [0, 앝) 씮 [0, 앝) such that 웂(0) ⫽ 0 and 0 ⬍ 웂(储x, u储) ⱕ L(x, u) for all (x, u) ⬆ (0, 0).

173

(106)

These lead to the following additional properties of L: L(x, u) ⬎ 0 ᭙(x, u) ⬆ (0, 0)

(107)

L(x, u) ⫽ 0 ⇔ (x, u) ⫽ (0, 0)

(108)

L(x, u) 씮 0 ⇒ (x, u) 씮 (0, 0).

(109)

Notice that the function J in Eq. (93) satisfies all of these conditions. As in the linear case, a proof for closed-loop stability can be constructed, if it can be guaranteed that x[k ⫹ p兩k] ⫽ 0.

(110)

Equation (110) can be satisfied if the moving horizon length, p, is chosen to be ‘‘large enough,’’ or if the constraint in Eq. (110) is directly incorporated in the on-line optimization problem. In either case, a closed-loop stability proof can be constructed as follows. a. A Prototypical Stability Proof for MPC with Nonlinear Model. As in the linear case, we assume perfect knowledge of f, full state information x[k], and absence of disturbances. The constraints are assumed to define nonempty (convex) regions containing the point (0, 0). Assume also that X[k⫹i] and U[k⫹i] are independent of k and i. Assume that there exists a solution Uoptkk⫹兩kp⫺1兩k] ⫽ 兵uopt[k兩k], ⭈ ⭈ ⭈ , uopt[k ⫹ p ⫺ 1兩k]其

(111)

to Eq. (101) at time k, corresponding to x[p] ⫽ 0. Consider the following candidate for control input at time k⫹1: Ukk⫹⫹p1兩兩kk⫹⫹11 ⴝ 兵uopt[k ⫹ 1兩k], ⭈ ⭈ ⭈ , uopt[k ⫹ p ⫺ 1兩k], 0其.

(112)

Ukk⫹⫹p1兩兩kk⫹⫹11 is feasible at time k ⫹ 1, because it contains inputs u that satisfied the same constraints at time k, the point (0, 0) has been assumed to be feasible, and f is assumed to be known perfectly. This feasible input results in a value of the objective function I[k⫹1] that satisfies Iopt[k ⫹ 1] ⱕ I [k ⫹ 1] ⫽ Iopt[k] ⫺ L(x[k], u[k]) ⱕ Iopt[k],

(113)

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MICHAEL NIKOLAOU

where the last inequality results from the positive semidefiniteness of L. Therefore, the sequence 兵I[k]其k앝⫽k0 is nonincreasing. It is also bounded from below by 0. Consequently, the sequence 兵I[k]其k앝⫽k0 converges to a limit b. To show that b ⫽ 0, rearrange Eq. (113) to get 0 ⱕ L(x[k]), u[k]) ⱕ Iopt[k] ⫺ Iopt[k ⫹ 1] ⇒ 0 ⱕ lim L(x[k], u[k]) ⱕ lim (Iopt[k] ⫺ Iopt[k ⫹ 1]) ⫽ 0 k씮앝

k씮앝

⇒ lim L(x[k], u[k]) ⫽ 0 k씮앝

⇒ lim (x[k], u[k]) ⫽ (0, 0), k씮앝

where the last equality follows from property (109) of the function L. This completes the proof of closed-loop stability.

3. The Stability Proof and MPC Practice The preceding prototypical stability proofs are relatively simple. They rely on the inequality 0 ⱕ (L(x[k], u[k]) ⱕ) Iopt[k] ⫺ Iopt[k ⫹ 1].

(114)

Proving this inequality hinges on a number of assumptions, of which the following are important: (i) The on-line optimization problem, Eq. (92) or (101), is feasible. (ii) The state can be steered to the setpoint in at most p steps, i.e., x[p] ⫽ 0. (iii) The controlled process model A and B in Eq. (91) or f in Eq. (100) is perfect. (iv) There are no external disturbances. (v) The state x is measurable. (vi) The input and state constraints, Eq. (94) and (95) or (103) and (104), are time-independent. (vii) The global optimum of the on-line optimization problem, including the terminal constraint x[k ⫹ p兩k] ⫽ 0, can be computed exactly. For stable processes, MPC practitioners have traditionally ensured that assumptions (i) and (ii) are satisfied by (a) selecting large enough p and (b) performing the optimization with respect to u[k兩k], . . . , u[k⫹m兩k], where m Ⰶ p.

MODEL PREDICTIVE CONTROLLERS

175

Rawlings and Muske (1993) have shown that this idea can be extended to unstable processes. In addition to guaranteeing stability, their approach provides a computationally efficient method of on-line implementation. Their idea is to start with a finite control (decision) horizon but an infinite prediction (objective function) horizon, i.e., m ⬍ 앝 and p ⫽ 앝, and then use the principle of optimality and results from optimal control theory to substitute the infinite prediction horizon objective by a finite prediction horizon objective plus a terminal penalty term of the form x[p]TPx[p],

(115)

corresponding to the optimal value of the truncated part of the original objective function. Chen and Allgo¨wer (1996) have presented an extension of this idea to MPC with nonlinear model and input constraints. They compute the terminal penalty term off-line as the solution of an appropriate Lyapunov equation. De Nicolao et al. (1998) present a similar idea for nonlinear time-varying systems. Genceli and Nikolaou (1995) have shown how to ensure feasibility and subsequently ensure robust stability for nonlinear MPC with Volterra models. Selecting large enough p is not the only way to guarantee the two assumptions (i) and (ii). One could directly include an end constraint x[p] ⫽ 0 in the on-line optimization problem, an idea proposed by several investigators (Kleinman, 1970; Thomas, 1975; Keerthi and Gilbert, 1988; Mayne and Michalska, 1990). This constraint does not pose any serious computational challenges in on-line implementation. Other options are also possible, based, for example, on constraining x(p) to belong to a small neighborhood of the set point (Mayne, 1997) or on state contraction arguments (Morari and de Oliveira, 1997; Mayne, 1997). Unstable processes pose the additional challenge that stabilization is possible only if the state x(k) lies in a certain domain, so that, even though the input may be constrained [Eq. (5)] enough control action can be available. If the state is not in the stabilizability domain, then nothing can be done to steer the state to the setpoint. The feasibility of state constraints is a common issue (see, e.g., Theorem 1 in Rawlings et al., 1994). For example, when simple output constraints have to be satisfied, such as in Eq. (7), then it might occur that not enough control action is available, because of constraints such as in Eq. (5). If such infeasibility is detected, one can use additional relaxation variables to modify output constraints as ymax ⫹ ␧ ⱖ y[k ⫹ i兩k] ⱖ ymin ⫺ ␧, i ⫽ 1, ⭈ ⭈ ⭈ , p, and add a term q␧2 to the objective function in Eq. (4).

(116)

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MICHAEL NIKOLAOU

The stability proof can be extended to handle bounded external disturbances [to address assumption (iv)] by additional ‘‘bookkeeping,’’ although the results may be conservative. Alternatively, one may introduce an integrator in the process output and show stability for an integrating system without disturbance.2 These issues and their implications for improving MPC are discussed in Section VI.

B. ROBUST STABILITY AND FRAGILITY OF CONSTRAINED MPC To show inequality (114), the preceding assumptions (i) through (vii) were made. When assumptions (i), (iv), and (v) are not satisfied, robustness issues arise, because the process behaves differently from the assumptions of the controller. When assumption (vi) is not satisfied, then fragility issues arise, because the controller behaves differently from the way it was designed. It should be mentioned that the issue of fragility is not confined to constrained MPC systems. Keel and Bhattacharyya (1997) showed that even in linear time-invariant control systems, there are controllers for which extremely small deviations of the controller parameter values from their designed values can result in instability (see Section IV,F).

1. Robust Stability To ensure robustness of stability in the presence of model uncertainty, one has to make sure that the inequality (114) is satisfied, even though the matrices A and B in Eq. (91) or the function f in Eq. (100) are not perfectly known. The inequality (114) also has to be satisfied when external disturbances enter the system. Two approaches have appeared in the literature. (a) Choose the parameters of the standard MPC on-line objective function I [k] (e.g., the matrices Q and R in J [k] for the linear case) in such a way that 0 ⱕ Iopt[k] ⫺ Iopt[k⫹1]. In other words, robustness is attempted through better tuning of MPC, with minor modifications of the MPC structure. (b) Modify the on-line MPC optimization problem, by adding constraints that help stabilize the closed-loop system, without critical 2 For example, the FIR model of Eq. (10) can be substituted by y[k] ⫽ y[k ⫺ 1] ⫹ ⌺jn⫽1 hiu[k⫺j], thereby eliminating the step disturbance d.

177

MODEL PREDICTIVE CONTROLLERS

dependence on tuning. For example, add to the on-line MPC optimization problem an additional constraint of the type 0 ⱕ Iopt[k ⫺ 1] ⫺ I [k].

(117)

The advantage of the first approach is that the form of the standard MPC optimization problem is retained. Its disadvantage is that the online objective may become unrealistically conservative, depending on the magnitude of process model uncertainty. The advantage of the second approach is that the on-line objective may be formulated in a way that reflects the true control objective, without regard to stability, because the latter is enforced by the constraint of Eq. (117). Its disadvantage is that the MPC on-line optimization problem may become complicated. a. MPC Tuning for Robust Stability. Consider the following MPC on-line optimization problem for an open-loop stable, multi-input–multi-output (MIMO) system, modeled by a finite-impulse-response (FIR) model: min

␧[k⫹1兩k],⭈⭈⭈,␧[k⫹nw兩k],⌬u[k兩k],⭈⭈⭈,⌬u[k⫹m兩k]

J [k],

(118)

where

冘 冘 ⫹冘 冘 r ⌬u [k ⫹ i兩k] , no

J [k] ⫽ ˆ

p

vj

j⫽1

i⫽1

ni

2 ( yj [k ⫹ i兩k] ⫺ y SP j ) ⫹

冘 冘 ␧ [k ⫹ i兩k] no

nw

wj

j⫽1

j

2

(119)

i⫽1

m

i,j

2

j

j⫽1 i⫽0

subject to the following: Process output prediction y[k ⫹ i兩k] ⫽

冘 G u[k ⫹ i ⫺ j兩k] ⫹ d[k ⫹ i兩k], i ⫽ 1, ⭈ ⭈ ⭈ , p N

[j]

j⫽1

(120) Disturbance prediction d[k ⫹ i兩k] ⫽ y[k] ⫺

冘 G u[k ⫺ j ], i ⫽ 1, ⭈ ⭈ ⭈ , p N

[j]

(121)

j⫽1

Input move constraints ⫺⌬umax ⱕ ⌬u[k ⫹ i兩k] ⱕ ⌬umax , i ⫽ 0, ⭈ ⭈ ⭈ , m (122)

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MICHAEL NIKOLAOU

Input constraints umin ⱕ u[k ⫹ i兩k] ⱕ umax , i ⫽ 0, ⭈ ⭈ ⭈ , m (123) Softened output constraints ymin ⫺ ␧[k ⫹ i兩k] ⱕ y[k ⫹ i兩k] ⱕ ymax ⫹ ␧[k ⫹ i兩k], i ⫽ 1, ⭈ ⭈ ⭈ , nw (124) End constraints u[k ⫹ m ⫹ i兩k] ⫽

冉冘 冊

⫺1

N

G[j]

j⫽1

(ySP ⫺ d[k ⫹ m ⫹ i兩k]), i ⱖ 0, (125)

where nj is the number of process inputs; no is the number of process outputs; nw is the number of inputs, nw is the number of time steps over which output constraints are enforced; G[j] are matrices of the FIR coefficients of the process model. The real process output is assumed to be y[k] ⫽

冘 H u[k ⫺ j ] ⫹ d[k]. N

[ j]

(126)

j⫽1

Notice that the model kernel 兵G[j]其jN⫽1 , is different from the true kernel 兵H[j]其jN⫽1 , with the modeling error bounded as 兩H[j] ⫺ G[j]兩 ⱕ E[j] max .

(127)

External disturbances are assumed to be bounded as dmin ⱕ d[k] ⱕ dmax

(128)

⫺⌬dmax ⱕ ⌬d[k] ⱕ ⌬dmax ,

(129)

and where ⌬dmax

再

ⱖ 0, k ⱕ M ⫽0

k⬎M

.

