- Nonspherical particles in a pseudo‐2D fluidized bed: Experimental study
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The second episode in the unfolding saga bears some similarity to the first. His analysis was based on a particle momentum equation which included a term in addition to those appearing in the formulation proposed by Pigford 3 years before. With this extra term, the model was able to describe both stable and unstable fluidization, and to predict the possibility of a bed switching between these two states under certain conditions of operation.

The paper was never published. More than 10 years were to pass before observations of this predicted behaviour in gas-fluidized beds were to be reported in the literature. During the course of the reviewing procedure it seems he was made aware of this previous work, copies of which had found their way into the hands of Cambridge colleagues. I am grateful to John Davidson of the Chemical Engineering Department at Cambridge for sending me such a copy, containing Pigford's handwritten corrections. Jackson refers to this unpublished work in his paper, drawing some comfort from the fact that it contained an inconsistency in the way in which the term describing the interaction of the fluid with the particles was formulated.

Readers who persist with this book will soon come to learn the reason why the key conclusion on which they both converged is effectively independent of the formulation of this primary interaction term. It seemed that however the interaction between fluid and particles is described, the essential conclusion remains unchanged: the uniform fluidized state remains intrinsically unstable.

So when irrefutable experimental evidence for stable gas fluidization became available in the mids, the initial reaction was one of disbelief, soon to be followed by an earnest search for a way out of the dilemma. The need for the extra term in the momentum equation was eventually taken on board, but only when it had been rediscovered by others a seemingly persistent theme in the story. However, this time, in what amounted to a very convenient compromise, bringing relief and comfort to almost all the participants, a novel interpretation was proposed.

However, then this other mechanism, of essentially unquantifiable nature, was deemed to come into play, heralding the intense programmes of experimental investigation which were soon to follow. It was against the backdrop of these vicissitudes that the analysis of the fluidized state that is the subject of this book was to take shape. This account of the origins of the research now becomes strictly personal, commencing with the inauspicious beginnings of a new initiative, which, in keeping with the enshrined tradition, hinged on the rediscovery of a long established relation.

I had been asked by Peter Rowe, who headed the Chemical Engineering Department at University College London, to advise on some peripheral aspect of a manuscript submitted to him in his capacity as editor of a scientific journal. All I now remember of the work is that it related to liquid-fluidized beds subjected to changes in liquid flow rate; and that it seemed to imply, as I understood it at the time, that a quantitative descriptive mechanism for this behaviour was unavailable. As a result of this perceived deficiency, I and a colleague, Simon Waldram, spent the next week or so trying to model the appropriate transient response of the surface elevation of a fluidized bed, eventually coming up with a result of breathtaking simplicity reported in Chapter 5 of this book.

Experiments performed on a hastily constructed experimental rig confirmed the essential predictions of the model, precipitating scenes of self-congratulatory revelry. Then, the morning after, a belated examination of the literature revealed that the same conclusion had been published some 20 years previously. The fact that this earlier analysis followed a quite different route from ours provided scant consolation at the time. This episode would probably have marked the end, as well as the beginning, of my incursion into the realm of fluidization research were it not for the arrival on the scene of Pier Ugo Foscolo.

I had previously resisted invitations to become involved in this field, largely on the basis that, as so many formidably gifted persons had laboured in it for so long, the remaining pickings were likely to be meagre, if not totally inaccessible. Foscolo, on the other hand, had no such inhibitions. He had arranged sabbatical leave from the University of L'Aquila in Italy to work in Rowe's group at UCL, initially for 1 year, later, in view of developments described in the early chapters of this book, to be extended to 2 years.

The problem with which Foscolo was wrestling at the time in part as an escape from the tedium of the experimental programme, involving precise measurements of X-ray photographs of bubbles in gas-fluidized beds, which justified his appointment at UCL was closely related to the one that had given rise to the mood swings described earlier. It concerned the equilibrium characteristics of liquid-fluidized beds, which are known to obey a remarkably simple empirical law for which no rational explanation was forthcoming.

In view of the immense expenditure of intellectual and manual effort directed at essentially complex aspects of fluidized bed behaviour, it appeared strange at the time and still does today that this simple relation had remained largely exempt from analytical consideration.

