chemical reactor modeling || fluidized bed reactors

10

Fluidized Bed Reactors

In this chapter the characteristics of fluidized gas-solid suspensions are de-scribed, and the basic designs of fluidized bed reactors are sketched. Severalmodeling approaches that have been applied to described these units are out-lined.

The term fluidization has been used in the literature to refer to dense-phase and lean-phase systems, as well as circulation systems involving fastfluidization, pneumatic transport or moving beds [56, 82]. The broad field offluidization engineering thus deals with all these modes of contacting, but thetwo major groups of fluidized bed reactors are the dense phase and lean-phasereactors. Among the dense phase reactors, the bubbling bed reactor designis most common. The lean-phase flow regimes are employed in circulatingbed reactors. The first industrial applications of the fluidized bed technologyconsidered gasification of coal and the chemical fluid catalytic cracking (FCC)process. Today, the FCC process and circulating fluidized bed combustion(CFBC) are the major technologies for circulating fluidized beds.

Moving packed beds normally consist of a stack of catalyst particles insidea tube thus resembling a fixed bed. In a moving packed bed reactor, as distinctfrom fixed bed, the gravity force is generally utilized to shift the catalyst fromtop to bottom. However, other arrangements like upwards, horizontal andinclined beds exist as well. Therefore, the moving bed reactors have manyof the same properties as fixed beds, but allow continuous regeneration ofdeactivated catalyst and lower pressure drop. Large scale operations of movingbeds can thus be employed for rapidly deactivated catalysts [82]. Temperaturegradients caused by extreme exothermic/endothermic reactions can also beminimized with appropriate solid circulation. Nevertheless, very little is knownabout the hydrodynamics, mixing, and transport characteristics of moving bedreactors. Moving beds are thus not considered further in this book.

H.A. Jakobsen, Chemical Reactor Modeling, doi: 10.1007/978-3-540-68622-4 10,c© Springer-Verlag Berlin Heidelberg 2008

868 10 Fluidized Bed Reactors

10.1 Solids Classification

When gas is passed through a bed of solid particles, various types of flowregime are observed. Operating conditions, solid flux, gas flux and systemconfiguration and the solid particle properties (e.g., mean size, size distri-bution, shape, density, and restitution coefficient) are factors that affect theprevailing flow regime. Geldart [49] investigated the behavior of solid particlesof various sizes and densities fluidized by gases. From this study a four groupclassification of solids was proposed to categorize the bed behavior based onparticle density and particle size:

• Group A: Solid particles having a small mean size 30 < dp < 100μm [52, 142] and/or low particle density <∼ 1.4g/cm3. These solidsfluidize easily, with smooth fluidization at low gas velocities and bub-bling/turbulent fluidization at higher gas velocities. Typical examples ofthis class of solid particles are catalysts used for fluid catalytic cracking(FCC) processes.

• Group B : Most solid particles of mean size 100μm < dp < 800μm [52]and density in the range 1.4g/cm3 < ρs < 4g/cm3. These solids fluidizevigorously with formation of bubbles that may grow in size. Sand particlesare representative for this group of solids.

• Group C : This class of solids includes very fine and cohesive powders.For most cases dp <∼ 20μm [52]. With these particles normal fluidationis extremely difficult because inter-particle forces are greater than thoseresulting from the action of gas. Cement, face powder, flour, and starchare representative for this group of solids.

• Group D : These solid particles are large dp >∼ 1mm [52] and/or dense,and spoutable. Large exploding bubbles or severe channeling may occur influidization of this type of solids. Drying grains and peas, roasting coffeebeans, gasifying coals, and some roasting metal ores are representative forthese solids.

Apart from density and particle size, several other solid properties, includingangularity, surface roughness and composition may also significantly affect thequality of fluidization. However, in many cases Geldart’s classification chartis still a useful starting point to examine fluidization quality of a specificgas-solid system.

10.2 Fluidization Regimes for Gas-Solid Suspension Flow

Most gas-solid systems experience a range of flow regimes as the gas velocityis increased. Several important gas-solid fluidization regimes for the chemicalprocess industry are sketched in Fig 10.1. In dense fluidized beds regions oflow solid density may be created. These gas pockets or voids are frequentlyreferred to as bubbles.

10.2 Fluidization Regimes for Gas-Solid Suspension Flow 869

Fig. 10.1. The primary gas-solid flow-regimes [52, 58]. Reprinted with permissionfrom the author 2007.

Each of the fluidization regimes has characteristic solids concentration pro-files. A plot of the profile showing the solids concentration versus the heightabove the distributor for the bubbling bed regime of fluidization takes a pro-nounced s-shape. With increasing gas velocity the s-shape profile becomesless pronounced and is almost upright or uniform for the pneumatic convey-ing regime.

The main characteristics of the pertinent gas-solid flow regimes are [144,56, 82, 47, 44]:

• Fixed bed : When a fluid is passing upward through a bed of fine particlesat a low flow rate, and the fluid merely seeps through the void spacesbetween stationary particles.

• Expanded bed : With an increase in flow rate, a few particles vibrate andmove apart in restricted ranges.

• Minimum fluidization: At a still higher velocity, a point is reached whereall the particles are just suspended by the upward-flowing gas. At thispoint the frictional force between particle and fluid just counterbalancesthe weight of the particles, and the vertical component of the compressiveforce between adjacent particles disappears. The pressure drop throughany section of the bed thus balance (approximately) the weight of fluidand particles in that section. Further increase in the gas velocity flow ratewill not change the pressure drop noteworthy.


• Smooth fluidization: In fine particle A beds, a limited increase in gas flowrate above minimum fluidization can result in smooth, progressive expan-sion of the bed. Bubbles do not appear as soon as the minimum fluidizationstate is reached. There is a narrow range of velocities in which uniform ex-pansion occurs and no bubbles are observed. Such beds are called a partic-ulate fluidized bed, a homogeneously fluidized bed, or a smoothly fluidizedbed. However, this regime does not exist in beds of larger particles of typeB and D, in these cases bubbles do appear as soon as minimum fluidizationis reached.

• Bubbling fluidization: An increase in flow rate beyond the point of mini-mum fluidization results in large instabilities with bubbling and channelingof gas. At higher flow rates, agitation becomes more violent and the move-ment of solids becomes more vigorous. Bubble coalescence and breakagetake place, and with increasing gas velocity the tendency of bubble coa-lescence is normally enhanced. However, the bed does not expand muchbeyond its volume at minimum fluidization.