(130)

For this MPC system, Eqs. (118) through (125), Vuthandam et al. (1995) developed sufficient conditions for robust stability with zero offset. These conditions can be used directly for calculation of minimum values for the prediction and control horizon lengths, p and m, respectively, as well as for the move suppression coefficients rji , which are not equal over the

MODEL PREDICTIVE CONTROLLERS

179

finite control horizon. Since the robust stability conditions are sufficient, they are conservative, particularly for very large modeling uncertainty bounds. The proof relies on selecting p, m, and rji to satisfy the inequality 0 ⱕ J ⬘opt[k] ⫺ J ⬘opt[k ⫹ 1]

(131)

with

冘 v ( y [k] ⫺ y ) ⫹ 冘 v 冘 ( y [k ⫹ i兩k] ⫺ y ⫹ 冘 w 冘 ␧ [k ⫹ i兩k] ⫹ 冘 冘 r ⌬u [k ⫹ i兩k] no

J⬘[k] ⫽ ˆ

j

SP 2 j

j

j⫽1

no

p

j

j

j⫽1

i⫽1

no

nw

j

j⫽1 ni

SP 2 j )

2

j

(132)

i⫽1

m

i,j

j

2

j⫽1 i⫽⫺N⫹1

⫹ f [k],

where the function f[k] is an auxiliary function that helps prove stability, as required by Eq. (136). Satisfaction of inequality (131) implies that the sequence 兵J⬘opt[k]其k앝⫽k0 is convergent. Then, the end condition, Eq. (125), is used to show that the sequence 兵J⬘opt[k]其k앝⫽k0 converges to 0. The proof starts with the inequality 0 ⱕ J⬘[k ⫹ 1] ⫺ J⬘opt[k ⫹ 1] ⇔ J⬘opt[k] ⫺ J⬘[k ⫹ 1] ⱕ J⬘opt[k] ⫺ J⬘opt[k ⫹ 1], (133) where J⬘[k⫹1] corresponds to a set of input values 兵⌬u[k ⫹ 1兩k ⫹ 1], ⭈ ⭈ ⭈ ⌬u[k ⫹ m兩k ⫹ 1], ⌬u[k ⫹ m ⫹ 1兩k ⫹ 1]其 to be selected such that 0 ⱕ J⬘opt[k] ⫺ J⬘[k ⫹ 1].

(134)

For the feasible input set ⌬u[k ⫹ 1兩k ⫹ 1] ⫽ ⌬uopt[k ⫹ 1兩k] ... ⌬u[k ⫹ m兩k ⫹ 1] ⫽ ⌬uopt[k ⫹ m兩k] ⌬u[k ⫹ m ⫹ 1兩k ⫹ 1] ⫽ to be determined for feasibility,

(135)

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MICHAEL NIKOLAOU

inequality (134) becomes

冘 v (y [k] ⫺ y ) ⫺ 冘 v ( y [k ⫹ 1 ⫹ p兩k ⫹ 1] ⫺ y ) ⫹ 冘 v [(y [k ⫹ 1兩k] ⫺ y ) ⫺ (y [k ⫹ 1] ⫺ y ) ] ⫹ 冘 v 冘 [( y [k ⫹ i兩k] ⫺ y ) ⫺ ( y [k ⫹ i兩k ⫹ 1] ⫺ y ⫹ 冘 w 冘 [␧ [k ⫹ i兩k] ⫺ ␧ [k ⫹ i兩k ⫹ 1] ] ⫹ 冘 冘 (r ⫺ r )⌬u [k ⫹ i兩k] no

0ⱕ

j

SP 2 j

j

j⫽1

no

j

SP 2 j

j

j⫽1 no

j

SP 2 j

j,opt

SP 2 j

j

j⫽1 no

p

j

i⫽2

no

nw

j

j⫽1 ni

SP 2 j

j,opt

j⫽1

2

j,opt

j

SP 2 j ) ]

(136)

2

j

i⫽1

m

i⫺1, j

i,j

j⫽1 i⫽⫺N⫹1

j,opt

2

⫹ f [k] ⫺ f [k ⫹ 1]. All terms in this inequality, after lengthy manipulations and strengthening of inequalities, can be expressed in terms of inputs ⌬u squared. The resulting expression is of the form

冘冘 ni

0ⱕ

m

j⫽1 i⫽⫺N⫹1

(ri,j ⫺ ri⫺1,j ⫹ ai, j)⌬uj,opt[k ⫹ i兩k]2,

(137)

where the positive constants ai, j depend on the model, uncertainty bounds, and input bounds. For that inequality to be true it is sufficient to have ri⫺1, j ⱕ ri, j ⫹ ai,j

(138)

r⫺N,j ⫽ 0.

(139)

with

Note that the inequality in Eq. (138) implies that weights of the input move suppression term containing ⌬u gradually increase. Details can be found in Vuthandam et al. (1995) and Genceli (1993). A similar result can be found in Genceli and Nikolaou (1993) for MPC with l1-norm based online objective. Variations for various MPC formulations have also been presented. Zheng and Morari (1993) and Lee and Yu (1997) have pre-

MODEL PREDICTIVE CONTROLLERS

181

sented results on MPC formulations employing on-line optimization of the form min max J [k, u, p], u

(140)

p

where the vector p refers to process model parameters that are uncertain. The idea of Eq. (140) is that the optimal input for the worst possible process model is computed at each time step k.

b. Modifying the MPC Algorithm for Robust Stability

(i) Stability Constrained MPC. Instead of tuning the MPC algorithm for robust stability, an inequality of the type (117) can be added to the online optimization, to ensure that the MPC controller automatically belongs to a set of stabilizing controllers. Through this modification, the designer can concentrate on formulating an on-line optimization objective that reflects the intended control objective. Cheng and Krogh (1996) developed that idea in what they call stability constrained receding horizon control. While their results are for unconstrained MPC, extension of their algorithm to constrained MPC is trivial. Following their work, consider a plant in controllable form x[k ⫹ 1] ⫽ Ax[k] ⫹ Bu[k]

(141)

where A⫽ ˆ

冋

0(N⫺M)⫻M I(N⫺M)

B⫽ ˆ

A1

冋

A2

册 冋 册

0(N⫺M)⫻M

x[k] ⫽ ˆ

B2

x(1)[k]

x(2)[k]

册

(142)

(143)

(144)

with B2⑀ᑬM⫻M invertible, x(1)[k] ⑀ ᑬM, x(2)[k] ⑀ ᑬN⫺M. The stability constrained receding horizon control algorithm is given by the following steps. At time step k, minimize an objective function H[k]

(145)

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MICHAEL NIKOLAOU

over a finite horizon of length p, subject to x[k ⫹ i ⫹ 1兩k] ⫽ Ax[k ⫹ i兩k] ⫹ Bu[k ⫹ i兩k], i ⫽ 0, ⭈ ⭈ ⭈ p ⫺ 1 x[k兩k] ⫽ x[k] 储x[k ⫹ i ⫹ 1兩k]储2 ⱕ (1 ⫺ ␴k)lˆk , i ⫽ 0, ⭈ ⭈ ⭈ , p ⫺ 1

␴k ⱖ ␧ k

(146)

(stability constraint) (147)

ˆlk ⫽max兵lk , 储x[k]储2其 lk ⫽

再

arbitrary, k ⫽ 0

max兵储x[k ⫺ 1 ⫹ i兩k ⫺ 1]储2, i ⫽ 1, ⭈ ⭈ ⭈ , p其, k ⱖ 1

储x(1)[k]储2 ␧k ⫽ clˆk

(148)

c ⱖ 1. Cheng and Krogh (1996) give a stability proof for the above algorithm. They extend their algorithm to include state estimation in Cheng and Krogh (1997). (ii) Robust-Stability-Constrained MPC. Badgwell (1997) has taken the idea of stability-constrained MPC a step further by developing an MPC formulation in which a constraint used in the on-line optimization problem guarantees robust stability of closed-loop MPC for stable linear processes. The trick, again, is to make sure that an inequality of the type (117) is satisfied for all possible models that describe the controlled process behavior. The set of these models is assumed to be known during controller design. Following Badgwell (1997), consider that the real process behavior is described by the stable state-space equations ¯ x[k] ⫹ B¯u[k] x[k ⫹ 1] ⫽ A

(149)

¯ , B¯) are where x[k] 僆 ᑬ , u[k] 僆 ᑬ , and the process parameters ␪¯ ⫽ ˆ (A not known exactly, but are known to belong to a set n

M

⍀⫽ ˆ 兵␪1 , ⭈ ⭈ ⭈ , ␪pm其 ⫽ 兵(A1 , B1), ⭈ ⭈ ⭈ , (Apm , Bpm)其

(150)

of pm distinct models. A nominal model ˜ , B˜) ␪˜ ⫽ (A

(151)

is used. For robust asymptotic stability, the state should be driven to the origin, while satisfying input, input move, and state constraints. No external

183

MODEL PREDICTIVE CONTROLLERS

disturbances are considered. Under these assumptions, the robust-stabilityconstrained MPC algorithm is as follows: min

␧[k⫹1兩k],⭈⭈⭈,␧[k⫹nw兩k],u[k 兩 k],⭈⭈⭈,u[k⫹m兩k]

⌫(x˜[k], 兵u[k ⫹ i ⫺ 1兩k]其ii⫽⫽앝1 , ␪˜ , 兵␧[k ⫹ i ⫺ 1兩k]其ii⫽⫽n1 w , Tk), (152)

where ⌫(x˜[k], 兵u[k ⫹ i ⫺ 1兩k]其ii⫽⫽앝1 , ␪˜ , 兵␧[k ⫹ i ⫺ 1兩k]其ii⫽⫽n1 w, Tk) ⫽ ˆ

冘 x˜[k ⫹ i兩k] Wx˜[k ⫹ i兩k] ⫹ 冘 u[k ⫹ i ⫺ 1兩k] Ru]k ⫹ i ⫺ 1兩k] 앝

i⫽1

⫹

앝

T

T

i⫽1

k⫹n 兩k k⫹n 兩k (ek⫹1兩wk )TT[k](ek⫹1兩wk ),

(153)

subject to the following: Process state prediction ˜ x˜[k ⫹ i ⫺ 1兩k] ⫹ B˜u[k ⫹ i ⫺ 1兩k], x˜[k ⫹ i兩k] ⫽ A x˜[k兩k] ⫽ x[k]

(154)

Input move constraints ⌬umin ⱕ ⌬u[k ⫹ i兩k] ⱕ ⌬umax , i ⫽ 0, ⭈ ⭈ ⭈ , m

(155)

Input constraints umin ⱕ u[k ⫹ i兩k] ⱕ umax , i ⫽ 0, ⭈ ⭈ ⭈ , m ⫺ 1 u[k ⫹ i兩k] ⫽ 0, i ⱖ m

(156)

Softened state constraints xmin ⫺ ␧[k ⫹ i兩k] ⱕ xᐉ[k ⫹ i兩k] ⱕ xmax ⫹ ␧[k ⫹ 兩k], ᐉ ⫽ 0, ⭈ ⭈ ⭈ , pm , i ⫽ 1, ⭈ ⭈ ⭈ , nw (157) ␧[k ⫹ i兩k] ⱖ ␧min ⬎ 0 Robust stability constraints ⌫(x[k], 兵u[k ⫹ i ⫺ 1兩k]其ii⫽⫽앝1 , ␪ᐉ , 兵␧[k ⫹ i ⫺ 1兩k]其ii⫽⫽n1 w, Tk) ⱕ ⌫(x[k], 兵uopt[k ⫹ i ⫺ 1兩k ⫺ 1]其ii⫽⫽앝1 , ␪ᐉ , 兵␧opt[k ⫹ i ⫺ 1兩k ⫺ 1]其ii⫽⫽n1 w , Tk) ᐉ ⫽ 0, ⭈ ⭈ ⭈ , pm ,

(158)

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where xᐉ[k ⫹ i兩k] ⫽ Aᐉxᐉ[k ⫹ i ⫺ 1兩k] ⫹ Bᐉu[k ⫹ i ⫺ 1兩k], xᐉ[k兩k] ⫽ x[k] T[k] ⫽

(159)