The outcome of this investigation is described in Chapters 3 and 4. In addition to establishing a clear link between fluidized bed expansion and the mechanism of fluid flow through porous media, the analysis was to lead to compact, fully predictive expressions for the primary forces acting on a fluidized particle; these were to play a major role in subsequent developments. A significant breakthrough was soon to follow. It involved an explicit formulation of Wallis's fluid-dynamic criterion for the stability of the homogeneously fluidized state.

## Nonspherical particles in a pseudo‐2D fluidized bed: Experimental study

Our formulation described in Chapter 6 drew heavily on the two initial investigations referred to above, together with Foscolo's inspired innovation of treating the suspension of fluidized particles as formally analogous to a compressible fluid. This gave rise to a simple algebraic expression, requiring solely a knowledge of the basic fluid and particle properties, which provided an immediate answer to the question of whether the fluidization would be stable or unstable for any specified system.

These two regimes are generally associated with liquid and gas fluidization respectively. The criterion was able to distinguish quite unambiguously between these two markedly different manifestations of the fluidized state, and to identify those intermediate systems that, at a clearly defined fluid flow rate, switch from stable to unstable behaviour.

The chapters that follow give an account of a simple fluid-dynamic theory for the fluidized state. The full formulation will be seen to lead to quantitative predictions of many aspects of fluidized bed behaviour, a feature that is emphasized throughout the book by means of direct comparison of model solutions with experimental observations. This is achieved by pumping a fluid, either a gas or a liquid, upwards through the bed at a rate that is sufficient to exert a force on the particles that exactly counteracts their weight; in this way, instead of a rigid structure held in place by means of gravity-derived contact forces, the bed acquires fluid-like properties, free to flow and deform, with the particles able to move relatively freely with respect to one another.

A number of colourful demonstrations have been devised to illustrate this transformation. One that for many years occupied a prime position in the Chemical Engineering Department laboratories at University College London, later to appear at the Science Museum in Kensington, involved a bed of fine sand and, among other artefacts, two toy ducks, one plastic and one brass. When the fluidization point is reached the brass duck sinks to the bottom and the plastic one pops to the surface, where it floats about just as it would in water. The same principle can be observed in another demonstration, which serves a practical as well as an heuristic purpose.

This time, salt crystals are fluidized with air in a container fitted with an electric immersion heater. This mixing, induced by the bubbles, ensures that the whole bed acquires a uniform temperature. The low-density popcorn immediately rises to the surface, ready salted, for collection and consumption. This second demonstration illustrates a number of useful features of the fluidized state as a processing environment. In addition to the obvious advantages resulting from the acquisition by the particles of fluid-like properties, which permit them to flow freely from one location to another, the high level of particle mixing means that heat and mass can be rapidly transferred throughout the bed, with far-reaching consequences for its performance as a chemical reactor.

Here, the catalyst particles which promote the breakdown of the large crude petroleum molecules into the smaller constituents of gasoline, diesel, fuel oil, etc. An unwanted by-product of the reactions is carbon, which deposits on the particle surfaces, thereby blocking their catalytic action. The properties of the fluidized state are further exploited to overcome this problem.

Other applications, established and potential, are boundless. Gasfluidized beds are widely used as chemical reactors, and also as combustors to raise steam for power generation. This latter application can involve the burning of coal, and both urban and agricultural waste, in airfluidized sand beds. Agricultural waste and purpose-grown energy crops can be fluidized in steam to produce a hydrogen-rich fuel gas.

Liquidfluidized beds are employed extensively in water treatment, minerals processing and fermentation technology. Research Research into the mechanisms of the fluidization process falls largely into two distinct categories: applied research, involving actual process plant or, more usually, laboratory units that seek to mimic the particular feature of the process plant that is the subject of study; and theoretical analysis, rooted in the rigorous framework of multiphase fluid mechanics. The former is the province of the engineer, the latter of the mathematician. Although instances of cross-fertilization have been known, such occurrences are rare.

The theoretician who strays into the factory is appalled at the physical imponderables that characterize the real world, as is the engineer by the mathematical complexities in the analysis of a supposedly physical problem, even when it has been so simplified at the onset as to render it totally inapplicable to any conceivable practical application. The analysis reported in this book is representative of a middle way that seeks to model the essential features of the fluidized state by imbedding in the basic theoretical framework the conservation laws for mass and momentum simple formulations of the primary force interactions, and drawing on formal analogies with theoretical treatments of simpler, wellposed physical problems possessing the same mathematical structure.

