Modeling of Fluidized-Bed Reactors - Werther

9.1. Modeling of Liquid–Solid Fluidized-Bed Reactors

An expansion formula of the Richardson–Zaki type, Equation (7), describes the hydrodynamics of liquid–solid ﬂuidized beds fairly well [218].

The difficulty in modeling this kind of reactor for bioreactions thus lies not so much in determining the flow and mixing conditions in the fluid as in describing the diffusion processes in the biofilm and the kinetics of the biological reactions [219].

In view of the small number of experimental studies reported thus far, no ﬁnal judgment can bemadeonthesuitabilityofvariousmodels[208].

9.2. Modeling of Gas–Solid Fluidized-Bed Reactors

Exhaustive literature surveys can be found in [2, 9, 10, 220]. [221]. Many models exist in the literature, which are classiﬁed in the cited

Figure 45. Solid-state fermentation of Aspergillus sojae in the ﬂuidized bed (adapted from [216])

a) Separator; b) Electrode; c) Agitator; d) Distributor;

e) Ejector

Figure 44. Flow sheet of Lurgi Circored process [206]

a) Preheater; b) Cyclone; c) First stage reactor; d) Second stage reactor; e) Briquetting unit; f) Gas cleaning unit

references under various schemes. The avail-able information can be summed up as follows:

No generally accepted model of the ﬂuidized-bed reactor exists; instead, many models have been proposed on the basis of more-or-less extensive experimental ﬁndings for various applications.

Any ﬂuidized-bed reactor model can be bro-ken down into separate components that de-scribe, with varying degrees of accuracy, the hydrodynamics (depending on solid properties, operating conditions, and geometry), gas–solid contact, and reaction kinetics. The essential point is that the reactor geometry effect, which is important for scale-up (Chap. 10), manifests it-self in the ﬂow conditions and must therefore be included in the hydrodynamic part of the model.

Before a reactor model found in the literature can be applied to a given problem, the designer must determine whether numerical values are available for all model parameters, that is, wheth-er the model is appropriate for design calcula-tions or is a ‘‘learning model’’ [222] in which the numerical values of important parameters can be determined only after the model is adapted to actual test results.

Reaction kinetics may be determined in a fixed-bed reactor, provided measurements are performed under conditions comparable to those that prevail in the fluidized-bed reactor (e.g., the same solids composition and particle-size distri-bution, the same activity state in the case of catalysts) [223]. However, the kinetic parameters can also be determined directly by measurements in a bench-scale fluidized-bed apparatus [224].

9.2.1. Bubbling Fluidized-Bed Reactors

By far the majority of ﬂuidized-bed reactor mod-els described in the literature deal with reactions in bubbling ﬂuidized beds [2, 9, 10, 225, 226].

For a specific application, modeling depends on the bubble flow regime. For slow-bubble systems (Fig. 7, left), the short-circuit flow of gas through the bubbles must be taken into account [227]. For fast-bubble systems (Fig. 7, right), the species have to be balanced separately in the bubble and suspension phases.

If models from the literature are employed, it should be taken into account that those devised in the past, when adequate computing hardware was

not available, often sought to obtain analytical expressions for the degree of conversion of a single reaction (usually taken as ﬁrst-order). The simplifying assumption of a single ‘‘effective’’

bubble size for the entire ﬂuidized bed was therefore made [2], or the mass-transfer area between the bubble and suspension phases was taken as uniformly distributed over the height of the bed (HTU or NTU concept, where HTU denotes height of transfer unit and NTU denotes number of transfer units [228]. Today, in view of the computing power available at the PC level, the recommended procedure is to start from local mass-transfer relations, write balance equa-tions for the differential volume element of the reactor, and then numerically integrate these equations.

Figure 46 presents a model used by WÊRTHER for a constant-volume reaction [224, 229]. Here the simplifying assumption is that flow through the suspension phase is at the minimum fluidiza-tion velocity u_mf. For a heterogeneous catalytic gas-phase reaction, the material balances for species i in the unsteady-state cases are as

In Equations (45) and (46) the following simpli-fying assumptions have been made:

1. Plug ﬂow through the suspension phase at an interstitial velocity (u_mf / e_mf)

2. Bubble phase in plug ﬂow, bubbles are solids free

3. Reaction in suspension phase only

4. Constant-volume reaction (see [224] for han-dling a change in number of moles)

5. Sorption effects neglected (see [229] for han-dling sorption)

Hereeiis the porosity of the catalyst particles;

a is the local mass-transfer area per unit of ﬂuidized-bed volume, which can be calculated as