(ekk⫹兩 kn⫺w1⫺1兩k⫺1)ToptT(k ⫺ 1](ekk⫹兩 kn⫺w1⫺1兩k⫺1)opt T[k ⫺ 1] eˆTT[k ⫺ 1]eˆ

eˆ ⫽ arg min储e ⫺ (ek兩⫹k⫺w1⫺ k n

1兩k⫺1

)opt储

subject to

(160)

e ⱖ emin ⬎ 0 xmin ⫺ e ⱕ xopt[k ⫹ i兩k ⫺ 1] ⱕ xmax ⫹ e, i ⫽ 1, ⭈ ⭈ ⭈ , nw , and ⌬umin ⬍ 0 ⬍ ⌬umax , umin ⬍ 0 ⬍ ⌬umax , xmin ⬍ 0 ⬍ xmax . The overall optimization problem is convex and has a feasible solution; therefore, it is guaranteed to have a unique optimal solution. The model linearity assumption, Eq. (154), is not critical. Badgwell (1997) has shown how these ideas can be readily extended to the case of stable nonlinear plants. A critical assumption in the preceding formulation is Eq. (150), which assumes that a set of distinct models captures modeling uncertainty. Ideally, one would like to have a continuum of models such that the real plant is one point in that continuum. The continuum could be approximated by considering a very large number of distinct models, with the obvious trade-off of increase in the dimensionality of the on-line optimization problem. 2. Fragility The stability proofs developed in the previous sections implicitly assume that an exact solution of the MPC on-line optimization can be obtained. However, an exact solution may not always be obtained in cases such as the following: • The on-line optimization problem is nonconvex; therefore, guarantees for reaching the global optimum may be hard to obtain. • The on-line optimization problem is nonlinear and involves equality constraints. Satisfaction of those constraints is not exact, but approximate (within ␧). In the first of these two cases, a local optimum may be obtained that is far from the global optimum. In such a case, stability analysis based on attain-

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ment of global optimum would entirely break down. In the second case, if the on-line optimization problem is convex, then the solution found numerically would be close to the exact solution. It might then be concluded that stability analysis would be valid, at least for small error in the approximation of the exact MPC system by the one approximately (numerically) computed on-line, provided that continuity arguments would be valid. It turns out, however, that this is not necessarily true. Keel and Bhattacharyya (1997) have shown that there exist linear time-invariant fragile controllers, i.e., such that closed-loop stability is highly sensitive to variations in controller parameters. In that context, the fragility properties of MPC should be rigorously examined. A number of authors (see, for example, Scokaert et al., 1998) have developed MPC variants and corresponding stability proofs, which overcome the above two problems by (a) requiring that the on-line optimization reaches a feasible (suboptimal) solution of a corresponding problem, and/or (b) substituting equality constraints of the type f(x) ⫽ 0

(161)

by inequality constraints of the type ⫺웃 ⱕ f(x) ⱕ 웃,

(162)

where 웃 is a vector with ‘‘small’’ entries. This ensures that the endconstraint can be satisfied exactly and, consequently, stability analysis can be rigorously valid.

C. PERFORMANCE AND ROBUST PERFORMANCE Rigorous results for the performance of constrained MPC are lacking. However, there are a number of propositions on how the performance of MPC could be improved. Such propositions rely on (a) modifying the structure of MPC for robust performance, (b) tuning MPC for robust performance, and (c) developing efficient algorithms for the numerical solution of the MPC on-line optimization problem, thus enabling the formulation of more complex and realistic on-line optimization problems that would in turn improve performance. The expected results of these propositions are difficult to quantify. Nevertheless, the proposed ideas have intuitive appeal and appear to be promising. One proposition is to formulate MPC in the closed-loop optimal feedback form (see Section II, B). The main challenge of this proposition is the

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difficulty of solving the on-line optimization problem. Kothare et al. (1996) propose a formulation that reduces the on-line optimization problem to semidefinite programming, which can be solved efficiently using interior point methods. A second proposition relies on the idea that the on-line optimization problem is unconstrained after a certain time step in the finite moving horizon. Where in the finite horizon that happens is determined by examining whether the state has entered a certain invariant set (Mayne, 1997). Once that happens, then closed-form expressions can be used for the objective function from that time point the end of the optimization horizon, p. The idea is particularly useful for MPC with nonlinear models, for which the computational load of the on-line optimization is substantial. A related idea was presented by Rawlings and Muske (1993), where the on-line optimization problem has a finite control horizon length, m, and infinite prediction horizon length, p, but the objective function is truncated, because the result of the optimization is known after a certain time point. Of course, as mentioned earlier, the mere development of more efficient optimization algorithms could indirectly improve performance. This could happen, for example, through the use of nonlinear instead of linear models in on-line optimization. As stated in the Introduction, numerical efficiency issues are beyond the scope of this discussion. A third proposition has been discussed in Section V,B,1,b. The idea is that if a robust stability constraint [Eqs. (147) or (158)], is used, MPC will be stabilizing; therefore, true performance objectives may translated into values for the tuning parameters of MPC, with no need to worry about potential instabilities resulting from poor tuning. However, that translation of performance objectives to values for MPC tuning parameters is not always straightforward. A fourth proposition was discussed by Vuthandam et al. (1995). Their idea is that the values of the MPC tuning parameters must satisfy robust stability requirements. It turns out that for the robust stability requirements developed by these authors, performance improves as the prediction horizon length, p, increases from its minimum value to larger values, but after a certain point performance deteriorates as p increases further. This happens because for very large p the input move terms in the on-line objective function must be penalized so much that the controller becomes very sluggish and performance suffers. Results such as these depend on the form of the robust stability conditions. If such conditions are only sufficient, as is the case with Vuthandam et al. (1995), then performance-related results may be conservative.

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VI. How Can Theory Help Develop Better MPC Systems? A. CONCEPTUAL UNIFICATION AND CLARIFICATION The theoretical developments of the past few years have provided a solid foundation for the analysis and synthesis of constrained MPC systems. We now have a tool, summarized perhaps by Eq. (100), by which we can study the stability (and other important properties) of constrained MPC and its variants in a framework that removes guessing or ambiguity and supports rigorous statements about the expected behavior of MPC. This framework is valuable for both researchers and practitioners of MPC. Of course, as with every theory, practitioners cannot rely solely on theory when they design real MPC systems. Real-world problems have their idiosyncratic peculiarities that may vary from case to case. It is imperative that the MPC designer understand the entire engineering framework within which an MPC system is going to function. But theory could help predict what could or would happen as a result of a specific design, thereby steering the design-in promising directions and reducing reliance on unnecessary improvisation. That help is valuable for the novice and reassuring for the expert. Theory can help augment the designer’s intuition in a very productive way. For example, theory could augment the value or shorten the duration of extensive simulations or plant testing frequently performed before the final commissioning of an MPC system. As another example, the concept of best achievable performance by a constrained MPC system is mentioned. Similarly to the minimum-work concept in thermodynamics, control theory should provide achievable targets of performance and should do so under practical conditions, e.g., in the presence of inequality constraints and model inaccuracies (Morari, 1988). Chmielewski and Manousiouthakis (1996) have provided such a result by proving that the best performance achievable by an MPC systems with input and state inequality constraints and objective function over an infinite horizon can be determined by solving a single, finite dimensional, convex program of known size. For researchers, the existence of a working theoretical framework provides impetus to attack MPC problems with confidence that was lacking before the seminal paper by Rawlings and Muske (1993). Despite the avalanche of recent results capitalizing on the use of Eq. (100), there appears to be significant room for extensions, improvements, and new ideas on MPC. Paralleling, in a way, the introduction and wide spread of other important frameworks in control theory (such as the universally recognized equations x˙ ⫽ Ax ⫹ Bu), the MPC stability framework could have signifi-

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cant implications for developments in control theory for systems different from the chemical process systems for which MPC was first extensively implemented. A factor whose importance for future developments cannot be overstated is the interplay between theory and practice. Abstraction of practical problems spurs theoretical research with industrial relevance, while consistent and systematic exploration of theoretical ideas (a painstaking process) can yield otherwise unimaginable or unexpected breakthroughs.

B. IMPROVING MPC The benefits of a framework for the rigorous study of MPC properties are not confined to the mere proof of MPC properties. More importantly, MPC theory can lead to discoveries by which MPC can be improved. In fact, the proofs of theoretical results frequently contain the seeds for substantial MPC improvements through new formulations. The existence of a theory that can be used to analyze fairly complex MPC systems allows researchers to propose high-performance new formulations whose properties can be rigorously analyzed, at least for working prototypes. The algorithmic complexity of such formulations might be high, but their functionality would also be high. It is reassuring to know that even when the analysis is not trivial, it is definitely feasible. In designing such systems the designer would have theory as an invaluable aid that could augment intuition. Moreover, theory could also provide guidelines for the efficient use of such systems by end users, by helping them predict what the effects of tweaking would be. The algorithmic complexity of on-line optimization would be hidden from the end user. The ensuing discussion shows some open issues and new ideas in constrained MPC. The list is neither complete nor time-invariant. Some of the following ideas were directly inspired by recent MPC theory. Others were developed rather independently, but knowledge of the fact that theory exists that can be used to study them makes those ideas more appealing from both a theoretical and practical viewpoint. 1. Process Models Before discussing issues related to the different classes of models that may be used with MPC, we should stress that the issue of model structure selection and model identification is at the heart of MPC. ‘‘The greatest contribution of Ziegler and Nichols (1942) is not the famous tuning rule but the ‘cycling’ identification technique: they understood precisely what

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minimum model information was necessary for tuning and developed a simple, reliable method for obtaining this information’’ (Morari, 1988). a. Linear vs Nonlinear Models. Although it is well recognized that all physical systems are, in principle, nonlinear (Section II, B), the challenge is when a nonlinear approach is necessary for the solution of a problem. It is then important to answer questions such as the following: • • • • •

When to use nonlinear models? For what processes? What is the structure of such models? What are the properties of MPC when particular models are used? How can such models be developed (from first principles and/or experimental data)? • What is the cost/benefit of using nonlinear models in MPC? b. Input–Output vs State-Space Models. Traditional MPC algorithms, such as the quadratic dynamic matrix control (QDMC) algorithm (Garcı´a and Morshedi, 1986) relied on FIR models. Although FIR models have several advantageous features (section II, B), they are not essential for the characterization of a control algorithm as MPC. DARMAX or state-space models have their own relative advantages and can be used equally well in MPC. With state-space models the issue of state estimation naturally arises. A new class of state estimators, introduced by Robertson et al. (1996), is discussed in the following section. c. Moving Horizon–Based State Estimation for State-Space Models. If a state-space model is used by an MPC system, then the problem of state estimation arises, since, usually, not all states can be measured. State estimation within a stochastic system framework has a rich history, the turning point being the celebrated work of Kalman (1960). The Kalman filter has limitations when applied to nonlinear systems (as the extended Kalman filter). In addition, the Kalman filter cannot explicitly handle constraints on estimated states. Yet, state variables such as mass fraction or temperature are constrained, and corresponding constraints should be satisfied by estimates of these variables. Robertson et al. (1996) presented a new class of state estimators. Their approach to the constrained state estimation problem relies on the moving horizon concept and on-line optimization. States are estimated by considering data over a finite past moving horizon and by minimizing a square-error criterion. In fact, the approach introduced by these authors can estimate both states and model parameters.