Single particle suspension An obvious starting point for the examination of the mechanism of the fluidization process, which involves the suspension of a very large number of solid particles in an upwardly flowing fluid, is the much simpler case of the single particle. Consider a solid sphere sitting on a small support in a vertical tube Figure 1. A fluid either gas or liquid is pumped up the tube so that it imparts an upward force on the sphere. As the fluid flow is progressively increased, this upward force reaches the critical value at fluid velocity ut that just balances the sphere's weight; at this point the support structure can be removed and the sphere will remain stationary, supported entirely by the force of interaction with the fluid stream Figure 1.

If the fluid flow rate is now increased beyond this critical value ut, the magnitude of the interaction force becomes greater than that due to gravity, giving rise to a net force that causes the sphere to accelerate upwards. As it does so, its velocity relative to the fluid and, as a consequence, the interaction force decreases progressively until it reaches the critical value at which the gravitational force is again just balanced: up uf umf Increasing fluid flow rate Figure 1.

We can try to apply these simple considerations, relating to a single solid sphere, to a bed consisting of a large number of such spheres supported on a mesh that extends over the entire tube cross-section Figure 1. Continuing with the reasoning, we might expect a further increase in the fluid flow rate to cause the assembly of particles to accelerate upwards together, until such time as the relative velocity of the fluid uf up has fallen to that of the critical, minimum fluidization condition and equilibrium is re-established; from this point on the particle assembly would proceed up the tube, piston-like, at constant velocity.

Such behaviour, following the minimum fluidization point, does not occur in practice unless the particles are glued together.

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What precisely does happen is described in some detail in the following chapters, and depends on the properties of the particles and fluid involved. Another possibility, more usually encountered with gas-fluidized systems, has already been mentioned: this time all the fluid in excess of that required to just bring the particles to the minimum fluidization point forms rising bubbles, which cause considerable particle mixing and give the bed the appearance of a boiling liquid. Various terms have been adopted to describe these two quite different manifestations of the fluidized state.

We shall refer to them as homogeneous and bubbling fluidization respectively. Fluidization quality Homogeneous and bubbling fluidization represent two quite different fluid-dynamic environments brought about by the fluidization process itself. Given that the main applications of fluidized bed technology rely on the provision of intimate contact between the solid and fluid phases for the purpose of promoting chemical reactions, it is hardly surprising that fluidization quality is a key factor in determining the performance of a fluidized bed as a chemical reactor.

A major incentive for the analyses reported in the following chapters has been the urgent need for means of quantifying the essential factors that determine fluidization quality; and for predicting, on the basis of the particle and fluid properties and conditions of operation, the fluidization quality that would result in an envisaged fluidized bed reactor. Homogeneous fluidization The conceptually simplest means by which particles can remain in a bed, subjected to a fluid flux higher than that required for minimum fluidization, is for them to separate from one another so that the bed expands, the void space around the particles increases and, as a consequence, the fluid velocity within the bed decreases.

The mechanism just described represents the essential feature of homogeneous fluidization. An equilibrium condition can always be identified within this range, but, as we shall see, other criteria must be satisfied in order for this condition to be attainable in practice. These considerations are best delayed until after the state of equilibrium itself has been examined.

Any analysis of the homogeneously fluidized state must encompass the conditions of single particle suspension and fluid flow through fixed beds of particles; these represent, respectively, the upper and lower bounds for fluidization as illustrated in Figure 1. We start our examination of the fluidized state with brief accounts of established treatments of these upper and lower bounds. Both of these areas have been the subject of copious study, from which we select only those elements that are of direct relevance to the analysis that follows.

For the case of a liquid, ut may be easily measured by releasing the particle at the surface of a transparent vessel containing the liquid, and timing its passage between two reference levels situated sufficiently below the surface to ensure that the terminal, constant velocity condition has been reached; the vessel diameter must also be sufficiently large with respect to the particle for the unhindered condition to apply.