¼

⁶_d^e_v^b

ð

⁴⁷

Þ

for spherical bubbles; r _j is the rate of partial reaction j per unit mass of catalyst; and nij is the stoichiometric number of species i in reaction j. The relation

k _G_;_i

¼

^u₃^mf

þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

^4 Diemf u_b pd v

r ð

⁴⁸

Þ

proposed by SIT and GRACE [230] has proved useful for describing the mass-transfer coefﬁ-cient k _G_i associated with component i in mass transfer between the bubble and suspension phases; D_i is the molecular diffusion coefﬁcient of species i.

The freeboard space above the bubbling ﬂu-idized bed must be considered in the reactor model if the entrainment rate is high and the reactions in the freeboard are not quenched, for example, by cooling.

Most ﬂuidized-bed models include concentra-tion proﬁles only for the vertical direcconcentra-tion. This one-dimensional modeling is acceptable when the reactants are admitted uniformly over the bed cross section. If, however, reactants are metered into the bed at individual feed points, three-dimensional modeling may become necessary.

Such models have been devised for the combus-tion of coal in bubbling ﬂuidized beds [232–234].

As a rule, the modeling of solids behavior in bubbling ﬂuidized-bed reactors is based on that in stirred tanks. Fluidized-bed combustion is one

of the few exceptions; here the model must take account of the propagation of coal from the feed point if the furnace emission behavior is to be described correctly [232, 234].

Temperature Homogeneity is a virtually fun-damental property of the ﬂuidized-bed reactor.

Even so, one exception is industrially important:

in high-pressure ﬂuidized-bed furnaces, the high energy density can cause local hot spots near the fuel injection points [235]. Reactor models that take care of this have been described, e.g., in [236].

9.2.2. Circulating Fluidized-Bed Reactors In the early days of circulating fluidized-bed reactor modeling, negligible axial dispersion and laterally uniform flow structure were believed to characterize these systems. Thus, simple plug-flow models were used [237]. This approach was found to oversimplify the behavior of circulating fluidized-bed reactors, because a significant amount of axial dispersion was observed. As a result, the plug-flow model has often been modi-fied by adding a dispersion term to the balance equations. Axial dispersion coefficients have been determined by many authors who measured the residence time distribution of tracer gases [238, 239]. Typical values of P^eclet numbers they found are on the order of 10.

By means of a model reaction it has been proved that in many cases circulating ﬂuidized-bed reactors cannot be characterized by solely considering mixing phenomena [240]. Instead,

Figure 46. Two-phase model of the ﬂuidized-bed reactor

the presence of mass-transfer limitations and bypassing was found to have a significant influ-ence. In analogy to low-velocity fluidized beds a detailed description of the local flow structure within the reaction volume must serve as a basis for appropriate reactor modeling.

The highly nonuniform flow structure of circulating fluidized beds described in Section 2.9.2 has led to reactor models which separately deal with different axial zones. The bottom zone–if it exists under the given operating conditions–can be described by models whose basic approaches were originally developed for modeling of bubbling fluidized beds [241].

Modeling of the upper section of the circulating fluidized bed is in most cases based on a proper description of the heterogeneous core–annulus flow structure [242–244]. These state-of-the-art models are one-dimensional and define two phases or zones which are present at every axial location:

1. A dense phase or annulus zone: high solids concentration, gas stagnant or moving downwards

2. A dilute phase or core zone: low solids con-centration, gas ﬂowing quickly upward.

Similar to the situation in bubbling ﬂuidized beds, the two phases exchange gas with each other and are modeled by separate equations which are obtained from mass balances for each component in each phase.

Today’s models still suffer from the problem that not all ﬂuid-mechanical variables can be predicted on the basis of the operating conditions.

Instead, reasonable estimations or measurements in cold-ﬂow models are used to obtain numerical values for many variables.

A common feature of all models for the upper part of circulating fluidized beds is the descrip-tion of the mass exchange between dense phase and dilute phase. In analogy to low-velocity fluidized beds the product of the local specific mass transfer area a and the mass-transfer coef-ficient k may be used for this purpose. Many different methods for determination of values for these important variables have been reported, such as tracer-gas backmixing experiments [241], non-steady-state tracer-gas experiments [245], model reactions [244] and theoretical calculations [243].