• Slugging fluidization: The gas bubbles coalesce and grow as they rise, andin a deep enough bed of small diameter they may eventually become largeenough to spread across the vessel. Fine particles flow smoothly down bythe wall around the rising void of gas. These voids are called axial slugs.For coarser particle beds, the portion of the bed above the bubble is pushedupward. Particles fall down from the slug, which finally disintegrates. Thenanother slug forms, and this unstable oscillatory motion is repeated. Thisis called a flat slug. Slugging normally occurs in long, narrow fluidizedbeds.

• Turbulent fluidization: This is often regarded as a transition regime frombubbling to lean phase fluidization. At relatively low gas velocities in thisregime, bubbles are present. Moreover, when fine Geldart A particles arefluidized at a sufficiently high gas flow rate, the terminal velocity of thesolids is exceeded, thus the upper surface of the bed becomes more diffusewith a large particle concentration in the freeboard, the solids entrainmentbecomes appreciable, and a turbulent motion of solid clusters and voids ofgas of various sizes and shapes occurs. In contrast to the bubbling regime,in this regime the tendency for bubble breakage is enhanced as the gasvelocity increases. For this reason the mean bubble size is significantlysmaller than in the bubbling regime, hence the suspension becomes moreuniform as the gas velocity further increases toward the lean phase fluidiza-tion regimes. However, for very high gas velocities within this regime pro-nounced radial gradients may occur, with a marked tendency for solids tobe present in much greater concentration in the wall region, while the coreof the column has a significantly smaller volume fractions of particles [56].

• Dense phase fluidization: Gas fluidized beds are considered dense phasefluidized beds as long as there is a clearly defined upper limit or surfaceto the dense bed. The dense-phase fluidization regimes include the smoothfluidization, bubbling fluidization, slugging fluidization, and turbulent

10.2 Fluidization Regimes for Gas-Solid Suspension Flow 871

fluidization regimes. In a dense-phase fluidized bed the particle entrain-ment rate is low but increases with increasing gas velocity.

• Spouting bed fluidization: Spouting can occur when the fluidizing gas isinjected vertically at a high velocity through a small opening into a bedof Geldart D particles. The gas jet penetrates the whole bed and form adilute flow in the core region. A solids circulation pattern is created, as theparticles carried upwards to the top of the core region by the fluidizinggas move downward in a moving bed mode in the annular region. Thegross scale particle circulation induced by the axial spout gives rise tomore regular and cyclic mixing behavior than in bubbling and turbulentfluidization.

• Channeling : In a fluidized bed channeling frequently occur if the gas distri-bution is nonuniform across the distributor. Channeling can also be causedby aggregation effects of cohesive Geldard C particles due to inter-particlecontact forces.

• Lean phase fluidization: As the gas flow rate increases beyond the pointcorresponding to the disappearance of bubbles, a drastic increase in theentrainment rate of the particles occurs such that a continuous feedingof particles into the fluidized bed is required to maintain a steady solidflow. Fluidization at this state, in contrast to dense-phase fluidization, isgenerally denoted lean phase fluidization. Lean phase fluidization encom-passes two flow regimes, these are the fast fluidization and dilute transportregimes.

• Fast fluidization: The fast fluidization regime is considered to be initiatedwhen there is no longer a clear interface between a dense bed and a moredilute freeboard region. Instead, there is a continuous, gradual decrease insolids content over the whole hight of the column. Particles are transportedout of the top of the vessel and must be replaced by adding solids nearthe bottom. Clusters of particles move downwards near the wall, while gasand entrained dispersed particles move upward in the core of the vessel.The term clustering refers to the phenomenon that solids coalesce to forma larger pseudo-particle.

• Dilute transport fluidization: The gas velocity is so large that all the par-ticles are carried out of the bed with the gas. This solid transport by gasblowing through a pipe is named pneumatic conveying. In vertical pneu-matic transport, particles are always suspended in the gas stream mainlybecause the direction of gravity is in line with that of the gas flow. Theradial particle concentration distribution is almost uniform. No axial vari-ation of solids concentration except i the bottom acceleration section [58].

The model predictions of the transition borders between the different gas-solid flow regimes shown in Fig 10.1 are still not reliable. The borders aregenerally sharp and fairly well correlated for the minimum fluidization (mf)and minimum bubbling (mb) transitions, whereas the transitions at highergas superficial velocities are diffuse and poorly understood.


The general flow regime diagram shown i Fig 10.2 illustrates the progres-sion of changes in behavior of a bed of solids as the gas velocity is progres-sively increased. The letters A, B, C and D refer to the Geldart classificationof solids.

10.3 Reactor Design and Flow Characterization

The fluidized bed reactors can roughly be divided into two main groups inaccordance with the operating flow regimes employed. These two categoriesare named the dense phase and lean phase fluidized beds.

Fig. 10.2. Flow regime map of gas-solid contacting. In the figure notation theordinate u∗ = Us

in[ρ2g/{μ(ρp − ρg)g}]1/3 is a dimensionless gas velocity, the abscissa

d∗p = dp[ρg(ρp − ρg)g/μ2]1/3 a dimensionless particle size, ut the terminal velocity

of a particle falling through the gas (m/s), and umf the gas velocity at minimumfluidization (m/s). Reprinted from [83] with permission from Elsevier.

10.3 Reactor Design and Flow Characterization 873

10.3.1 Dense-Phase Fluidized Beds

A dense phase fluidized bed generally consist of a gas distributor, a cyclone,a dipleg, a heat exchanger, an expanded section, and baffles [44]. A schematicrepresentation of a dense phase fluidized bed reactor is shown in Fig 10.3.