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The system whose states and parameters are estimated is represented by the difference equations

冋

x[k ⫹ 1]

册 冋 ⫽

册 冋 册

f(x[k], p[k], u[k])

p[k ⫹ 1] p[k] y[k] ⫽ g(x[k], p[k] ⫹ v[k],

⫹

wx[k]

wp[k]

(163)

where wx , wp , are v are white noise vectors with zero mean, and the vectorvalued vector function f represents the solution x[k ⫹ 1] ⫽ x[k] ⫹

冕

tk⫹1

tk

␾(x(t), p[k], u[k]) dt ⫹

冕

tk⫹1

tk

웆(t) dt

(164)

of the differential equation dx/dt ⫽ ␾(x(t), p[k], u[k]) ⫹ 웆(t). The leastsquares estimate of the state z[k] ⫽ ˆ

冋 册

(165)

w[k] ⫽ ˆ

冋 册

(166)

x[k]

p[k]

and noise vectors wx[k]

wp[k]

and v[k] at time point k is obtained by the following constrained minimization min

z[k⫺m⫹1],v,w

(z[k ⫺ m ⫹ 1] ⫺ z[k ⫺ m ⫹ 1兩k ⫺ m])TP[k ⫺ m ⫹ 1兩k ⫺ m]⫺1

(z[k ⫺ m ⫹ 1] ⫺ z[k ⫺ m ⫹ 1兩k ⫺ m])

⫹

冘 k

i⫽k⫺m⫹1

v[i]TR⫺1v[i] ⫹

冘 k

(167)

w[i]TQ⫺1w[i]

i⫽k⫺m⫹1

subject to the equality constraints of Eq. (163) and the inequality constraints vmin ⱕ v[i] ⱕ vmax , k ⫺ m ⫹ 1 ⱕ i ⱕ k xmin ⱕ x[i] ⱕ xmax , k ⫺ m ⫹ 1 ⱕ i ⱕ k pmin ⱕ p[i] ⱕ pmax , k ⫺ m ⫹ 1 ⱕ i ⱕ k

(168)

wmin ⱕ w[i] ⱕ wmax , k ⫺ m ⫹ 1 ⱕ i ⱕ k ⫺ 1. The matrices P, Q, and R can be interpreted as covariance matrices of corresponding random variables, for which probability density functions are normal subject to truncation of their tail ends, dictated by the constraints of Eq. (168). The on-line optimization problem posed by Eq. (167) can be solved by

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nonlinear programming algorithms, if the model in Eq. (163) is nonlinear, or by standard QP algorithms, for a linear model. Note that the moving horizon keeps the size of the optimization problem fixed, by discarding one old measurement for each new one received. The effect of the initial estimate z[k⫺m ⫹ 1兩k⫺m] becomes negligible as the horizon length m increases. This is the duality counterpart of the MPC moving horizon requirement that the state should reach a desired value at the end of the moving prediction horizon. In fact, the duality between this state estimation approach and MPC parallels the duality between LQR and Kalman filtering. As in the case of MPC, the performance of the proposed approach depends on the accuracy of the model used. d. MPCI: Expanding the MPC/On-Line Optimization Paradigm to Adaptive Control. To maintain the closed-loop performance of an MPC system, it may become necessary to update the process model originally developed off-line. Control objectives and constraints most often dictate that this update has to take effect while the process remains under MPC (Cutler, 1995; Qin and Badgwell, 1997). This task is known as closed-loop identification. Frequently occurring scenarios where closed-loop identification is desired include the following: • Because of equipment wear, a modified process model is needed (without shutting down the process), for tight future control. • The process has to operate in a new regime where an accurate process model is not available, yet the cost of off-line identification experiments for the development of such a model is prohibitively high, thus making closed-loop identification necessary. • Process identification is conducted off-line, but environmental, safety, and quality constraints still have to be satisfied. Closed-loop identification has been addressed extensively in a linear sto˚ stro¨m and Wittenmark, 1989). Good discussions chastic control setting (A of early results from a stochastic control viewpoint are presented by Box (1976) and Gustavsson et al. (1977). Landau and Karimi (1997) provide an evaluation of recursive algorithms for closed-loop identification. Van den Hof and Schrama (1994), Gevers (1993), and Bayard and Mettler (1992) review research on new criteria for closed-loop identification of state space or input–output models for control purposes. The main challenge of closed-loop identification is that feedback control leads to quiescent process behavior and poor conditions for process identification, because the process is not excited (see, for example, Radenkovic and Ydstie, 1995, and references therein). Traditional methods for excitation of a process (So¨derstro¨m et al., 1975; Fu and Sastry, 1991; Van Der

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Klauw et al., 1994; Ljung, 1987, 1993; Schrama, 1992) under closed-loop control through the addition of external dithering signals to the process input or setpoint have the weaknesses that controller performance is adversely affected in a way that may be difficult to predict, because it depends on the very process being identified under closed-loop control. To remedy these problems, Genceli and Nikolaou (1996) introduced the simultaneous Model Predictive Control and Identification (MPCI) paradigm. MPCI relies on on-line optimization over a finite future horizon (Fig. 20). Its main difference from standard MPC is that MPCI employs the well known persistent excitation (PE) condition (Goodwin and Sin, 1984) to create additional constraints on the process inputs in the following kind of on-line optimization problem, solved at each sampling instant k: minimize

[control objective over optimization horizon] (169)

process input values over control horizon

subject to Standard MPC constraints

(170)

Persistent excitation constraints on inputs over finite horizon. (171) The form of the PE constraint depends only on the model structure considered by the identifier and is independent of the behavior of the identified plant. Of course, the model structure should be ‘‘close’’ (but not necessarily contain) the real plant structure. The above formulation defines a new class of adaptive controllers. By

FIG. 20. The MPCI moving horizon. Notice the nonsettling projected plant output and the periodicity of the manipulated input.

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placing the computational load on the computer-based controller that will perform on-line optimization, MPCI greatly simplifies the issue of closedloop model parameter convergence. In addition, constraints are explicitly incorporated in the MPCI on-line optimization. By contrast, most of the existing adaptive control theory requires the controller designer to make demanding assumptions that are frequently difficult to assert. To explain MPCI quantitatively, consider, for simplicity, a single-inputsingle-output (SISO) process modeled as y[k] ⫽

冘 a u[k ⫺ i] ⫹ 冘 b y[k ⫺ 1] ⫹ d [k] m

n

i

i

i⫽1

(172)

i⫽1

⫽ ␾[k ⫺ 1]T␪ ⫹ w[k],

(173)

where y[k] is the process output; u[k] is the process input; d[k] is a constant disturbance, d, plus white noise with zero mean, w[k];

␪⫽ ˆ [a1 ⭈ ⭈ ⭈ amb1 ⭈ ⭈ ⭈ bnd]T

(174)

is the parameter vector to be identified; and

␾[k ⫺ 1]T ⫽ [u[k ⫺ 1] ⭈ ⭈ ⭈ u[k ⫺ m] y [k ⫺ 1] ⭈ ⭈ ⭈ y [k ⫺ n]1]. (175) Using the strong PE condition and Eqs. (169) to (171), one can formulate an MPCI on-line optimization problem at time k as

冘 冋웆 (y [k ⫹ i兩k] ⫺ y M

min

u[k兩k],⭈⭈⭈,u[k⫹M⫺1兩k],애,␧,␳ i⫽1

i

) ⫹ ri⌬u[k ⫹ i ⫺ 1兩k]2]

SP 2

(Objective) (176)

⫹ q 1u2⫹ q 2␧2 ⫺ q 3␳ subject to

umax ⱖ u[k ⫹ i ⫺ 1兩k] ⱖ umin , i ⫽ 1, 2, ⭈ ⭈ ⭈ , M (Input constraint) (177) ⌬umax ⱖ ⌬u[k ⫹ i ⫺ 1兩k] ⱖ ⫺⌬umax , i ⫽ 1, 2, ⭈ ⭈ ⭈ , M (Input move constraint) (178) ymax ⫹ ␧ ⱖ y [ y ⫹ i兩k] ⱖ ymin ⫺

␧

output constraint softening variable

, i ⫽ 1, 2, ⭈ ⭈ ⭈ , M

(Output constraint) (179) y(k ⫹ i) ⫽ ␾(k ⫹ i ⫺ 1)T␪¯ (k), i ⫽ 1, 2, ⭈ ⭈ ⭈ , M (Future output prediction) (180)

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␪¯ [k] ⫽

冉冘 s

j⫽1

␾[k ⫺ j ]␾[k ⫺ j ]T

冊

⫺1

[␰[k ⫺ 1] ⭈ ⭈ ⭈ ␾[k ⫺ s]]y[k] (Parameter estimate) (181)

冘 ␾[k ⫹ i ⫺ j 兩k]␾[k ⫹ i ⫺ j 兩k]
(␳ ⫺ 애)I Ɑ 0 s

T

j⫽1

␳ ⬎

i ⫽ 1, 2, ⭈ ⭈ ⭈ , M,

ⱖ0

其

其 PE level

애 PE softening variable

(PE constraints) (182) where y[k] ⫽ ˆ [ y[k] ⭈ ⭈ ⭈ y [k ⫺ s ⫹ 1]]T

␾[k ⫺ j ⫺ 1]T ⫽ ˆ [u[k ⫺ j ⫺ 1] ⭈ ⭈ ⭈ ⭈ u[k ⫺ j ⫺ m] y [k ⫺ j ⫺ 1] ⭈ ⭈ ⭈ y[k ⫺ j ⫺ n] 1],

(183) (184)

with all past values of inputs u and outputs y assumed to be known. Note that Eq. (182) for i ⫽ 1 ensures that the closed-loop input u is persistently exciting. In typical MPC fashion, this optimization problem is solved at time k, and the optimal u(k) is applied to the process. This procedure is repeated at subsequent times k⫹1, k⫹2, etc. For the numerical solution of the MPCI on-line optimization problem, Genceli and Nikolaou (1996) have developed a successive semidefinite programming algorithm, with proven convergence to a local optimum.

2. Objective a. Multiscale MPC. Perhaps the most compelling impetus behind computer integration of process operations is the opportunity to closely coordinate (integrate) a range of individual activities, in order to achieve overall corporate objectives. In optimization jargon, the ultimate target of computer integration is to relate dispersed individual activities to an overall corporate objective function that could, in turn, be used in optimal decision making over time. So far, the development of a manageable all-inclusive objective function has been practically beyond reach, because of the enormous complexity of the problem. As a remedy, hierarchical decomposition of the problem and optimal decision making at each level are employed. This decomposition is usually realized according to the hierarchical structure of

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Fig. 21 (Bassett et al., 1994). Note that early applications of computer-based on-line optimization worked at the top levels of the process operations hierarchy, where decisions are made less frequently than at lower levels and, consequently, the limited speed, input/output, and storage capacity of early computers were not an impediment. The implicit assumption in this decomposition is that the aggregate of the individually optimal decisions will be close to the overall optimal decision at each point in time. Frequently, this is not the case. Therefore, there exists a strong incentive to establish a framework for the formulation and solution of optimization problems that integrate as many levels as possible above the chemical process level of the Process Operations Hierarchy (Prett and Garcia, 1988; Kantor et al., 1997). Why is it not trivial to perform an integrated optimization, by merely combining the individual optimization problems at each level of the Process Operations Hierarchy to form a single optimization problem? There are a number of reasons: • Dimensionality. Each level in Fig. 21 is associated with a different time scale (over which decisions are made) that can range from split seconds, at the regulatory control level, to years, at the capacity planning and design level. The mere combination of individual level optimization problems into one big problem that would span all time scales would render the dimensionality of the latter unmanageable. • Engineering/Business concepts. Although engineering considerations dominate the lower levels of the Process Operations Hierarchy, business concepts emerge at the higher levels. Therefore, a variety of

FIG. 21. Process Operations Hierarchy in the chemical process industries.

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individual objectives of different nature emerge that are not trivial to combine, at either the conceptual or the implementation level. • Optimization paradigms. Various optimization paradigms have found application at each level (e.g., stochastic programming, mixed integer– nonlinear programming, quadratic programming, linear programming). However, it is not obvious what would be a promising paradigm for the overall optimization problem. • Software implementation. The complexity of the integrated optimization problem is exacerbated when implementation issues are considered. A unifying framework is needed that will allow both software and humans involved with various levels of the Process Operations Hierarchy to seamlessly communicate with one another in a decisionmaking process over time.