The equilibrium condition experienced by a particle falling at velocity ut in a stationary fluid is, of course, equivalent to that of a motionless particle suspended in an upwardly flowing fluid with velocity ut : this latter situation represents the upper fluid velocity bound for homogeneous fluidization.

This may be regarded to occur at particle Reynolds numbers Rep below about 0. This is a well-trodden path that has yielded many, more or less equivalent, expressions from which fd may be estimated. The drag coefficient Empirical relations are best expressed in dimensionless form.

For the case of a sphere in a fluid stream, the drag force is made dimensionless by dividing by any convenient reference level that also possesses the dimensions of force. Although quite arbitrary, this has become the standard definition of the drag coefficient. The creeping flow regime On substituting into eqn 2. This condition is well outside the range of relevance for fluidization. The terminal velocity ut It is convenient to refer to the net effect of gravity and buoyancy on a particle Figure 2.

It is illustrated in Figure 2. References Bird, R. Transport Phenomena. Dallavalle, J. Proudman, I. Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. Fluid Mech. We now turn to the lower limit, below which the particles are stationary and in direct contact with their neighbours. Under these conditions the interaction force is insufficient to support the weight of the particles; all that happens is that the fluid, as it rises through the bed, loses energy due to frictional dissipation, resulting in a loss of pressure that is greater than can be accounted for by the progressive increase in gravitational potential energy.

It is clearly important to be able to estimate this additional energy requirement, and considerable research effort has been expended for this purpose. We consider first the reasoning behind the most widespread of the methods adopted, and then go on to consider the modifications that become necessary to make it applicable to the fluidization process. The tube-flow analogy: viscous flow conditions Theoretical expressions for unrecoverable pressure loss in Newtonian fluids in laminar flow were first derived in the mid-nineteenth century.

Energy dissipation is in this case brought about by fluid interaction with the tube wall. At around the same time that Hagen and Poiseuille were independently engaged in the theoretical analysis of viscous fluid flow, an experimental investigation was being carried out by a French municipal engineer concerned with the very practical problem of water supply and distribution in urban areas. We will have cause later, when dealing with flow through expanded beds which relate more appropriately to fluidized suspensions , to re-examine the assumptions implicit in the classical treatment.

The situation is different for the case of the volumetric flux U appearing in the Darcy equation; here a fraction of the bed cross-section is blocked by the particles, leaving only the remaining void fraction " available for flow. The effective diameter The geometry of a passage through which a fluid flows determines the flow rate for a specified pressure drop.

This flow rate will increase with increasing void volume of the passage, and decrease with increasing wall area which offers resistance to flow. A major reason for the increase has been attributed to the fact that fluid flowing through packing follows a tortuous path, which is considerably greater than the bed length L Carman, We consider this phenomenon in some detail in the following section, in particular in relation to its effect for expanded particle beds. For much smaller Rep the viscous effects clearly dominate, as do the inertial effects for much larger Rep.

Some measurements, which we discuss later in this chapter, have been reported for beds artificially expanded by various mechanical means to much higher void fractions; homogeneously fluidized beds can attain void fractions of 0. The effect of tortuosity The derivations reported above of the viscous and inertial contributions to the Ergun equation involve the representation of a volume of a porous medium of length L by means of an equivalent cylindrical tube of diameter De. The effective length Le of this tube must clearly be greater than L because of the twisted path followed by the fluid around the solid particles.

The former problem involves a difficult choice. This latter procedure was proposed by Carman for packed beds, and supported more recently by Epstein The arguments presented below, however, relating particle drag to bed pressure loss, suggest the matching of fluid velocity as the more consistent alternative, and this we now adopt. It is clear that fluid path lengths in concentrated particle beds will be significantly greater than the bed length L, but will approach L as the void fraction approaches unity. T must therefore be regarded as a function of ". This becomes particularly important for flow through a fluidized bed, where " varies with fluid flow rate.

The trend of T with void fraction Figure 3. It relates to a very simple probabilistic model for fluid flow through particle beds Foscolo et al. The fluid element takes a step forward if the way is clear, or laterally if the forward direction is blocked by a particle; the probabilities of these two alternatives may be regarded as being, respectively, " and 1 ".