At the bottom of the vessel is the gas distributor which yields the desireddistribution of fluidizing gas and supports particles in the bed. A distrib-utor with sufficient flow resistance to obtain a uniform distribution of gasacross the bed is required, and sometimes caps are used to avoid gas jettingeffects and plate clogging by fine particles. There are two basic designs ofdistributors, some for use when the inlet gas contains solids and others foruse when the inlet gas is clean. For clean gases the distributor is designedto prevent back flow of solids during normal operation, and in many cases itis designed to prevent back flow during shutdown. Perforated plate distribu-tors are widely used in industry because they are cheap and easy to fabricate[82]. However, the perforated plate distributors cannot be used under severeoperating conditions, such as high temperature or highly reactive environ-ment. The tuyere type of distributors are normally used in these situations,

Fig. 10.3. A typical dense phase bubbling bed reactor design. The reactor consists ofa gas distributor, an internal cyclone with solids recycle through a dipleg, and a heatexchanger. The freeboard section of the vessel is expanded and the heat exchangermay also function as a baffle. Reprinted with permission from [136]. Copyright 2004American Chemistry Society.


but these are more expensive than perforated plates. The tuyere distributordesign prevents solids from falling through the distributor. The high pressuredrop porous plate distributors are commonly used in laboratory scale units toensure even distribution of gas across the bed entrance.

A dense phase fluidized bed vessel has two zones, a dense phase having adistinct upper surface separating it from an upper dilute phase. The section ofthe vessel between the surface of the dense phase and the exit of the gas streamis called the freeboard zone. In vessels containing fluidized beds, the gas leavingthe dense bed zone carries some suspended particles. Particle entrainmentrefers to the ejection of particles from the dense bed into the freeboard bythe fluidizing gas. Particle elutriation refers to the separation of fine particlesfrom a mixture of particles and their ultimate removal from the freeboard.The flux of solids leaving the freeboard with the gas is called carry-over. Theentrained solids are normally separated from the outlet gas by internal orexternal cyclones and returned to the bed. In many cases several cyclones arecombined to form a multistage cyclone system. A dipleg returns the particlesseparated by the internal cyclones into the vessel. The outlet of a dipleg maybe located in the freeboard or immersed in the dense bed. A standpipe can beused to return the particles separated by the external cyclones into the densebed. Sometimes a heat exchanger device is placed in the dense bed or thefreeboard to control the temperature. The heat exchanger removes generatedheat from or adds required heat to the fluidized bed by a cooling or heatingfluid. An expanded freeboard section on the top of the vessel may be used toreduce the local gas velocity in the freeboard so that settling of the particlescarried by the fluidizing gas can be efficiently achieved. Any internals otherthan diplegs can be employed as baffles to restrict flow, enhance the breakageof bubbles, promote gas-solid contact, and reduce particle entrainment.

The primary bubbling fluidized bed consists of gas bubbles flowing througha dense emulsion phase with gas percolating through the bed of solids. Afluidized bed reactor of this type designed for catalytic reactions operatedin the bubbling bed regime has been shown in Fig 10.3. In many cases theparticle entrainment rate is so low that the cyclone is considered superfluous.

In the turbulent fluidized bed larger amounts of particles are entrainedprecluding steady state operations. To maintain steady state operation theentrained particles have to be collected by cyclones and returned to the bed.For vessels operating in the turbulent fluidization regime internal cyclonesmay deal with the moderate rate of entrainment. This fluidization system, assketched in Fig 10.4, is sometimes called a fluid bed. Since smooth and steadyrecirculation of solids through a solid trapping device is crucial for optimaloperations of these units, Kunii and Levenspiel [82] included the turbulentfluidized beds in the reactor classification named circulating fluidized beds(i.e., the main lean-phase reactor design). However, this is not a conventionalclassification of the turbulent bed operation mode.

Bubbling and turbulent fluidized beds are operated with small granular orpowdery non-friable catalysts. Rapid deactivation of the solids can then be


handled, and the efficient temperature control allows large scale operations.The main advantages of a turbulent fluidized bed over a standard bubblingfluidized bed are a more homogeneous fluidization that provides better con-tacting between gas and catalyst (i.e., low gas bypassing), and higher heattransfer coefficients between the suspension and heat transfer surfaces.

In design of fluidized bed systems the cross sectional area is determinedby the volumetric flow of gas and the allowable or required fluidizing velocityof the gas at operating conditions. Generally, bed heights are not less than0.3 m or more than 15 m [111]. For fluidized bed units operated at elevatedtemperatures refractory-lined steel is the most economical material.

Fig. 10.4. A schematic representation of a turbulent fluidized bed. The illustrationshows that in a turbulent fluidized bed entrainment is significant and an internalcyclone with solids recycle through a dipleg is required. Reprinted from [82] withpermission from Elsevier.

10.3.2 Lean-Phase Fluidized Beds

The primary exploitation of the lean-phase fluidized beds is associated withthe circulating fluidized bed (CFB) reactors.

The operation of circulating fluidized bed systems requires that both thegas flow rate and the solids circulation rate are controlled, in contrast to thegas flow rate only in a dense phase fluidized bed system. The solids circulationis established by a high gas flow.


The integral parts of a CFB loop are the riser, gas-solid separator, down-comer, and solids flow control device [44]. The CFB is thus a fluidized bedsystem in which solid particles circulate between the riser and the downcomer,as illustrated in Fig 10.5. The riser is the main component of the system. Thename riser is generally used to characterize a tall vessel or column that pro-vides the principal reaction zone. On average, the particles travel upwards(or rise) in the riser, though the motion at the wall may be downwards. Thefluidized gas is introduced at the bottom of the riser, where solid particlesfrom the downcomer are fed via a control device and carried upwards in theriser. The fast fluidization regime is the principal regime under which theCFB riser is operated. The particles exit at the top of the riser into the gas-solid separators which are normally cyclones. In lean-phase fluidized beds, therate of entrainment is far larger than in turbulent fluidized beds, and biggercyclone collectors outside the bed are usually necessary. The separated par-ticles then flow to the downcomer and return to the riser. The entrance andexit geometries of the riser often significantly affect the gas and solid flowbehavior in the reactor. The efficiency of the cyclones determine the particlesize distribution and solids circulation rate in the system. The downcomerprovides hold volume and a static pressure head for particle recycling to theriser. The downcomer can be a large reservoir which aids in regulating thesolids circulation rate, a heat exchanger, a spent solid regenerator, hopper ora standpipe. The main task in achieving smooth operation of a CFB systemis to control the solids recirculation rate to the riser. Several designs of valvesfor solids flow control are used. The solids flow control device serves two mainfunctions, namely to seal the riser gas flow to the downcomer and to con-trol the solids circulation rate. Rotary valves are effective sealing devices forsolids discharge. The L-valve can act as a seal and as a solids-flow controlvalve. There are many other valve designs available to suit specific conditions.The riser cannot be considered as an isolated entity in the CFB loop becausethe pressure drop over the riser must be balanced by that imposed by theflow through its accompanying components such as the downcomer and therecirculation device.