These reasons that make the overall problem difficult suggest that a concerted attack is needed, from both the engineering and business ends of the problem. It should be stressed that, while there may be some common mathematical tools used in both engineering and business, the bottleneck in computer integration of process operations is not the lack of solution to a given mathematical problem, but rather the need for the formulation of a mathematical problem that both corresponds to physical reality and is amenable to solution. Stephanopoulos et al. (1997) have used a wavelet-transform based formalism to develop process models at multiple scales and use them in MPC. That formalism hinges on using transfer functions that localize both time and scale, unlike standard (Laplace or z-domain) transfer functions, which localize scale (frequency), or standard difference or differential equation models, which localize time. Based on this formalism, these authors address MPC-related issues such as simulation of linear systems, optimal control, state estimation, optimal fusion of measurements, closed-loop stability, constraint satisfaction, and horizon length determination. In relation to the last task, Michalska and Mayne (1993) have proposed a variable horizon algorithm for MPC with nonlinear models. Their approach addresses the difficulty of global optimum requirements in stability proofs. The moving horizon length, p, is a decision variable of the on-line optimization. Closedloop stability is established by arguments such as Eq. (113).

b. Dynamic Programming (Closed-Loop Optimal Feedback). As discussed in Sections II,B and V,C, the main reason for not implementing the closedloop optimal feedback MPC form is the difficulty of the associated optimization problem. If inequality constraints are not present, then an explicit

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closed-form controller can be determined, as Lee and Cooley (1998) have shown. This is an area where significant developments can be expected. 3. Constraints a. MPC with End Constraint. Perhaps the most important practical outcome of our new understanding of MPC stability properties is the importance of the end constraint in Eqs. (63) or (64). Such a constraint has already been incorporated in certain commercial packages with minimal effort, either heuristically or following theoretical research publications. MPC theory has made it clear that including an end constraint in MPC online optimization is not merely a matter of company preference or software legacy, but rather an important step toward endowing the MPC algorithm with improved properties. b. Chance Constrained MPC: Robustness with Respect to Output Constraint Satisfaction. Although MPC constraints that bound process inputs can be easily ensured to be satisfied by the actual system, constraints on process outputs are more elusive. That is because future process outputs within an MPC moving horizon have to be predicted on the basis of a process model (involving the process and disturbances). Because the model involves uncertainty, process output predictions are also uncertain. This uncertainty in process output predictions may result in adverse violation of output constraints by the actual closed-loop system, even though predicted outputs over the moving horizon might have been properly constrained. Consequently, a method of incorporating model uncertainty into the output constraints of the on-line optimization is needed. This would improve the robustness of constrained MPC. In this paper we introduce an approach to achieving that goal. The proposed approach relies on formulating output constraints of the type ymin ⱕ y ⱕ ymax as chance constraints of the type Pr兵 ymin ⱕ y ⱕ ymax其 ⱖ 움,

(185)

where Pr兵A其 is the probability of event A occurring, y is the process output bounded by ymin and ymax , and 움 is the specified probability, or confidence level, that the output constraint would be satisfied. Under the assumption that the process output y is predicted by a linear model with normally distributed coefficients, this chance constraint can be reformulated as a convex, deterministic constraint on process inputs. This new constraint can then be readily incorporated into the standard MPC formulation. The resulting on-line optimization problem can be solved using reliable convex optimization algorithms such as FSQP (Lawrence et al., 1997; Zhou et al., 1997), or interior-point methods (Kassmann et al., 2000).

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VII. Future Needs A. IS BETTER MPC NEEDED? A seasoned practitioner would probably be in a better position to answer this question. But then, a more relevant question might be, ‘‘Is better MPC possible?’’ We claim that the answer is affirmative (to both questions!). In our discussions with MPC practitioners (certainly not with a statistically representative sample), the most frequently expressed improvement need has been to decrease the time that MPC is in the manual mode. The reasons, however, why MPC is switched to ‘‘manual’’ vary widely. It appears that improvements are needed in various areas (e.g., model development, computation, programming, communications, user interface), not just MPC theory. But theory is important, as the preceding sections of this work tried to explain. Of course, as MPC matures to a commodity status, the particular algorithm included in a commercial MPC software package, albeit very important, becomes only one of the elements that can make an MPC product successful in the marketplace. As with many products, the cost associated with MPC development, implementation, and maintenance has to be compared against technical and economical benefits. The term ‘‘better MPC’’ need not imply a new variant of the traditional MPC algorithm, but rather a better way of using computers in computer-aided process operations. For example, Section VI, B, 2, a made the case about integrating various levels of the process operations hierarchy. Although expressing the need for such integration is relatively easy, the complexity of the problem is great enough not to allow a simple solution as a matter of implementation. Indeed, understanding the practical limitations as well as the theoretical properties of a complex computer-integrated system is a formidable challenge. Because of that, it appears that collaboration between academic and industrial forces would be beneficial for the advancement of computeraided process operations technology.

B. IS MORE MPC THEORY NEEDED? Yes! Although there has been significant progress, there are still several open issues related to MPC robustness, adaptation, nonlinearity handling, performance monitoring, model building, computation, and implementation. In general terms, there are two theoretical challenges associated with advancing MPC technology: (a) development of new classes of control strategies, and (b) rigorous analysis of the properties of control strategies.

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Practice has shown that both challenges are important to address (Morari, 1991). MPC is only one tool in the broader area of computer-aided process engineering. With computer power almost doubling every year and widespread availability of highly interconnected computers in process plants, the long-term potential for dramatic developments in computerassisted process operations is enormous (Boston et al., 1993; Ramaker et al., 1997; Rosenzweig, 1993). Although improved MPC systems may be internally complex, the complexity of the design (e.g., translation of qualitative engineering requirements to design parameter specifications), operation, and maintenance of such a systems by process engineers and operators should be low, to ensure successful implementation (Birchfield, 1996). ‘‘[In the past] complexity of design and operation were traded for the simplicity of the calculation [performed by the controller]. If control engineers had the computing devices of today when they began to develop control theory, the evolution of control theory would probably have followed a path that simplified the design and operation and increased the complexity of control calculations’’ (Cutler, 1995). Our opinion is that effective use of computers will rely on integration of several different entities, performing different functions and effectively communicating with one another as well as with humans (Minsky, 1986; Stephanopoulos and Han, 1994). A broadening spectrum of process engineering activities will be delegated to computers (Nikolaou and Joseph, 1995). Although MPC will remain at the core of such activity, peripheral activities and communication around regulatory control loops (e.g., process and controller monitoring, controller adaptation, communication among different control layers) will grow. Although no single dominant paradigm for such activities exists at present, it appears that MPC has a very important role to play. ACKNOWLEDGMENTS Three anonymous reviewers suggested numerous improvements on the original manuscript. The author acknowledges their contribution with gratitude. Partial support from the National Science Foundation is also acknowledged.

REFERENCES Allgo¨wer, F., and Doyle, F. J. III, Nonlinear process control: Which way to the promised Land?, Fifth International Conference on Chemical Process Control (Kantor, J. C., Garcia, C. E., and Carnahan B., Eds.), AIChE Symposium Series, Vol. 93, pp. 24–45 (1997).

200

MICHAEL NIKOLAOU

Astro¨m, K. J., and Wittenmark, B., ‘‘Computer Control Systems: Theory and Design.’’ Prentice Hall, 1984. Astro¨m, K. J., and Wittenmark, B., ‘‘Adaptive Control.’’ Addison-Wesley, 1989. Badgwell, T. A., A robust model predictive control algorithm for stable nonlinear plants, Preprints of ADCHEM 97, Banff, Canada (1997). Bassett, M. H., Doyle, F. J. III, Gudva, G. K., Pekny, J. F., Reklaitis, G. V., Subrahmanyam, S., and Zentner, M. G., Perspectives on model based integration of process operations, Proceedings of ADCHEM 94, Kyoto, Japan (1994). Baxley, R. A., and Bradshaw, J., personal communication. Texas A&M University (1998). Bayard, D. S., Yam, Y., and Mettler, E., A criterion for joint optimization of identification and robust control, IEEE Trans. on Autom. Control 37, 986 (1992). Bequette, B. W., Nonlinear control of chemical processes: a review. Ind. Eng. Chem. Res. 30, 1391-1413 (1991). Birchfield, G. S., Trends in optimization and advanced process control in the refinery industry, Chemical Process Control V, Tahoe City, CA (1996). Bitmead, R. R., Gevers, M., and Wertz, V., ‘‘Adaptive Optimal Control—The Thinking Man’s GPC.’’ Prentice-Hall, 1990. Boston, J. F., Britt, H. I., and Tayyabkhan, M. T., Computing in 2001. Software: Tackling tougher tasks, Chem. Eng. Prog. 89, 11, 7 (1993). Box, E. P. G., Parameter estimation with closed-loop operating data, Technometrics 18, 4 (1976). Buckley, P. S., Second Eng. Found. Conf. on Chem. Proc. Contr., Sea Island, GA, Jan. (1981). Chen H., and Allgo¨wer, F., A quasi-infinite horizon nonlinear predictive control scheme with guaranteed stability, Report AUT96-28, ETH, http://www.aut.ee.ethz.ch/cgi-bin/ reports.cgi (1996). Cheng, X., and Krogh, B. H., A new approach to guaranteed stability for receding horizon control, 13th IFAC World Congress, San Francisco, 433–438 (1996). Cheng, X., and Krogh, B. H., Stability-constrained model predictive control with state estimation, ACC Proceedings (1997). Chmielewski, D., and Manousiouthakis, V., On constrained infinite time linear quadratic optimal control, Sys. Cont. Let. 29, 121–129 (1996). Choi, J., and Manousiouthakis V., Bounded input initial state bounded output stability over ball, AIChE Annual Meeting, paper 191a (1997). Clarke, D. W., Mohtadi, C., and Tuffs, P. S., Generalized predictive control. Part 1: The basic algorithms, Automatica 23, 2, 137–148 (1987a). Clarke, D. W., Mohtadi, C., and Tuffs, P. S., Generalized predictive control. Part 2: extensions and interpretation, Automatica 23, 2, 149–160 (1987b). Cutler, C. R., An industrial perspective on the evolution of control technology in Methods of Model Based Process Control, R. Berber, Ed. Kluwer, 1995, pp. 643–658. Darby, M. L., and White, D. C., On-line optimization of complex process units, Chem. Eng. Prog. 51–59 (October 1998). De Nicolao, G., Magni L., and Scattolini R., Stabilizing receding-horizon control of nonlinear time-varying systems, IEEE Trans. AC 43, 7 (1998). Economou, C. G., ‘‘An Operator Theory Approach to Nonlinear Controller Design,’’ Ph.D. Thesis, Chemical Engineering, California Institute of Technology (1985). Edgar, T. F., Current problems in process control, IEEE Control Systems Magazine, 13–15 (1989). Fleming, W. H., (Chair), ‘‘SIAM Report of the Panel on Future Directions in Control Theory: A Mathematical Perspective,’’ SIAM Publication (1988). Foss, A. S., Critique of chemical process control theory, AIChE J. 199, 209–215 (1973).

MODEL PREDICTIVE CONTROLLERS

201

Fu, L. C., and Sastry, S., Frequency domain synthesis of optimal inputs for on-line identification and adaptive control, IEEE Trans. on Autom. Control 36, 353 (1991). Garcı´a, C. E., and Morari, M., Internal model control I. A unifying review and some new results, Ind. Eng. Chem. Process Des. Dev., 21, 308–323 (1982). Garcı´a, C. E., and Morshedi, A. M., Quadratic programming solution of dynamic matrix control (QDMC), Chem. Eng. Comm. 46, 73–87 (1986). Garcı´a, C. E., and Prett, D. M., Design methodology based on the fundamental control problem formulation, Shell Process Control Workshop, Houston, TX (1986). Genceli, H., ‘‘Robust Stability Analysis of Constrained l1-Norm Model Predictive Control.’’ PhD Thesis, Texas A&M University (1993). Genceli, H. and Nikolaou, M., Design of robust constrained nonlinear model predictive controllers with Volterra series, AIChE J. 41, 9, 2098–2107 (1995). Genceli, H., and Nikolaou, M., New approach to constrained predictive control with simultaneous model identification, AIChEJ. 42, 10, 2857–2869 (1996). Genceli, H., and Nikolaou, M., Robust stability analysis of constrained l1-norm model predictive control, AIChE J. 39(12), 1954–1965 (1993). Gevers, M., Towards a joint design of identification and control?, in ‘‘Essays on Control: Perspectives in the Theory and Its Applications’’ (H. L. Trentelman and J. C. Willems, Eds). 1993, pp. 111–151. Goodwin, G. C., and Sin, K. S., ‘‘Adaptive Filtering: Prediction and Control.’’ Prentice Hall, 1984. Gustavsson, I., Ljung, L., and So¨derstro¨m, T., Identifiability of processes in closed loop— identifiability and accuracy aspects, Automatical 13, 59–75 (1977). Kalman, R. E., A new approach to linear filtering and prediction problems, Trans. ASME, Ser. D: J. Basic Eng. 82, 35 (1960). Kane, L., How combined technologies aid model-based control, IN CONTROL, Vol VI, No. 3, 6–7 (1993). Kantor, J. C., Garcı´a, C. E., and Carnahan, B. (Eds.) ‘‘Fifth International Conference on Chemical Process Control,’’ AIChE Symposium Series Vol. 93 (1997). Kassman, D. E., Badgwell, T. A., and Hawkins, R. B., Robust steady-state target calculation for model predictive control, AIChe J. 46(5), 1007–1024 (2000). Keel, L. H., and Bhattacharyya, S. P., Robust, fragile, or optimal?, IEEE Trans. AC 42, 8, 1098–1105 (1997). Keerthi, S. S., and Gilbert, E. G., Optimal, infinite horizon feedback laws for a general class of constrained discrete time systems: Stability and moving-horizon approximations, JOTA, 57, 265–293 (1998). Kleinman, B. L., An easy way to stabilize a linear constant system, IEEE Trans. AC 15, 12, 693 (1970). Kothare, M. V., Balakrishnan, V., and Morari, M., Robust constrained model predictive control using linear matrix inequalities, Automatica 32 (10) 1361–1379 (1996). Landau, I.D., and Karimi, A., Recursive algorithms for identification in closed loop: A unified approach and evaluation, Automatica 33, 8, 1499–1523 (1997). Lawrence, C. T., Zhou, J. L., and Tits, A. L., ‘‘User’s guide for CFSQP Version 2.5: A C code for solving (large scale) constrained nonlinear (minimax) optimization problems, generating iterates satisfying all inequality constraints,’’ Institute for Systems Research, University of Maryland, Technical Report TR-94-16r1, College Park, Maryland, 20742 Lee, J. H., and Cooley, B. L., Optimal feedback control strategies for state-space systems with stochastic parameters, IEEE Trans. AC, in press (1998). Lee, J. H., and Yu, Z., Worst-case formulation of model predictive control for systems with bounded parameters, Automatica 33, 763–781 (1997).