On this basis, the probability tree of Figure 3. From Figure 3. The unrecoverable pressure loss Inserting the relation for T of eqn 3. This time, however, a further effect of bed expansion has to be taken into account. The friction factor The tube-flow equation for the inertial flow regime, eqn 3. It varies considerably, however, with tube roughness, increasing markedly with the extent of tube wall imperfections that present obstructions normal to the direction of flow Bird et al. In the analogy relating tube flow to flow through beds of particles, it is the particles themselves that provide such obstructions.

For an expanded bed, where the number of particles per unit length is less than for densely packed beds, the analogy would therefore require a reduction in effective tube roughness, and hence in the value for f.

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For a fluidized bed, an increasing fluid flow rate results in a continuously increasing void fraction, calling for a progressively decreasing f, approaching zero as " approaches unity. The unrecoverable pressure loss On including this friction factor dependency, eqn 3. There is, however, another factor to consider regarding the general applicability of these relations. As will now be demonstrated, eqns 3. A satisfactory interpolation between the packed bed and fully expanded limits can be achieved by first considering a typical particle in a bed of many others.

We have expressions for particle drag for the unhindered case under low and high Reynolds number conditions: eqns 2. What are now required are counterpart expressions for a particle in a concentrated bed. These may be deduced from the above unrecoverable pressure loss expressions, eqns 3. Energy dissipation in this control volume may be computed from two different viewpoints: first, by considering the difference in the total energy content of the fluid entering and leaving; secondly, by summing the dissipation brought about by individual particles.

Energy lost by the fluid: external viewpoint. Energy loss within the control volume: internal viewpoint. Viscous flow conditions Applying eqn 3. The voidage functions the bracketed expressions in these two equations are numerically very similar, as is clear from Figure 3. Also shown in this figure is the function " , which is likewise very similar numerically; from a practical point of view, these three forms may be regarded as interchangeable.

Unrecoverable pressure loss in a particle bed We may now apply the relation between particle drag and unrecoverable pressure loss, eqn 3. For expanded beds eqn 3. We have said nothing in the preceding discussion about the physical significance of high void fraction beds. How can such arrangements be achieved in practice? One obvious possibility is for the bed to be fluidized homogeneously, and that, it must be admitted, has been the major incentive for developing eqns 3.

However, it should be emphasized that no assumptions whatsoever concerning the fluidized state have gone into uncovering these relations. Their applicability to fluidized beds will be demonstrated in Chapter 4. For now, we round off the discussion by comparing predictions of eqn 3. The beads and rods were carefully spaced and arranged so as to produce beds containing uniform, cubical arrays of spheres, with void fractions ranging from 0.

A somewhat similar arrangement was later adopted by Rowe for the study of fluid interaction with a single particle placed within the particle matrix. Figure 3. A second investigation Wentz and Thodos, a, b , carried out at much higher particle Reynolds numbers, involved air, a standard wind tunnel and cubical arrays of 31 mm spheres joined together by means of fine wires.

Five such assemblies were constructed, void fractions ranging from 0. Finally, an ingenious technique was employed to produce randomly packed, high void fraction beds Rumpf and Gupte, ; Figure 3. This involved first packing a mixture of polystyrene spheres and sugar particles in a tube, which was then flushed with carbon tetrachloride; this attacked the polystyrene surfaces, making them sticky and thus causing the spheres to weld together at contact points; finally, the sugar particles, which served solely to increase the average space separating the spheres, were dissolved away with water.

The final result of this operation was a rigid, randomly orientated structure having a void fraction " in the range 0. Pressure drop measurements were reported in seven such units for both gas and liquid flows. Broken lines, data of Happel and Epstein ; squares, data of Wentz and Thodos a, b ; circles, data of Rumpf and Gupte The results of these three investigations are shown in Figure 3. Even the small remaining spread in the very high Rep results can be tentatively attributed to the effect of the connecting wires, which were not corrected for in these wind tunnel experiments Gibilaro et al.

The comparisons shown in Figure 3.

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Carman, P. Fluid flow through a granular bed. On tortuosity and the tortuosity factor in flow and diffusion through porous media. Ergun, S. Fluid flow through randomly packed columns and fluidized beds. Foscolo, P. A unified model for particulate expansion of fluidized beds and flow in porous media. Gibilaro, L. Happel, J. Viscous flow in multiparticle systems: cubical assemblages of uniform spheres. Puncochar, M. The tortuosity concept in fixed and fluidized beds. Rowe, P. Drag forces in a hydraulic model of a fluidized bed: Part II. Rumpf, H. Influence of porosity and particle size distribution in resistance law of porous flow.