In general, the high operating gas velocities for lean phase fluidizationyield a short contact time between the gas and solid phases. Fast fluidizedbeds and co-current pneumatic transport are thus suitable for rapid reactions,but attrition of catalyst may be serious.

However, it is not always easy to distinguish between the flow behav-ior encountered in the fast fluidization and the transport bed reactors [56].The transport reactors are sometimes called dilute riser (transport) reactorsbecause they are operated at very low solids mass fluxes. The dense risertransport reactors are operated in the fast fluidization regime with highersolids mass fluxes. The transition between these two flow regimes appears tobe gradual rather than abrupt. However, fast fluidization generally applies toa higher overall suspension density (typically 2 to 15% by volume solids) andto a situation wherein the flow continues to develop over virtually the entire


Fig. 10.5. A schematic representation of a circulating fluidized bed. The CFB loopconsists of a riser, gas-solid cyclone separators, standpipe type of downcomer, and anon-mechanical solids flow control device. Reprinted from [82] with permission fromElsevier.

height of the reactor, whereas the flow usually associated with transport bedreactors tends to be more dilute (typically 1 to 5 % by volume solids) anduniform. However, in practice the differences in operation are generally small,hence the names are often used in an indistinguishable manner1. This regimeoverlap is also indicated in the regime map shown in Fig 10.2. The pertinentcharacteristics that distinguish the CFB from the dense phase fluidized bedsand from the riser (transport) reactors are summarized in Table 10.1.

Advantages of the fast fluidization regime, relative to the dense phase flu-idization regimes, include higher gas throughput per unit area, adjustableretention time of solids, limited axial dispersion of gas coupled with nearuniformity of temperature and solids composition, reduced tendency for par-ticles to undergo agglomeration, and possibility of staged addition of gaseous

1 Berruti et al [13], for example, used the term CFB to generically describe systemslike fast fluidized bed, riser reactor, entrained bed, transport bed, pneumatictransport reactor, recirculating solid riser, highly expanded fluid bed, dilute phasetransported bed, transport line reactor and suspended catalyst bed in co-currentgas flow.


reactants at different levels. Gas-solid contacting also tends to be very fa-vorable. However, by increased overall reactor height and added complexityin designing and operating the recirculating loop the CFB systems tend tohave higher capital costs than low-velocity systems, so that one or more ofthe above advantages must be very significant for this option to be viable.One of the most important factors inhibiting the commercialization of novelprocesses operated in risers is scale-up uncertainties arising from the complexhydrodynamic behavior of the CFB reactors.

Table 10.1. The key features that distinguish circulating fluidized bed reactorsfrom low-velocity fluidized beds and from lean-phase transport reactors [58].

Low-velocity FB Reactor CFB Reactor Transport Reactors

Particle residence Minutes or hours Seconds Once throughtimes in Reactor systemFlow regime Bubbling, slugging Fast fluidization Dilute transport

or turbulent,distinct upper interface

Superficial Less than 2 m/s 3 to 16 m/s 15 to 20 m/sgas velocityMean particle 0.03 to 3 mm 0.05 to 0.5 mm 0.02 to 0.08 mmdiameterNet circulation 0.1-5 kg/m2s 15-1000 kg/m2s Up to ∼ 20 kg/m2sof solidsVoidage 0.6-0.8 0.8-0.98 less than 0.99Gas mixing Intense Intermediate Little

A combination of a circulating fluidized bed riser reactor operating in thefast fluidization regime and a bubbling fluidized bed regenerator is frequentlyused in industry for heterogeneous catalyzed gas phase reactions in cases wherethe catalyst rapidly deactivates and has to be regenerated continuously. Sucha catalytic circulating fluidized bed reactor design is sketched in Fig 10.6.

The most prominent chemical reactions operated within such a reactordesign is the FCC process, which is widely used in the modern petroleumrefinery industry. In this catalytic chemical process vaporized heavy hydro-carbons crack into lower-molecular-weight compounds. To explain the princi-pal operating principles of this particular CFB unit, a FCC riser reactor canbe divided into four parts from bottom to top according to their functions[45]: The prelift zone, the feedstock injection zone, the reaction zone, andthe quenching zone. In the prelift zone, catalysts enter the riser reactor fromthe regenerator and are then conveyed by the prelift gas. In the feedstockinjection zone, feed oil is introduced into the riser through the feed nozzles,and the heavy oil comes in contact with the high-temperature catalyst. Rapidreactions are then taking place in the reaction zone.


Apart from the mentioned application of CFB to the fluid catalyticcracking (FCC) process, circulating fluidized beds utilizing the fast fluidiza-tion regime have been used for a number of gas-solid reactions includingcalcination, combustion of a wide variety of fuels, gasification, and dry scrub-bing of gas streams [56]. Applications for catalytic reactions can be taken toinclude the transport reactors employed in modern catalytic cracking opera-tions in the petroleum industry and certain Fischer-Tropsch synthesis reactors.

Fig. 10.6. A typical catalytic circulating bed reactor design. This CFB loop consistsof a riser, gas-solid cyclone separators, and a downcomer. In this particular casethe downcomer consists of a spend solid regenerator. Reprinted from [135] withpermission from Elsevier.

Generally, to maximize profitability, the gas and solids residence times inCFBs are chosen to achieve the highest product yield per unit volume [14]. InFCC units, for example, a short and uniform catalyst residence time in theriser reactor, with reduced back-mixing, leads to better reactor performance byreducing the inventory of the deactivated catalyst in the riser. In other words,a uniform radial profile of solids velocity and little solids back-mixing in theriser is preferred, leading to shorter and more uniform solids residence times.In a FCC unit axial mixing is disadvantageous. For other catalytic reactions,lower gas velocities may be preferred because this gives higher solids holdup,thus maximizing the specific activity per unit volume [13].