202

MICHAEL NIKOLAOU

Lee, J. H., Gelormino, M. S., and Morari, M., Model predictive control of multi-rate sampled data systems, Int. J. Control 55, 153–191 (1992). Lee, J. H., Morari, M., and Garcı´a, C. E., State space interpretation of model predictive control, Automatica 30, 707–717 (1994). Ljung, L., ‘‘System Identification: Theory for the User.’’ Prentice-Hall, 1987. Ljung, L., Information contents in identification data from closed-loop operation, Proc. 32nd Conference on Decision and Control, San Antonio, TX (1993). Longwell, E. J., Chemical processes and nonlinear control technology, Proceedings of CPC IV, 445–476 (1991). Marlin, T. E., and Hrymak, A. N., Real-time operations optimization of continuous processes, in ‘‘Fifth International Conference on Chemical Process Control’’, (Kantor, J. C., Garcı´a, C. E., and Carnahan B., Eds.), AIChE Symposium Series, Vol. 93, pp. 156–164 (1997). Mayne, D. Q., Nonlinear model predictive control: An assessment, in ‘‘Fifth International Conference on Chemical Process Control’’ (Kantor, J. C., Garcı´a, C. E., and Carnahan, B., Eds.), AIChE Symposium Series, Vol. 93, pp. 217–231 (1997). Mayne, D. Q., and Michalska, H., Receding horizon control of non-linear systems, IEEE Trans. AC 35, 5, 814–824 (1990). Mayne, D. Q., Rawlings, J. B., Rao, C. V., and Scokaert, P. O. M., Constrained model predictive control: Stability and optimality, Automatica, 36, 789–814 (2000). Meadows, E. S., Henson, M. A., Eaton, J. W., and Rawlings, J. B., Receding horizon control and discontinuous state feedback stabilization, Int. J. Control 62 (5), 1217–1229 (1995). Michalska, H., and Mayne, D. Q., Robust receding horizon control of constrained nonlinear systems, IEEE Trans. AC 38, 1623–1632 (1993). Minsky, M., ‘‘The Society of Mind.’’ Simon and Schuster, 1986. Morari, M., Three critiques of process control revisited a decade later, Shell Process Control Workshop, Houston, TX (1986). Morari, M., Advances in process control theory, Chem. Eng. Prog. 60–67 (October, 1988). Morari, M., Model predictive control—the good, the bad and the ugly, Proc. Conf. on Chem. Process Control IV, South Padre Island, TX (1991). Morari, M., and de Oliveira, S. L., Contractive model predictive control for constrained nonlinear systems, IEEE Trans. AC, in press (1997). Morari, M., and Zafiriou, E., ‘‘Robust Process Control.’’ Prentice Hall, 1989. Mosca, E., ‘‘Optimal Predictive and Adaptive Control.’’ Prentice Hall, 1995. National Research Council Committee, ‘‘Frontiers in Chemical Engineering: Research Needs and Opportunities.’’ National Academy Press, 1988. Nikolaou, M., and Joseph, B., Intelligent control, ISPE 95 Proceedings, 68–69 (1995). Nikolaou, M., and Manousiouthakis, V., A hybrid approach to nonlinear system stability and performance, AIChE J. 35(4) 559–572 (1989). Nour-Eldin, H. A., ‘‘Optimierung linearer Regelsysteme mit quadratischer Zielfunktion.’’ Springer-Verlag, 1971. Ogunnaike, B., and Wright, R., Industrial applications of nonlinear control, in Fifth International Conference on Chemical Process Control (Kantor, J. C., Garcı´a, C. E., and Carnahan B., Eds.), AIChE Symposium Series, Vol. 93, pp. 46–59 (1997). Prett, D. M., and Garcı´a, C. E., ‘‘Fundamental Process Control.’’ Butterworths, Stoneham, MA (1988). Prett, D. M., and Morari M. (Eds.), ‘‘Shell Process Control Workshop.’’ Butterworths (1987). Prett, D. M., Garcı´a, C. E., and B. L. Ramaker, ‘‘The Second Shell Process Control Workshop.’’ Butterworths (1990). Propoi, A. I., Use of LP methods for synthesizing sampled-data automatic systems, Automation and Remote Control, 24, 837 (1963).

MODEL PREDICTIVE CONTROLLERS

203

Qin, J., and Badgwell, T., An overview of industrial model predictive control technology, in ‘‘Fifth International Conference on Chemical Process Control’’ (Kantor, J. C., Garcı´a, C. E., and Carnahan, B., Eds.), AIChE Symposium Series, Vol. 93, pp. 232–256 (1997). Radenkovic, M. S., and Ydstie, B. E., Using persistent excitation with fixed energy to stabilize adaptive controllers and obtain hard bounds for the parameter estimation error, SIAM J. Contr. and Optimization 33(4), 1224–1246 (1995). Rafal, M. D., and Stevens, W. F., Discrete dynamic optimization applied to on-line optimal control, AIChE J., 14(1), 85–91 (1968). Ramaker, B. L., Lau, H. K., and Hernandez, E. Control technology challenges for the future, in Fifth International Conference on Chemical Process Control (Kantor, J. C., Garcı´a, C. E., and Carnahan, B., Eds.), AIChE Symposium Series, Vol. 93, pp. 1–7 (1997). Rawlings, J. B., and Muske, K. R., The stability of constrained receding horizon control, IEEE Trans. Ac AC-38, 1512–1516 (1993). Rawlings, J. B., Meadows, E. S., and Muske, K. R., Nonlinear model predictive control: A tutorial and survey, Proceedings of ADCHEM ’94, 203–214, Kyoto, Japan (1994). Richalet, J. A., Rault, A. Testud, J. D., and Papon, J., Model predictive heuristic control: Application to industrial processes, Automatica 13, 413 (1978). Robertson, D. G., Lee, J. H., and Rawlings, J. B., A moving horizon-based approach for least-squares estimation, AIChE J. 42 (8), 2209–2224 (1996). Rosenzweig, M., Chemical engineering computing in 2001, Chem. Eng. Prog. 89(11) 7 (1993). Schrama, R. J. P., Accurate identification for control: The necessity of an iterative scheme, IEEE Trans. on Autom. Control 37, 991 (1992). Schwarm, A., and Nikolaou, M., Chance constraint formulation of model predictive control, AIChE Annual Meeting (1997). Scokaert, P. O. M., Mayne, D. Q., and Rawlings, J. B., Suboptimal model predictive control, IEEE Trans. AC 44(3), 648–654 (1999). Shinskey, F. G., ‘‘Process Control Systems.’’ 2nd ed., McGraw-Hill (1979). So¨derstro¨m, T., Gustavsson, I., and Ljung, L., Identifiability conditions for linear systems operating in closed loop, Int. J. Control 21, 243 (1975). Stephanopoulos, Geo., ‘‘Chemical Process Control.’’ Prentice Hall, (1984). Stephanopoulos, Geo., and Han, C., Intelligent systems in process engineering: A review, Proceedings of PSE ’94, 1339–1366 (1994). Stephanopoulos, Geo., Karsligil, O., and Dyer, M., A multi-scale systems theory for process estimation and control, preprints, NATO-ASI Series, Antalya, Turkey (1997). Thomas, Y. A., Linear quadratic optimal estimation and control with receding horizon, Electron. Lette. 11, 19–21 (1975). Van den Hof, P. M. J., and Schrama, R. J. P., Identification and control—closed loop issues, 10th IFAC Symposium on System Identification, Copenhagen, Denmark (July, 1994). Van Der Klauw, A. C., G. E. Van Ingen, A. Van Rhijn, S. Olivier, P. P. J. Van Den Bosch, R. A. de Callafon, Closed-loop identification of a distillation column, 3rd IEEE Conference (1994). Vidyasagar, M., ‘‘Control System Synthesis: A Factorization Approach.’’ MIT Press, Cambridge, MA (1985). Vidyasagar, M., ‘‘Nonlinear Systems Analysis,’’ 2nd ed. Prentice Hall, 1993. Vuthandam, P., Genceli, H., and Nikolaou, M., Performance bounds of robust model predictive control, AIChE J. 41(9), 2083–2097 (1995). Williamson, D., ‘‘Digital Control and Implementation—Finite Wordlength Considerations.’’ Prentice Hall, 1991.

204

MICHAEL NIKOLAOU

Zafiriou, E., On the effect of tuning parameters and constraints on the robustness of model predictive controllers, Proceedings of Chemical Process Control—CPC IV, 363–393 (1991). Zafiriou, E., Stability of model predictive control with Volterra series, AIChE Annual Meeting, St. Louis (1993). Zheng, Z. Q., and Morari, M., Robust stability of constrained model predictive control, Proceedings of the American Control Conference, session WM7, 379–383, San Francisco (1993). Zheng, A., and Morari, M., Control of linear unstable systems with constraints, Proceedings of the American Control Conference, Seattle, WA, 3704–3708 (1995). Zhou, J. L., Tits, A. L., and Lawrence, C. T., ‘‘User’s guide for FFSQP Version 3.7: A Fortran code for solving optimization programs, possibly Minimax, with general inequality constraints and linear equality constraints, generating feasible iterates,’’ Institute for Systems Research, University of Maryland, Technical Report SCR-TR-92-107r5, College Park, Maryland, 20742 (1997). Ziegler, J. G., and Nichols, N. B., Optimum settings for automatic controllers, Trans. ASME, 64, 759 (1942).

INDEX A

virtual mass coefficient, 46 Buoyancy in fluidized bed, 116–117 in multiphase systems, 22–23 from pressure gradients, 121–122 Business concepts, in multiscale MPC, 195–196

Adaptive control, MPC, 191–194 Algorithms, MPC control algorithm, 135 description, 136 GPC, 145 robust stability constraints, 182–184 stability constraints, 181–182 Asymptotic stability, MPC, 146

C Chance constraint, MPC, 197 Closed-loop stability for MPC nonminimum-phase process, 157–159 for MPC performance, 185–186 sensitivity, 167–169 Coefficients dispersion coefficient, 22, 41, 45–46, 53– 58, 63 virtual mass coefficient, 22, 41–42, 46, 58, 61–63 Constraints, MPC chance constraint, 197 end constraints, 178, 197 inequality constraints, 142–145 input constraints, 178, 183 input move constraints, 177, 183 output constraints, 178 robust stability constraints, 182–184 softenend state constraints, 183 stability, 181–182 system constraints, 169–170 Continuous feedback, for MPC nonlinear process, 162–163 Continuous stirred tank reactor, 148–149 Contraction solid–liquid fluidized bed, 110 three-phase fluidized beds, 105–109, 111–113 Control algorithm, in MPC conception, 135 Control theory, classical, and MPC, 135–136 CSTR, see Continuous stirred tank reactor

B Behavior f behavior, 18–22 hysteresis, 25–26 MPC system, 134, 156–157 BIBO stability, see Bounded-input– bounded-output stability Bounded dispersions bubble columns, 91–100 fluidized beds, 91, 99 mathematical model, 73–90 model parameter estimation, 90–91 Bounded-input–bounded-output stability, 149–150 Bubble columns bounded dispersions, 91–92 bubble rise velocity effect, 94 column diameter effect, 96 dispersion coefficient, 45–46 dispersion effect, 95 dispersion height effect, 96–98 f behavior, 21–22 gas density effect, 94 gas phase, 2–3 heterogeneous regime, 3 hold-up parameter effect, 95 mathematical model, 73–90 model vs. experimental values, 99–100 Sparger resistance effect, 92–94 terminal rise velocity, 42–44 unbounded bed analysis, 68–69 205