Similar to the bubbling fluidized-bed reactor, the solids behavior of the circulating fluidized-bed reactor can usually be descrifluidized-bed as complete-ly mixed. This does not hold for riser reactors with very high gas velocities, such as those used in FCC risers (u > 10 ms^¹⁾. Here, better model-ing results will be obtained by assummodel-ing dis-persed plug flow of solids [239].

Like for bubbling fluidized beds, it can be assumed that circulating fluidized beds exhibit a high degree of temperature homogeneity even in the case of highly exothermic reactions. How-ever, in the case of very large circulating fluid-ized beds for coal combustion, significant hori-zontal and vertical temperature profiles have been observed inside the combustion chambers [246].

Despite the many uncertainties, circulating fluidized bed reactors have been modeled suc-cessfully. For example, three-dimensional gas and solids concentration profiles were calculated in circulating fluidized-bed boilers with local injection of reactants [247] and coal feeding via discrete feeding points [248].

9.3. New Developments in Modeling Fluidized-Bed Reactors

9.3.1. Computational Fluid Dynamics

The models described above follow the ‘‘classi-cal’’ chemical engineering approach which re-places the complex particle–fluid interaction in the fluidized bed by idealized configurations (plug flow, stirred tank, either valid overall or in regions) with mixing and mass-transfer coef-ficients describing the transport of matter. How-ever, more recently, there has been a strong tendency to model the fluid mechanics of fluid-ized-bed reactors from first principles. The prob-lem of computational fluid dynamics (CFD) modeling in this area is that the particle–particle and particle–fluid interaction must be considered on the particle scale, while the reactor perfor-mance must be described on a much larger scale, typically on the order of several meters. This leads to computational difficulties and problems with available computing capacities. At present (ca. 2006) there is no generally accepted CFD model of the fluidized-bed reactor available, but rapid progress can be seen in this area [249–253].

A promising approach appears to be multiscale modeling strategy [254].

The idea essentially is that fundamental mod-els which take into account the relevant details of ﬂuid–particle (lattice Boltzmann model) and par-ticle–particle (discrete-particle model) interac-tions are used to develop closure laws to feed continuum models which can then be used to simulate the ﬂow structures on a larger scale.

Figure 47 illustrates this approach, which finally leads to the discrete-bubble model and should be applicable to the large industrial scale of the bubbling fluidized-bed reactor. The multiscale methodology [255] still requires development work but provides a good chance to arrive at more realistic fluidized-bed reactor models in the not too far future.

9.3.2. Modeling of Fluidized-Bed Systems Another line of development is the modeling of ﬂuidized-bed reactor systems. Whereas previ-ously the isolated ﬂuidized bed was modeled,

the focus now is on the coupling between the ﬂuidized bed and the cyclone for catalyst recov-ery and recycle [256] or even on the coupling between two ﬂuidized-bed reactors [257], e.g., reactor–regenerator systems as are used in the FCC and maleic anhydride processes.

As an example, Figure 48 shows a ﬂuidized-bed coupled with a cyclone and its translation into the model system. Attrition leads to a loss of material from the system, which requires the addition of fresh catalyst after some time (Fig.

49). A population balance model which considers the changes in the catalyst particle size intervals allows the change in the catalyst inventory with time to be followed. We see that it takes several weeks for the system to reach a quasisteady state.

As a consequence of attrition and incomplete separation in the cyclone, the mean particle diameter in the bed increases with time, and this leads to larger bubbles and a reduced area of mass transfer between bubbles and the surrounding suspension in the bed. As a further consequence the conversion rate of a simple ﬁrst-order reaction falls off with time. Finally, Figure 50

Figure 47. The multiscale approach for CFD modeling of ﬂuidized-bed reactors [254].

shows that improvements in the efﬁciency of the solids-recovery system are able to increase the conversion rate again, which is in agreement with large-scale industrial experience [258, 259].

10. Scale-up

Typical diameters of bench-scale ﬂuidized-bed reactors are roughly 30–60 mm, and of

Figure 48. Fluidized-bed reactor model system [256]

Figure 49. Reactor behavior as a function of operating time [256]

pilot-scale units 450–600 mm, which should allow a reliable scale-up [9]. Full-scale fluid-ized-bed reactors used in the chemical industry have diameters up to ca. 10 m. Circulating fluidized bed combustors are even bigger with bed cross-sectional areas reaching 200 m² [261]. As equipment size increases, character-istic changes take place in the gas–solid flow that can decisively affect reactor performance.