Scale-up of CFBs is generally less of a problem than with bubbling beds[111]. Moreover, the higher velocity in CFB means higher gas throughput,which can minimize the reactor costs. Several CFB loop designs have beenproposed for getting smooth steady state circulation of solids. Basically, thereare two basic types of solids circulation loops distinct in that some includea reservoir of solids while others do not. The solids circulation loops whichdo not include a reservoir of solids (hopper) are less flexible in operationcompared to the circulation systems with reserviors.

10.3.3 Various Types of Fluidized Beds

Numerous types of fluidized bed reactor designs exist within each of the twocategories mentioned in the previous subsection, some of them are illustratedin Fig 10.7. The key issues leading to re-design of the primary bubbling bedare also indicated.

10.3.4 Experimental Investigations

The first experimental investigations of bubbling bed fluidization led to theflow interpretation that the bubbles are flowing evenly through an essentiallystagnant emulsion phase without affecting the flow of the emulsion phase. Thispicture of the bubbling bed hydrodynamics is named the two-phase theory[130, 29].

The subsequent experimental investigations on solids mixing and circula-tion in bubbling beds revealed that the bubbling phenomenon creates particlecirculation patterns [5, 53, 82, 116]. Moreover, the axial and radial transportof solids within the fluidized bed influence many parameters governing thechemical process performance in these units. Most important, the heat trans-port within the bed is efficient due to the chaotic motion of solid particleshaving the property of high heat capacity compared to the fluidizing gas,making the bed close to isothermal.

Lin et al [90] were among the first to use the modern non-invasive Com-puter Automated Radioactive Particle Tracer (CARPT) technique to mea-sure the Lagrangian solid particle motion in gas-solid bubbling fluidized beds.This advanced measuring technique determines the time-average solids ve-locity components in all three space dimensions simultaneously so that thederived Eulerian flow field map and various turbulence fields can be visu-alized. A number of experiments were conducted measuring mean velocitydistributions for Geldart B glass beads of density 2.5 (g/cm3) and diametersranging from 0.42 − 0.60 (mm) in an air fluidized bed of 13.8 (cm) diameter,at various superficial air velocities ranging from 32−89 (cm/s). The mean cir-culating flow patterns observed are summarily presented in Figs 10.8 (a)-(d).For bubbling fluidized beds consisting of Geldart A and B particles operatingwell above the minimum fluidization conditions the bubble flow will cause thesolid particles to develop certain characteristic flow patterns. For a shallow


Fig. 10.7. Types of fluidized bed reactors [77]. Reprinted with permission from theauthors.

The primary fluidized bed The key issues leading to re-design ofreactor types: the conventional bubbling fluid bed:

1: Bubbling Fluidized Bed A: Higher gas velocity2: Turbulent Fluidized Bed B: Counter-current contacting is beneficial3: Circulating Fluidized Bed C: Incompatible differences in desired environment4: Riser D: Dusty environment5: Downer E: Large particles/low gas load6: Cross-current Fluid Bed7: Counter-current Fluid Bed8: Spouted Fluidized Bed9: Floating Fluidized Bed10: Twin Fluidized Bed

bed there is normally an upflow of particles near the wall and a downflow inthe center of the bed at low gas velocities, as shown in Figs 10.8 (a) and (b).Increasing the gas velocity may reverse this flow pattern. In this case the bedapparently consists of one circulating cell just above the distributor similarto the one observed in shallow beds with upflow near the wall and down flowin the center. In the upper part of the bed the solids have an upward motion


in the center and downwards near the wall. This flow pattern is shown inFigs 10.8 (c) and (d). An additional advantage associated with the CARPTtechnique is that it can be used to characterize dense high flux suspensionsas emphasized in industrial practice such as fluid catalytic cracking (FCC)units and CFB combustors. The CARPT measuring technique has thus beenapplied by Bhusarapu et al [14] investigating the solids velocity field in gas

Fig. 10.8. Particle circulation patterns at various fluidizing velocities for a gasfluidized bed ID 13.8 (cm) consisting of 0.42 − 0.6 (mm) diameter glass beads [90].L∗ denotes the static bed height. Case a) Us

in = 32 (cm/s) and Usin/Us

mf = 1.65,Case b) Us

in = 45, 8 (cm/s) and Usin/Us

mf = 2.36, Case c) Usin = 64, 1 (cm/s) and

Usin/Us

mf = 3.41, and Case d) Usin = 89, 2 (cm/s) and Us

in/Usmf = 4.6. Reprinted

with permission of John Wiley & Sons Inc, Copyright 1985.

10.4 Fluidized Bed Combustors 883

solid risers. Measurements were performed in two different risers at low andhigh solid fluxes at varying superficial gas velocity spanning both the fast-fluidized and dilute phase transport regimes.

Based on the experimental investigations reported in the literature, usingboth invasive and non-invasive techniques [129], it is recognized that mostCFBs operating in the fast fluidization flow regime are subject to predom-inantly downflow of relatively dense suspensions along the outer wall whilethere is a net dilute upflow in the core [62, 99, 133, 73]. The fast fluidizationregime is also characterized by a dense region at the bottom of the riser and adilute region at the top. Due to the large reflux and density of the suspension,the temperature gradients are normally very small.

In vertical pneumatic transport the radial particle concentration distribu-tion is almost uniform, but some particle strands may still be identified nearthe wall. Little or no axial variation of solids concentration except in the bot-tom acceleration section is observed [58]. The flow associated with transportbed reactors tends to be dilute (typically 1 to 5 % by volume solids) anduniform. By virtue of the smaller reflux and density of the suspension withinthe dilute pneumatic conveying regime, there might be larger temperaturegradients than within the fast fluidization regime [56].

Optical techniques like laser doppler anemometry (LDA) can be used toobtain knowledge about the local solids hydrodynamics in CFB units close towalls at low solids fluxes [14]. Such LDA measurements of FCC particles in ariser in circulating fluidized bed have been reported by [119, 120].

An overview of sources of experimental data in the open literature charac-terizing the hydrodynamics of CFB risers can be found in [13]. These investi-gations might be useful for CFB riser model validation. However, despite thedevelopment of novel experimental techniques and many experimental inves-tigations, there is still considerable uncertainty and disagreement with regardto the dependence of fine scale structures on the operating conditions. Thisdependency is important in scale-up, design, and optimization of these units.