206

INDEX D

Dimensionality, in multiscale MPC, 195 Dispersed phase, solid–liquid fluidized beds, 3 Dispersion coefficient bubble columns, 45–46 in fluidized beds, 41, 53–58 in gas–liquid bubble columns, 63 in multiphase systems, 22 Dispersions bounded, 73–100 unbounded, 6–22 Disturbance, MPC prediction, 177 DMC, see Dynamic matrix control Drag force, in multiphase systems, 23 Dynamic matrix control, MPC systems, 159 Dynamic programming, MPC, 196–197

E Elasticity, fluidized beds, 26, 28–29 Elastic wave velocity equations, 30–32 End constraints, MPC, 178, 197 Energy dissipation, in fluidized bed, 117–121 Engineering concepts, in multiscale MPC, 195–196 Equations bubble column mathematical model, 73–75 elastic wave velocity, 30–32 linear stability continuity, 82–84 linear stability momentum, 84–88 solid–liquid fluidized beds, 7–12 Expansion solid–liquid fluidized bed, 110 three-phase fluidized beds, 105–109, 111–113

F f behavior bubble columns, 21–22 fluidized beds, 18–20 Feedback, continuous, MPC, 162–163 FG stability, see Finite-gain stability Finite-gain stability, 150–152, 155

Finite-impulse-response model, 138–139, 177–181 Finite prediction horizon, 163–164 FIR, see Finite-impulse-response model Fluid density, in fluidized beds, 48–49 Fluidized beds bounded analysis, 100–103 bounded dispersions, 91 dispersion coefficient, 41 elasticity, 26 elastic wave velocity, 30–32 f behavior, 18–20 force balance equations, 27 gas–solid, see Gas–solid fluidized beds model vs. experimental values, 99 nomenclature, 122–127 particle forces, 116–122 slip velocity, 41 solid–liquid, see Solid–liquid fluidized beds stability vs. instability, 26–27 three-phase, 103–113 transition, 28–29 unbounded analysis, 65, 68–69, 100–103 virtual mass coefficient, 41–42 voidage propagation velocity, 29–30 Fluid viscosity, in fluidized beds, 47–48 Force, particle in fluidized bed buoyancy, 116–117 energy dissipation, 117–121 pressure gradients, 121–122 Fragility, MPC systems, 165–169, 184–185

G Gas bubbles diameter, 61 rise velocity, 94 solids-free liquid wake, 110–111 wake volume, 108 Gas density, in bubble columns, 94 Gas–liquid bubble columns bubble diameter effect, 61 dispersion coefficient effect, 63 hysteresis behavior, 26 Richardson–Zaki index effect, 63–65 slip velocity, 44–45 stability analysis, 36–40 stability maps, 61

207

INDEX terminal rise velocity effect, 61 unbounded dispersions, 18 virtual mass coefficient effect, 61–63 Gas phase, in bubble columns, 2–3 Gas–solid fluidized beds dispersion coefficient effect, 53–58 fluid density effect, 48–49 fluid viscosity effect, 47–48 particle density effect, 49–53 particle diameter effect, 53 particle phase, 3 stability analysis, 24–26, 32–36 stability maps, 46–47 unbounded dispersions, 15–18 virtual mass coefficient effect, 58 Generalized predictive control, for unconstrained MPC algorithm, 145 Global asymptotic stability, MPC, 147 GPC, see Generalized predictive control

L Linearization, solid–liquid fluidized beds, 12–15 Linear models, MPC, 170–172, 189 Linear–quadratic regulator, and MPC, 136–137 Linear stability analysis, bubble column batch operation, 88 boundary condition linearization, 81–82 continuity equations, 82–84 momentum equations, 84–88 Y direction averaging, 78–81 Liquid–liquid spray columns, unbounded dispersions, 18 Liquids, terminal bubble velocity, 43–44 LQR, see Linear–quadratic regulator

M H Heavy oil fractionator, stability, 165 Heterogeneous regime, bubble columns, 3 Heuristic models, three-phase fluidization, 105–109 Hold-up pressure correlation model, 115–116 Hydrodynamic models, for fluidized bed stability, 26 Hysteresis behavior gas–liquid bubble columns, 26 gas–solid fluidized beds, 25–26

I Inequality constraints, absence in MPC, 142–145 Input constraints, MPC, 178, 183 Input move constraints, MPC, 177, 183 Input-output models, MPC, 189 Integrators, MPC systems, 159–161

K Kinematic wave velocity, stability, 27

Mathematical model, bubble columns boundary conditions, 75–77 cylindrical columns, 89–90 equations, 73–75 linear stability analysis, 77–88 MIMO, see Multi-input–multi-output system Model parameters bounded dispersions, 90–91 bubble columns, 42–46 fluidized beds, 41–42 Model predictive control algorithms, 135–136, 145, 181–184 in classical control theory, 135–136 conceptual unification, 187–188 constraints, 169–170, 197 dynamic programming, 196–197 enhancements, 198 fragility, 165–169, 184–185 industrial origins, 134–135 integrators, 159–161 linear model, 170–172 and LQR, 136–137 model uncertainty, 165 multiscale MPC, 194–196 nonlinearity, 162–164 nonlinear process model, 140, 172–174 nonminimum phase, 157–159 on-line optimization, 156–157

208

INDEX

performance, 185–186 process models, 188–194 as real-time problem, 133–134 robust performance, 185–186 robust stability, 176–184 short horizons, 157–159 stability proof, 174–176 stability regions, 160–161 stochastic disturbance model, 140 stochastic objective function, 140–142 system behavior, 134 theory development, 198–199 traditional formulation, 137–139 unstable process model, 140 unstable units, 159–161 without inequality constraints, 142–145 Models bubble columns, 73–90 FIR, 138–139, 177–181 for fluidized bed stability, 26 hold-up pressure correlation, 115–116 MPC input-output, 189 linear model, 170–172, 189 nonlinear process, 140, 172–174, 189 process, 188–194 state-space, 189–191 stochastic disturbance, 140 unstable process, 140 pressure–hold-up correlation, 115–116 three-phase fluidization, 105–109 Moving horizon-based state estimation, MPC models, 189–191 MPC, see Model predictive control Multi-input–multi-output system, for MPC, 177–181 Multiphase systems bounded vs. unbounded analysis, 100–103 buoyancy force, 22–23 dispersion coefficient, 22 drag force, 23 pressure, 22 regime transition, 23–24 virtual mass coefficient, 22 N Nonlinear process MPC models, 140, 172–174, 189

MPC systems, 162–164 Nonminimum phase, MPC, 157–159

O On-line optimization, MPC, 156–157, 177– 181, 186, 191–194 Optimization paradigms, in multiscale MPC, 196 Output constraints, softened, MPC, 178

P Particles, in fluidized beds buoyancy, 116–117 density, 49–53 diameter, 53 energy dissipation, 117–121 phase, 3 pressure gradients, 121–122 Performance, MPC, 185–186 Phase transition, in fluidized beds, 28–29 p-norms, 149 Point of transition, solid–liquid fluidized bed, 110 Pressure gradients in fluidized bed, 121–122 hold-up correlation model, 115–116 in multiphase systems, 22 Process models, MPC, 140, 172–174, 188–194 Process state MPC output prediction, 177 MPC prediction, 183 MPC SISO, 137–138

R Real-time problem, MPC, 133–134 Regime transition, theoretical analysis, 23–24 Reynolds averaging, for solid–liquid fluidized beds, 8–9 Richardson–Zaki index, in gas–liquid bubble columns, 63–65 Robust stability, MPC constraint prediction, 183–184

209

INDEX S Short horizons, MPC, 157–159 Single-input–single-output process, in MPC, 137–138 SISO, see Single-input–single-output process Slip velocity fluidized beds, 41 gas–liquid bubble columns, 44–45 Softenend state constraints, MPC, 183 Software, for multiscale MPC, 196 Solid–liquid fluidized beds contraction–expansion prediction, 111–112 dispersion coefficient effect, 53–58 equations, 7–12 expansion and contraction, 110 fluid density effect, 48–49 fluid viscosity effect, 47–48 heterogenous–heterogeneous stability, 112–113 homogeneous–heterogeneous stability, 112 homogeneous–homogeneous stability, 113 linearization, 12–15 particle density effect, 49–53 particle diameter effect, 53 particle phase, 3 stability analysis, 32–36 stability maps, 46–47 steady-state conditions, 12 virtual mass coefficient effect, 58 Solids-free liquid wake gas bubbles, 110–111 three-phase fluidized bed model, 107–108 Sparger pressure drop, estimation, 90–91 Sparger resistance, in bubble columns, 92–94 Stability bubble columns, 42–46, 78–88 fluidized beds, 26–27, 41–42, 91 gas–liquid bubble column maps, 61–65 gas–liquid bubble columns, 36–40 gas–solid fluidized bed maps, 46–60 gas–solid fluidized beds, 24–26, 32–36 MPC asymptotic stability, 146 BIBO stability, 149–150

closed-loop stability, 157–159, 167–169, 185–186 definition, 145–146 FG stability, 150–152 finite-gain–initial conditions stability, 155 global asymptotic stability, 147 heavy oil fractionator, 165 input dependence, 153–155 inputs for characterization, 152–153 linear model, 170–172 nonlinear model, 172–174 nonlinear process, 162–163 p-norms, 149 proof and practice, 174–176 regions, 160–161 robust stability, 176–184 role in system, 155–156 uniform asymptotic stability, 147 uniform stability, 146 unstable CSTR, 148–149 unstable system, 147–148 multiphase systems, 23–24, 100–103, 114–115 solid–liquid fluidized bed maps, 46–60 solid–liquid fluidized beds, 32–36, 112–113 State-space models, MPC, 189–191 Steady-state conditions, solid–liquid fluidized beds, 12–15 Stochastic disturbance model, MPC, 140 Stochastic objective function available measurements, 141 constraints, 141 MPC formulation, 140–141 sampling period, 141–142

T Terminal rise velocity, bubbles in contaminated liquids, 43–44 estimation, 42 in gas–liquid bubble columns, 61 in pure liquids, 43 Theory MPC, 198–199 MPC classical control, 135–136 Three-phase fluidized beds characterization, 103–104

210

INDEX

contraction–expansion prediction, 111–113 definition, 3–4 heuristic models, 105–109 parameter effects, 110–111

U Unbounded dispersions bubble columns, 21–22 criterion, 6–7 fluidized beds, 18–20 gas–liquid bubble columns, 18 gas–solid fluidized beds, 15–18 liquid–liquid spray columns, 18 solid–liquid fluidized beds, 7–15 Uniform asymptotic stability, MPC, 147 Uniform stability, MPC, 146 Unstable process model, MPC, 140 Unstable system, MPC with bounded output, 147–148 Unstable units, MPC systems, 159–161

V Velocity, in beds and columns slip velocity, 41, 44–45 terminal rise velocity, 42–44, 61 voidage propagation velocity, 29–30 wave velocity, 27, 30–32 Virtual mass coefficient bubble columns, 46, 61–63 fluidized beds, 41–42, 58 in multiphase systems, 22 Voidage propagation velocity, fluidized beds, 29–30

W Wake model, three-phase fluidized beds, 105–109 Wave velocity, in beds and columns elastic, equations, 30–32 kinematic, stability, 27

CONTENTS OF VOLUMES IN THIS SERIAL Volume 1 J. W. Westwater, Boiling of Liquids A. B. Metzner, Non-Newtonian Technology: Fluid Mechanics, Mixing, and Heat Transfer R. Byron Bird, Theory of Diffusion J. B. Opfell and B. H. Sage, Turbulence in Thermal and Material Transport Robert E. Treybal, Mechanically Aided Liquid Extraction Robert W. Schrage, The Automatic Computer in the Control and Planning of Manufacturing Operations Ernest J. Henley and Nathaniel F. Barr, Ionizing Radiation Applied to Chemical Processes and to Food and Drug Processing

Volume 2 J. W. Westwater, Boiling of Liquids Ernest F. Johnson, Automatic Process Control Bernard Manowitz, Treatment and Disposal of Wastes in Nuclear Chemical Technology George A. Sofer and Harold C. Weingartner, High Vacuum Technology Theodore Vermeulen, Separation by Adsorption Methods Sherman S. Weidenbaum, Mixing of Solids