Such changes result either directly from the geometry or indirectly from design changes made as the unit is enlarged. In particular, experience has shown that the following factors affect the performance of bubbling ﬂuidized beds during scale-up [262]:

Bed Diameter. According to Equation (22), the mean upward bubble velocity increases as the bed diameterd _tincreases. As a result, the bubbles have a shorter residence time in the bed; hence the exchange area between the bubble and sus-pension phases is smaller, so conversion is reduced [263]. In case of circulating ﬂuidized-bed combustors, measurements have shown that the downwards velocity of solids in the wall zone increases drastically with increasing size of the combustor [260].

Grid Design. In the laboratory, porous plates are the preferred type of gas distributor because of the ease of working with them. Gas distribution becomes worse when these are re-placed by industrial distributor designs; thus the exchange area between the bubble and suspen-sion phases is reduced, again with consequently lower conversion [43].

Internals. Whereas the laboratory fluidized bed is generally operated with no internals, plant equipment often must contain bundles of heat-exchanger tubes. Screens, baffles, or similar internals are frequently used to redisperse the bubble gas in industrial reactors. The mass-trans-fer area is thus increased relative to the fluidized bed without internals; the extra area can be utilized to partially offset the conversion-reduc-ing effects of bed diameter and gas distributor [263].

Catalyst Particle-Size Distribution. Bub-ble growth is influenced by the proportion of fines in the particle-size distribution of the bed (usually measured as the weight fraction< 0.044 mm) or by the mean grain size d _p (via u_mf, Eq. 18). If the content of fines increases, bubbles collapse sooner and the equilibrium bubble size becomes smaller, with a resultant greater bub-ble–suspension mass-transfer area. This effect generally is fully developed only in the plant-scale reactor, where bubbles can grow without the hindrance of vessel walls. Thus, in principle, the performance of catalytic fluidized-bed reac-tors can be controlled by modifying the catalyst particle-size distribution [112, 264]. The recom-mended content by weight of fines (< 0.044 mm) for ‘‘good fluidization’’ is 30–40 % [265], but maintaining this high a fines content in the system over a long span of time requires a very efficient solids recovery system.

Lateral Mixing of Reactants. On a labora-tory scale, reactants experience compulsory uni-form distribution over the bed cross section. In plant equipment, on the other hand, reactants often arrive in the reactor via individual feed points. The resulting uneven distribution of reactants can have a marked effect on reactor performance, which has been shown for the effect of coal feeding on the emission properties of ﬂuidized-bed furnaces [248].

Secondary Reactions in the Freeboard. In a bench-scale apparatus, the ﬂuidized gas is rapidly cooled by the vessel wall in the freeboard space after leaving the bed, so secondary reac-tions in the freeboard are often negligible. Such is not the case in the plant-scale reactor. The action of wall cooling is not signiﬁcant here, and the entrainment rate is high because of the higher

Figure 50. Inﬂuence of the Sauter diameter on the chemical conversion of a simple ﬁrst-order reaction [256].

fluidization velocities common in full-scale equipment. Both effects – lack of cooling and high solids concentration in the freeboard – may lead to marked secondary reactions in the free-board of industrial fluidized-bed reactors. In the case of a system of consecutive reactions where the desired product is formed as an intermediate, the freeboard reactions will generally lower the selectivity. The effect of freeboard reactions has been demonstrated for the example of NO and CO emissions from a fluidized-bed furnace [232].

Catalyst Attrition. Catalyst attrition is min-imal in laboratory apparatus, because of the use of porous plates as gas distributors, as well as the low gas velocities and bed depths. Attrition is necessarily greater in industrial reactors. To re-duce this risk in scale-up, the attrition tests described in Section 2.11 should be carried out and the results converted to the full-scale condi-tions with the aid of Equacondi-tions (34), (35) and (36).

Other Factors. In addition to the factors just listed, many other effects become apparent when a fluidized-bed reactor is scaled up that are difficult to calculate. Examples are the risk of nonuniform gas distribution over very large cross sections in shallow fluidized beds; the formation of deposits in the bed; the fouling of heat-exchange surfaces; and catalyst aging and poi-soning. On the whole, accordingly, the scale-up of fluidized-bed reactors is a complex process, commonly requiring a large amount of pilot-scale experimentation. Current knowledge about the fluid mechanics in the fluidized bed, however, enables simulation calculations of many of the scale-up effects, so the amount of testing during

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