10.4 Fluidized Bed Combustors

Although the scope of this book is the fluidized bed vessels used as chem-ical reactors, a brief outline of the combustor units representing the mostwidespread use of this technology is considered very useful in understand-ing the chemical reactor operation but also to orientate oneself in the vastliterature on fluidization technology. Recent reviews of fluidized bed combus-tion systems are given by Anthony [2], Leckner [86], Longwell et al [93] andIssangya et al [67].

The popularity of fluidized bed combustion is due largely to the technol-ogy’s fuel flexibility and the capability of meeting sulfur dioxide and nitrogenoxide emission requirements without the need for expensive flue-gas treatment.


In the early 1960s engineers in Britain and China considered fluidizedbed combustion (FBC) to be an alternative future combustion technologyto enable utilization of low-grade coal and oil shale fines, fuels that can-not be burned efficiently in conventional boiler furnaces [82]. Several powergeneration cycles utilizing the fluidization technology were commercialized[86, 18, 87]. The working principles of the conventional cycles can be out-lined as follows. The fluidized bed boilers supply steam to a Rankine cycle.The efficiency of the electric power production has been further increased in acombined cycle with a pressurized FBC serving as the heat source for both thesteam and the gas turbine cycle. More advanced cycles, such as the air blowngasification cycle and the integrated gasification combined cycle (IGCC), arecurrently being developed.

In a fluidized bed combustor the bed is made up primarily of inert materialwhich may be ash, absorbent, or some other inert material such as sand. Thetechnology’s fuel flexibility arises from the fact that the fuel is present in thecombustor at a low level and are burnt surrounded by these inerts [2]. Thesolid fuel normally represent between 0.5 and 5% of the total bed material.In general, almost any solid, liquid, slurry or gas containing carbon, hydrogenand sulfur can be used as fuels for energy production.

The first fluidized bed gasifiers were designed for burning coal. The secondgeneration units were utilizing petroleum fuels. However, in order to compen-sate for the shortage of petroleum, the utilization of coal for combustion andgasification has again become dominating. Currently almost half of the totalworldwide FBC capacity is primarily fueled by coals [87]. Other fossil fu-els like oil and natural gas can also be burned effectively and efficiently ina CFB unit. Nevertheless, other combustable materials like petroleum coke,biomass and municipal waste are gaining in popularity. In particular, wasteand biomass are used to replace a part of the coal as CO2-neutral fuels. Inaddition, co-firing coal and petroleum coke can also be beneficial.

The first fluidized bed applications employed bubbling bed boilers, butproblems with erosion of in-bed cooling tubes diverted the mainstream ofdevelopment towards the circulating fluidized bed boiler [86]. The first circu-lating fluidized bed combustion (CFBC) systems were developed in the late1970s by Ahlstrom Pyropower in Findland, Lurgi in Germany, and StudsvikEnergiteknik in Sweden [18]. However, bubbling beds remain important forparticular applications for which they have cost advantages.

Today the circulating fluidized bed (CFB) has become the dominating de-sign for combustors operated at atmospheric pressure. Pressurized circulatingfluidized bed combustors are under development for combined power cycleapplications, but so far no clear advantages have been revealed yet. For thisreason the existing commercial pressurized fluid bed systems are bubblingbeds.

During the last decades several FBC technology’s have become availablefor the combustion of coal and alternative fuels, with the trend that circulating


fluidized bed combustion (CFBC) is prevailing over bubbling fluidized bedcombustion (BFBC) [4, 2].

One of the most attractive features of FBC, employing bubbling and cir-culating beds, is the potential to use a low cost sorbent to capture sulfur (insitu) within the fluidized bed in cases where high sulfur-fuels are burnt. Thesorbent is typically limestone or dolomite (minerals composed of calcium andmagnesium carbonates) and is fed to the bed either together with the fuel oras a separate solid stream [88, 3]. More than 90 % of the sulfur pollutants incoal can be captured by the sorbent. Low NOx emissions is enforced since thefuel is burnt at temperatures of about 750 to 950◦C, well below the thresholdwhere nitrogen oxides form (i.e., nitrogen oxide pollutants are produced atabout 1400◦C). The environmental pollution by combustion ash containingresidual sorbents must also be treated properly [80].

It should be mentioned that the combustion technology is not limited tothese major designs. In catalytic fluidized bed combustion of low-sulfur naturalgas, for example, powder catalysts are operated in the turbulent flow regimewhere the gas-solid contact is optimal so as to maintain a high combustionefficiency [46].

Fig. 10.9. A typical circulating fluidized bed combustor design [63]. The furnace(riser) is normally operated in the fast fluidization regime. The ash which is entrainedfrom the furnace is separated from the flue gas in the cyclone. Most of the ashparticles are sent into the siphon. The siphon is a small bubbling fluidized bed actingas a pressure lock. From the siphon the ash flows back into the riser. Reprinted withpermission from Elsevier, copyright Elsevier 2007.


The fast fluidization regime is most often encountered in circulating flu-idized beds where provision for continuous return of a significant flow of en-trained solids is an integral part of the equipment. In combustion systems,the return is accomplished by capturing the entrained solids in one or moreexternal cyclones or in impingement separators. These key features of a circu-lating fluidized bed combustion system are shown schematically in Fig 10.9.The captured particles are sent back to the base of the reactor (riser) througha vertical standpipe (downcomer), and then through a non-mechanical sealor a non-mechanical valve (mechanical valves are more common in FCC in-stallations). The bottom section of the riser might also be tapered to preventsolids from sitting and agglomerating in the bottom section. In some cases,the solids pass through a separate low-velocity fluidized bed heat exchangeror a siphon only (equivalent to a catalyst regenerator in a FCC reactor instal-lation) during their journey from cyclones capture to re-injection.

The design of CFB employed in chemical reactor engineering and circulat-ing fluidized bed combustion may be distinguished by the aspect ratio (H/D)of the riser [106, 67]. For chemical process analysis tall and narrow riser unitswith an aspect ratio of the order of 20 or higher is normally used. A chemi-cal reactor utilizing the fast fluidization regime typically operates at high netsolids fluxes for the purpose of producing product chemicals. A typical indus-trial scale CFB combustor is designed as larger units, with a lower aspect ratiotypically less than 10, for the purpose of producing heat, electricity, fuel-gasor combinations of these.