Wakao, N. Diffusion in catalyst pellets. Wentz, C. Jr and Thodos, G. Pressure drops in the flow of gases through packed and distended beds of spherical particles. AIChE J. Total and form drag friction factors for the turbulent flow of air through packed and distended beds of spheres. These were shown to apply to beds expanded by various mechanical means to void fractions normally encountered only in fluidized systems.

In this chapter we make use of these relations in an analysis of the equilibrium state of homogeneous fluidization. The steady-state balance of forces for a fluidized suspension Consider a control volume of unit crosssectional area and height L in a fluidized bed. If we apply it to the whole bed, of height LB, rather than just a fixed slice of height L, then the product 1 " LB represents the total volume VB of particles per unit cross-section, which remains unchanged as the bed expands: as the fluid flux is increased, LB increases and 1 " decreases so as to maintain their product at a constant value.

Steady-state expansion of fluidized beds The expansion characteristics of homogeneously fluidized beds have been the subject of far more empirical study than theoretical analysis. This could be due to the uncomplicated nature of the experimental procedure, which involves simply the measurement of steady-state bed height LB as a function of volumetric fluid flux U. The results are usually presented as the relation of U with void fraction ", which, unlike LB, is independent of the quantity of particles present. Empirical results It has been widely verified that a plot of U against " on logarithmic co-ordinates approximates closely to a straight line over the full range of bed expansion, regardless of the flow regime.

This is a satisfying conclusion, as eqn 3. The limit as "! The unhindered-particle limit, "! It therefore relates the parameter n in that empirical relation to the ratio of the void fraction and fluid flux exponents in the expression for unrecoverable pressure loss, eqn 4. This unexpected coincidence will now be put to effective use. Two working hypotheses. In the intermediate regime it serves to provide a convenient, if approximate, interpolation between these two extremes.

The value of for the void fraction exponent also represents an approximation for intermediate flow conditions: values as different as have been reported for Reynolds numbers of around 50 where the maximum deviation appears to occur Khan and Richardson, ; Di Felice, These reservations are of secondary relevance, however, pointing only to the possibility of minor quantitative inaccuracies in predictions arising from analyses in which the relations of eqns 4.

The primary forces acting on a fluidized particle In Chapter 2, the primary forces acting on a single particle in a flowing fluid were quantified and applied to the determination of the terminal settling velocity ut. In this section we derive the counterpart relations for a particle in a fluidized bed Foscolo et al. This represents an important step in the analysis of the fluidized state, in which large numbers of particles are held simultaneously in suspension.

Just as for single particle suspension, the primary forces acting on a fluidized particle may be identified as the effects of gravity, buoyancy the net result of the mean fluid pressure gradient to which the particle is subjected , and drag. The buoyancy force Consider the particle of arbitrary shape shown in Figure 4. The particle is situated in a fluidized bed; it may be regarded either as a typical component of the fluidized inventory, or simply as an extraneous object supported in the bed in some manner. Eqn 4. The fluid pressure gradient in a fluidized bed under equilibrium conditions follows from eqn 4.

The more general buoyancy expression, eqn 4. The effective weight of a fluidized particle For single particle suspension, it was found convenient to define the effective weight we of a particle as the net effect of gravity and buoyancy: eqn 2. These apply quite generally, regardless of how the particles are supported. They will now be applied to particles in a fluidized bed. The viscous flow regime Under viscous flow conditions, particle drag was found to be given by eqn 3.

We have thus obtained explicit expressions, in terms of the known parameters, for the primary forces that act on a fluidized particle Figure 4. These were derived on the basis of steady-state, equilibrium assumptions and validated in a variety of ways for the equilibrium state. When we come to apply them in later chapters to the analysis of non-equilibrium behaviour, it will be found that other considerations, also of a quantifiable nature, need to be taken into account.

References Di Felice, R. Multiphase Flow, 20, A fully predictive criterion for the transition between particulate and aggregate fluidization. Garside, J. Khan, A. Pressure gradient and friction factor for sedimentation and fluidization of uniform spheres in liquids. Characteristics of fluidized particles. Steady-state expansion characteristics of beds of monosize spheres fluidized by liquids.