Volume 3 C. S. Grove, Jr., Robert V. Jelinek, and Herbert M. Schoen, Crystallization from Solution F. Alan Ferguson and Russell C. Phillips, High Temperature Technology Daniel Hyman, Mixing and Agitation John Beck, Design of Packed Catalytic Reactors Douglass J. Wilde, Optimization Methods

Volume 4 J. T. Davies, Mass-Transfer and Interfacial Phenomena R. C. Kintner, Drop Phenomena Affecting Liquid Extraction Octave Levenspiel and Kenneth B. Bischoff, Patterns of Flow in Chemical Process Vessels Donald S. Scott, Properties of Concurrent Gas–Liquid Flow D. N. Hanson and G. F. Somerville, A General Program for Computing Multistage Vapor– Liquid Processes

Volume 5 J. F. Wehner, Flame Processes–Theoretical and Experimental J. H. Sinfelt, Bifunctional Catalysts S. G. Bankoff, Heat Conduction or Diffusion with Change of Phase 211

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CONTENTS OF VOLUMES IN THIS SERIAL

George D. Fulford, The Flow of Liquids in Thin Films K. Rietema, Segregation in Liquid–Liquid Dispersions and Its Effect on Chemical Reactions Volume 6 S. G. Bankoff, Diffusion-Controlled Bubble Growth John C. Berg, Andreas Acrivos, and Michel Boudart, Evaporation Convection H. M. Tsuchiya, A. G. Fredrickson, and R. Aris, Dynamics of Microbial Cell Populations Samuel Sideman, Direct Contact Heat Transfer between Immiscible Liquids Howard Brenner, Hydrodynamic Resistance of Particles at Small Reynolds Numbers Volume 7 Robert S. Brown, Ralph Anderson, and Larry J. Shannon, Ignition and Combustion of Solid Rocket Propellants Knud Østergaard, Gas–Liquid–Particle Operations in Chemical Reaction Engineering J. M. Prausnitz, Thermodynamics of Fluid–Phase Equilibria at High Pressures Robert V. Macbeth, The Burn-Out Phenomenon in Forced-Convection Boiling William Resnick and Benjamin Gal-Or, Gas–Liquid Dispersions Volume 8 C. E. Lapple, Electrostatic Phenomena with Particulates J. R. Kittrell, Mathematical Modeling of Chemical Reactions W. P. Ledet and D. M. Himmelblau, Decomposition Procedures for the Solving of Large Scale Systems R. Kumar and N. R. Kuloor, The Formation of Bubbles and Drops Volume 9 Renato G. Bautista, Hydrometallurgy Kishan B. Mathur and Norman Epstein, Dynamics of Spouted Beds W. C. Reynolds, Recent Advances in the Computation of Turbulent Flows R. E. Peck and D. T. Wasan, Drying of Solid Particles and Sheets Volume 10 G. E. O’Connor and T. W. F. Russell, Heat Transfer in Tubular Fluid–Fluid Systems P. C. Kapur, Balling and Granulation Richard S. H. Mah and Mordechai Shacham, Pipeline Network Design and Synthesis J. Robert Selman and Charles W. Tobias, Mass-Transfer Measurements by the LimitingCurrent Technique Volume 11 Jean-Claude Charpentier, Mass-Transfer Rates in Gas–Liquid Absorbers and Reactors Dee H. Barker and C. R. Mitra, The Indian Chemical Industry–Its Development and Needs Lawrence L. Tavlarides and Michael Stamatoudis, The Analysis of Interphase Reactions and Mass Transfer in Liquid–Liqid Dispersions Terukatsu Miyauchi, Shintaro Furusaki, Shigeharu Morooka, and Yoneichi Ikeda, Transport Phenomena and Reaction in Fluidized Catalyst Beds

CONTENTS OF VOLUMES IN THIS SERIAL

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Volume 12 C. D. Prater, J. Wei, V. W. Weekman, Jr., and B. Gross, A Reaction Engineering Case History: Coke Burning in Thermofor Catalytic Cracking Regenerators Costel D. Denson, Stripping Operations in Polymer Processing Robert C. Reid, Rapid Phase Transitions from Liquid to Vapor John H. Seinfeld, Atmospheric Diffusion Theory Volume 13 Edward G. Jefferson, Future Opportunities in Chemical Engineering Eli Ruckenstein, Analysis of Transport Phenomena Using Scaling and Physical Models Rohit Khanna and John H. Seinfeld, Mathematical Modeling of Packed Bed Reactors: Numerical Solutions and Control Model Development Michael P. Ramage, Kenneth R. Graziano, Paul H. Schipper, Frederick J. Krambeck, and Byung C. Choi, KINPTR (Mobil’s Kinetic Reforming Model): A Review of Mobil’s Industrial Process Modeling Philosophy Volume 14 Richard D. Colberg and Manfred Morari, Analysis and Synthesis of Resilient Heat Exchanger Networks Richard J. Quann, Robert A. Ware, Chi-Wen Hung, and James Wei, Catalytic Hydrometallation of Petroleum Kent David, The Safety Matrix: People Applying Technology to Yield Safe Chemical Plants and Products Volume 15 Pierre M. Adler, Ali Nadim, and Howard Brenner, Rheological Models of Suspensions Stanley M. Englund, Opportunities in the Design of Inherently Safer Chemical Plants H. J. Ploehn and W. B. Russel, Interations between Colloidal Particles and Soluble Polymers Volume 16 Perspectives in Chemical Engineering: Research and Education Clark K. Colton, Editor Historical Perspective and Overview L. E. Scriven, On the Emergence and Evolution of Chemical Engineering Ralph Landau, Academic–Industrial Interaction in the Early Development of Chemical Engineering James Wei, Future Directions of Chemical Engineering Fluid Mechanics and Transport L. G. Leal, Challenges and Opportunities in Fluid Mechanics and Transport Phenomena William B. Russel, Fluid Mechanics and Transport Research in Chemical Engineering J. R. A. Pearson, Fluid Mechanics and Transport Phenomena

214

CONTENTS OF VOLUMES IN THIS SERIAL

Thermodynamics Keith E. Gubbins, Thermodynamics J. M. Prausnitz, Chemical Engineering Thermodynamics: Continuity and Expanding Frontiers H. Ted Davis, Future Opportunities in Thermodynamics Kinetics, Catalysis, and Reactor Engineering Alexis T. Bell, Reflections on the Current Status and Future Directions of Chemical Reaction Engineering James R. Katzer and S. S. Wong, Frontiers in Chemical Reaction Engineering L. Louis Hegedus, Catalyst Design Environmental Protection and Energy John H. Seinfeld, Environmental Chemical Engineering T. W. F. Russell, Energy and Environmental Concerns Janos M. Beer, Jack B. Howard, John P. Longwell, and Adel F. Sarofim, The Role of Chemical Engineering in Fuel Manufacture and Use of Fuels Polymers Matthew Tirrell, Polymer Science in Chemical Engineering Richard A. Register and Stuart L. Cooper, Chemical Engineers in Polymer Science: The Need for an Interdisciplinary Approach Microelectronic and Optical Materials Larry F. Thompson, Chemical Engineering Research Opportunities in Electronic and Optical Materials Research Klavs F. Jensen, Chemical Engineering in the Processing of Electronic and Optical Materials: A Discussion Bioengineering James E. Bailey, Bioprocess Engineering Arthur E. Humphrey, Some Unsolved Problems of Biotechnology Channing Robertson, Chemical Engineering: Its Role in the Medical and Health Sciences Process Engineering Arthur W. Westerberg, Process Engineering Manfred Morari, Process Control Theory: Reflections on the Past Decade and Goals for the Next James M. Douglas, The Paradigm After Next George Stephanopoulos, Symbolic Computing and Artificial Intelligence in Chemical Engineering: A New Challenge The Identity of Our Profession Morton M. Denn, The Identity of Our Profession

Volume 17 Y. T. Shah, Design Parameters for Mechanically Agitated Reactors Mooson Kwauk, Particulate Fluidization: An Overview

CONTENTS OF VOLUMES IN THIS SERIAL

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Volume 18 E. James Davis, Microchemical Engineering: The Physics and Chemistry of the Microparticle Selim M. Senkan, Detailed Chemical Kinetic Modeling: Chemical Reaction Engineering of the Future Lorenz T. Biegler, Optimization Strategies for Complex Process Models Volume 19 Robert Langer, Polymer Systems for Controlled Release of Macromolecules, Immobilized Enzyme Medical Bioreactors, and Tissue Engineering J. J. Linderman, P. A. Mahama, K. E. Forsten, and D. A. Lauffenburger, Diffusion and Probability in Receptor Binding and Signaling Rakesh K. Jain, Transport Phenomena in Tumors R. Krishna, A Systems Approach to Multiphase Reactor Selection David T. Allen, Pollution Prevention: Engineering Design at Macro-, Meso-, and Microscales John H. Seinfeld, Jean. M. Andino, Frank M. Bowman, Hali J. L. Forstner, and Spyros Pandis, Tropospheric Chemistry Volume 20 Arthur M. Squires, Origins of the Fast Fluid Bed Yu Zhiqing, Application Collocation Youchu Li, Hydrodynamics Li Jinghai, Modeling Yu Zhiqing and Jin Yong, Heat and Mass Transfer Mooson Kwauk, Powder Assessment Li Hongzhong, Hardware Development Youchu Li and Xuyi Zhang, Circulating Fluidized Bed Combustion Chen Junwu, Cao Hanchang, and Liu Taiji, Catalyst Regeneration in Fluid Catalytic Cracking Volume 21 Christopher J. Nagel, Chonghun Han, and George Stephanopoulos, Modeling Languages: Declarative and Imperative Descriptions of Chemical Reactions and Processing Systems Chonghun Han, George Stephanopoulos, and James M. Douglas, Automation in Design: The Conceptual Synthesis of Chemical Processing Schemes Michael L. Mavrovouniotis, Symbolic and Quantitative Reasoning: Design of Reaction Pathways through Recursive Satisfaction of Constraints Christopher Nagel and George Stephanopoulos, Inductive and Deductive Reasoning: The Case of Identifying Potential Hazards in Chemical Processes Keven G. Joback and George Stephanopoulos, Searching Spaces of Discrete Solutions: The Design of Molecules Processing Desired Physical Properties Volume 22 Chonghun Han, Ramachandran Lakshmanan, Bhavik Bakshi, and George Stephanopoulos, Nonmonotonic Reasoning: The Synthesis of Operating Procedures in Chemical Plants Pedro M. Saraiva, Inductive and Analogical Learning: Data-Driven Improvement of Process Operations

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CONTENTS OF VOLUMES IN THIS SERIAL

Alexandros Koulouris, Bhavik R. Bakshi and George Stephanopoulos, Empirical Learning through Neural Networks: The Wave-Net Solution Bhavik R. Bakshi and George Stephanopoulos, Reasoning in Time: Modeling, Analysis, and Pattern Recognition of Temporal Process Trends Matthew J. Realff, Intelligence in Numerical Computing: Improving Batch Scheduling Algorithms through Explanation-Based Learning Volume 23 Jeffrey J. Siirola, Industrial Applications of Chemical Process Synthesis Arthur W. Westerberg and Oliver Wahnschafft, The Synthesis of Distillation-Based Separation Systems Ignacio E. Grossmann, Mixed-Integer Optimization Techniques for Algorithmic Process Synthesis Subash Balakrishna and Lorenz T. Biegler, Chemical Reactor Network Targeting and Integration: An Optimization Approach Steve Walsh and John Perkins, Operability and Control in Process Synthesis and Design Volume 24 Raffaella Ocone and Gianni Astarita, Kinetics and Thermodynamics in Multicomponent Mixtures Arvind Varma, Alexander S. Rogachev, Alexandra S. Mukasyan, and Stephen Hwang, Combustion Synthesis of Advanced Materials: Principles and Applications J. A. M. Kuipers and W. P. M. van Swaaij, Computational Fluid Dynamics Applied to Chemical Reaction Engineering Ronald E. Schmitt, Howard Klee, Debora M. Sparks, and Mahesh K. Podar, Using Relative Risk Analysis to Set Priorities for Pollution Prevention at a Petroleum Refinery Volume 25 J. F. Davis, M. J. Piovoso, K. A. Hoo, and B. R. Bakshi, Process Data Analysis and Interpretation J. M. Ottino, P. DeRoussel, S. Hansen, and D. V. Khakhar, Mixing and Dispersion of Viscous Liquids and Powdered Solids Peter L. Silveston, Li Chengyue, Yuan Wei-Kang, Application of Periodic Operation to Sulfur Dioxide Oxidation