The group B solids normally used in large scale CFBC and circulatingfluidized bed gasification (CFBG) units consists of silica sand and/or primaryashes, and sorbent in the case of coal-fired units. We reiterate that the com-mon FCC particles belongs to the Geldart group A. In particular, the particlesizes applied to fluid bed combustion are normally in the range 150 − 250(μm), whereas for catalytic cracking and other chemical processes finer parti-cles with sizes in the range 60−70 (μm) are used [67]. The flow pattern in CFBgasifiers and combustors is similar since the overall riser geometry, fluidizationconditions and properties of the solids used are similar [106]. However, minordifferences in the flow behavior can occur because of corner effects since com-bustors generally have a square or rectangular cross section and may partlyhave bare membrane tube walls, whereas the riser of a gasifier normally hasa circular cross section with plane walls. Nevertheless, the flow pattern of thelarge CFB combustors and the gasifier units differs significantly from thoseof tall and narrow CFB reactor units (which normally have a circular crosssection) due to the dissimilar operating conditions employed, so the abundantliterature on CFBC and CFBG is seldom applicable for CFB reaction technol-ogy and visa versa. The important features of the two principal applicationsof the fast fluidizization regime, the fluid catalytic cracking riser reactor andthe solid fuel combustion vessel, are compared in Table 10.2.

Although considerable work has been done to understand the flow dynam-ics of CFBs, much of the CFB data reported are for low density CFB systems


representative of CFB combustors. Further work is thus required on high den-sity systems to better understand the riser reactor behavior leading to morereliable scale-up of these units.

Table 10.2. Comparison of typical operating conditions for the two principal ap-plications of fast fluidization: fluid catalytic cracking and circulating fluidized bedcombustion [56, 67].

FCC Riser Reactor CFB Combustor

Particle density, (kg/m3) 1100-1500 1800-2600Mean particle diameter, (μm) 60-70 150-250Particle size distribution Broad BroadGeldart powder group A BInlet Superficial gas velocity, (m3/m2s) 8-18 5-9Exit temperature, (C) 500-550 850-900Temperature uniformity Gradients UniformPressure, (kPa) 150-300 110-120Net Solid Flux, (kg/m2s) 400-1400 10-100Suspension density, (kg/m3) 50-80 at the top 10-40 at the topExit geometry Various AbruptRiser cross-section geometry circular rectangular/squareRiser diameter (m) 0.7-1.5 8-10Height-to-diameter-ratio > 20 < 5 − 10Average solids residence time per pass (s) 2-4 20-40

Many of the modern combustion processes can be characterized by rela-tively low reaction rates compared to the modern catalytic processes operatedin chemical reactors [67]. Therefore, these combustion processes do requirelower gas velocities and higher solids circulation rates. On the other hand,many catalytic gas-phase reactions, including FCC, Fischer-Tropsch synthe-sis and oxidation of butane, utilize a relatively high gas velocity in the riserto promote plug flow operating conditions and short contact times betweenthe gas and solids.

The solids residence time distribution (RTD) in the riser may thus beimportant in non-catalytic gas-solid reactions, as in a combustor, since thisdistribution characterizes the degree of solids mixing and provides informationabout the physical properties of the solid particles in the riser [14]. Moreover,lateral mixing and internal recirculation of solids in a CFB combustor arenecessary to maintain uniform temperatures over the entire length of the riser.Hence, lateral and longitudinal mixing is advantageous in a CFB combustor.

Prediction of the flow and transport processes is crucial in modeling theheat transfer and combustion/gasification gas produced. The conventionalmodeling of bubbling and circulating fluidized bed coal combustors were out-lined by Arena et al [4].


Saraiva et al [121] presented an extended model for a circulating atmo-spheric fluidized bed combustor (CAFBC) which included hydrodynamics forthe fast section at the top of the bed as well a bubbling bed section at thebottom of the CAFBC. For the fast section of the bed, one dimensional mo-mentum and energy balances were used to predict the temperature and veloc-ity profiles for gas and particles throughout the reactor. The model containspecies mass balances for five gas species including SO2, as well as a model ofSO2 retention by limestone particles. A bubbling bed model was consideredto simulate the chemical process at the bottom of the combustor.

Recently, Pallares and Johnsson [106] presented an overview of the macro-scopic semi-empirical models used for the description of the fluid dynamics ofcirculating fluidized bed combustion units. They summarized the basic mod-eling concepts and assumptions made for each model together with the majoradvantages and drawbacks. In order to make a structured analysis of the pro-cesses involved, the CFBC unit is often divided into 6 fluid dynamical zoneslike the bottom bed, freeboard, exit zone, exit duct, cyclone and downcomerand particle seal, which have been shown to exhibit different fluid dynamicalbehavior.

10.5 Milestones in Fluidized Bed Reactor Technology

Fuel conversion in a fluidized bed was first introduced by Winkler whopatented a gasifier in 1922 [137]. The first large-scale use of fluidized beds, theWinkler gas generator, was thus established for the process of gasification ofcoal in 1926 [82, 47]. These units were 13 (m) high, 12 (m2) in cross sectionand fed by powered coal to produce synthesis gas for the chemical indus-tries2. A sketch of the pioneering Winkler gas generator is shown in Fig 10.10(a). A number of more efficient bubbling bed and lean-phase CFB fluidiza-tion technologies for gasifying coal have been developed over the years [37].An informative overview of the gasification chemistry, gasifier types and coalgasification reactor models is given by Denn and Shinnar [32].

In chemical reactor engineering, on the other hand, the fluidization tech-nique is considered initiated by the cooperative work of the Standard OilDevelopment Co, the MW Kellogg Co, and Standard Oil of Indiana devel-oping the first FCC unit. With war threatening in Europe and the Far Eastaround 1940, the chemical engineering community in USA was urged to findnew ways of transforming kerosene and gas oil into high-octane gasoline fuels.The real breakthrough of the fluidized bed technology was thus associatedwith the catalytic cracking of gas-oil into gasoline, first practiced in 1942 atthe Baton Rouge refinery of Standard Oil of New Jersey (now Exxon) [82, 146].

2 The gasification process involves conversion of coal and air (or oxygen) into agaseous mixture of mainly CO, CO2, H2, H2O, CH4 and N2.