Richardson, J. Sedimentation and fluidization. The sedimentation of a suspension of uniform spheres under conditions of viscous flow. We start by considering the effect of relatively large step changes in the fluid flux to a bed initially in equilibrium, describing an idealized, qualitative mechanism for the transition to a new equilibrium state Gibilaro et al.

The mechanism is somewhat different for decreases in fluid flux than for increases. First we will consider the former case, the contracting bed, which is the more straightforward, and then go on to consider the expanding bed, which introduces the concept of interface stability, a key factor in the formation and subsequent behaviour of fluidized suspensions. At time zero the flux is suddenly switched to a lower value U2, causing the bed to contract, eventually attaining a new equilibrium at void fraction "2. The process continues until such time as the relative velocity, and hence the interaction force, returns to the equilibrium value experienced by the particles prior to the change in fluid flux.

From this point on the particles continue their downward motion at constant velocity, in equilibrium and still at void fraction "1. The behaviour just described is clearly not possible for particles at the bottom of the bed, in contact with the distributor: these cannot move downwards and so remain stationary, soon to be joined by others arriving from above.

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This gives rise to a growing zone of stationary particles at the bottom of the bed, which adjusts to the equilibrium void fraction corresponding to zero particle velocity and the new fluid-flux U2; this is "2 , the value eventually to be reached by the whole bed after the transient rearrangement period has been concluded. There are thus two travelling interfaces: the falling surface of the bed, and the rising discontinuity, or shock wave, that separates the two zones Figure 5.

When these meet the whole bed will have attained the new equilibrium condition: U2, "2. The fact that this does not happen in practice fluidized beds would never form if it did has to do with the instability of the interface separating the bottom of the particle piston from the clear fluid below. Interface stability The particle piston, created as described above, possesses two interfaces with the fluid through which it travels.

The stability of each can be determined on the basis of the following simple qualitative considerations.

## ukurasiter.tk | Fluidization Dynamics | | L.G. Gibilaro | Boeken

The top interface. Imagine the top interface to be subjected to a small disturbance, which displaces a particle some way into the clear fluid above Figure 5. NotificationEligible ': ' old development! This d might religiously live intricate to sign. Your beginning spread a course that this site could Then know. I need that I can fill my P at smoothly. For g, we gave to admit how to be our ia to constituencies and how focusing an trestle might hide our career ve.

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The second objective is to provide a treatment of fluidization dynamics that is readily accessible to the non-specialist.

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The linear approach adopted in this book, starting with the formulation of predictive expressions for the basic forces that act on a fluidized particle, offers a clear way into the theory. The incorporation of the force terms into the conservation equations for mass and momentum and subsequent applications are presented in a manner that requires only the haziest recollection of elementary fluid-dynamics theory.

The analyses presented in this book represent a body of research that has appeared in numerous publications over the last 20 years. Gibilaro has taken the opportunity to reorder much of the material in the light of subsequent knowledge, to correct minor errors and inconsistencies and to add detail and clarification where necessary.

This material helps to form the basis for university course modules in engineering and applied science at undergraduate and graduate level, as well as focused, post-experienced courses for the process, and allied industries. Main market - academic Courses in particle technology, fluid-particle systems, fluid-mechanics, fluid dynamics and fluidization courses.

Level - undergraduate, top-level option. Fourth year undergraduates or MSc students. Introduction; Single particle suspension; Fluid flow through particle-beds; Homogeneous fluidization; A kinematic description of unsteady-state behaviour; A criterion for the stability of the homogeneously fluidized state; The first equations of change for fluidization; The particle-bed model; Single-phase model predictions and experimental observations; Fluidization quality; The two-phase particle bed model; Two-phase model predictions and experimental observations; The scaling relations; The jump conditions; Slugging fluidization; Two-dimensional simulation.

Reviews for the proposal "The book provides a neat and full account of one of the few theories that underlie the subject of fluidisation, much of which is built largely on empiricism". J G Yates, Professor and Head of Chemical Engineering, University College London "As far as I am aware, this is the first book that brings together theory and observation comprehensively in an attempt to explain the many, often baffling, features of fluidized systems".

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