10.5 Milestones in Fluidized Bed Reactor Technology 889

Before that, the catalytic cracking was carried out in fixed bed reactors3 ascommercialized in 1937. Catalytic cracking deposits carbonaceous productson the catalyst, causing rapid deactivation of the latter. To maintain the pro-duction capacity, the coke had to be burned off. This regeneration requiredswitching the fixed bed reactor out of production. In order to eliminate thecycling, attempts were made to circulate the catalyst and burn off the coke ina separate vessel, the re-generator. Both the reactor and the regenerator wereoperated under transport conditions. A sketch of the pioneering FCC reactoris shown in Fig 10.10 (b). Today this reactor is classified as a CFB, but it wasthen called an upflow unit [97]. The term circulating fluidized bed was firstapplied to alumina calciners by Reh [113] in 1971.

The high turbulence created in the fluid-solid mixture leads to much higherheat transfer coefficients than those which can be obtained in fixed beds.The resulting uniformity of the fluidized bed makes it applicable for effectingcatalytic reactions, especially highly exothermic and temperature sensitivereactions. However, the fluidization technology is much more complicated thanthat associated with fixed bed reactors.

Fig. 10.10. Two pioneering fluidized bed reactors: (a) the Winkler gas generator;(b) the first large-scale pilot plant for fluid catalytic cracking. Reprinted from [82]with permission from Elsevier.

3 This fixed bed process itself was already representing a major improvement overthe earlier thermal-cracking methods yielding more gasoline of higher octane rat-ing and less low-value, heavy fuel oil by-product [146].


For the low activity FCC catalysts then available, the bubbling bed designwas a decided improvement over the first CFB reactor. Until the mid-1970s,virtually all FCC units maintained a dense phase bubbling or turbulent bed inthe reactor vessel. A few of the second generation bubbling bed FCC reactorsare still in operation [97].

The contemporary commercial reactors used for sulphide ore roasting,Ficher-Tropsch synthesis and acrylonitrile manufacture were routinely oper-ated in the bubbling and turbulent fluidization regimes [56, 112].

With the introduction of zeolites in the early 1960s, the FCC catalystactivity began to increase steadily [97]. By 1980 many units were again oper-ating in a CFB mode to reduce the residence time of the gas reactants in thereactor. Today, by far the greatest use of CFB reactors is for the FCC processin petroleum refining.

There are processes in which the total amount of catalyst is entrainedby the gas. The reactors then belong to the category of transport reactors.Examples are some of the present Fischer-Tropsch reactors for the productionof hydrocarbons from synthesis gas and the modern catalytic cracking units.Fig 10.11 shows the Synthol circulating solids reactor. In the dilute side ofthe circuit, reactant gases carry suspended catalyst upward, and the fluidizedbed and stand-pipe on the other side of the circuit provide the driving forcefor the smooth circulation of the solid catalyst. For the removal of heat, heatexchangers are positioned in the reactor.

Fig 10.12(a) shows a Fluid Catalytic Cracking (FCC) unit (Exxon’s modelIV), in which the catalyst is circulating through a pair of U-tubes. Liquid oilis fed to the riser under the reactor, and on vaporization it reduces the bulkdensity of the up-going mixture and promotes the circulation of catalyst. Thestacked unit in Fig 10.12(b) is an alternative design by Universal Oil ProductsCompany. It uses a higher pressure in the re-generator than in the reactor,a single riser and a micro-spherical catalyst. Some of the synthetic crystallinezeolite catalyst introduced were so active that the cracking mainly or entirelytook place in the riser, so that the reaction vessel caused over-cracking intoundesired light gases and coke. In recent versions of the catalytic cracker,the catalyst is completely entrained in the riser-reactor to reduce the contacttime.

Table 10.3 presents a few examples of industrial applications of fluidizedbeds for synthesis reactions. Other examples are given by [146, 82, 53, 58].

Other fluid bed applications have also used CFBs in preference to densephase fluidized beds, but the use of CFBs is limited to situations where thehigher capital and operational costs of higher gas velocity can be justified bysignificant process advantages. In many applications, a well designed densephase fluidized bed may suffice and be less costly to construct and operatethan a CFB.

10.5 Milestones in Fluidized Bed Reactor Technology 891

Fig. 10.11. Synthol circulating fluid bed Fischer-Tropsch reactor. Reprinted from[82] with permission from Elsevier.

Fig. 10.12. FCC units in their middle stage of development. (a) Exxon’s reactormodel IV. (b) Alternative design by Universal Oil Products Company. Reprintedfrom [82] with permission from Elsevier.


Table 10.3. Industrial Applications of Fluidized Bed Catalytic Reactors [82].

Product or Reaction Type

Fluidized bed catalytic cracking (FCC) Riser reactor: FFBRegenerator: BB/FB

Phthalic anhydride FBFischer-Tropsch synthesis FFBVinyl acetate FBAcrylonitrile BB/FBEthylene dichloride BB/FBChloromethan FBMaleic anhydride FBPolymerization of olefines: Polyethylene (low density) BBPolymerization of olefines: Polypropylene FBo-cresol and 2,6-xylenol FBCalcination/roasting of ores BB/FBInclineration of solid waste BB/FB

FB = fluidized bed; FFB = fast fluidized bed; BB = bubbling fluidized bed

10.6 Advantages and disadvantages

In general, fluidized beds are of special interest when a high degree of gas tosolid contact coupled with large throughput of gas at fairly low pressure dropis needed.

In industry many different reactor designs are employed for the catalyticgas-solid processes, most important are the fixed bed, moving bed and flu-idized bed designs. In fixed bed reactors the catalyst particles are arranged ina vessel, generally a vertical cylinder, with the flux of reactants and productspassing through the stagnant bed. In moving bed reactors the bed can beremoved either continuously or periodically in portions. The fluid circulationis similar to that in a fixed bed.

Among these vessels, the fixed bed reactors are the conventional workhorsesfor these processes. The fixed bed reactors are generally used for very slow ornon-deactivating catalysts. For some of these processes serious temperaturecontrol problems limit the size of the reactor units. In the fixed bed units thecatalyst particles must be fairly large and uniform, and with poor temperaturecontrol the catalyst may sinter and clog the reactor. Bubbling and turbulentfluidized beds are more suitable for small granular or powdery non-friablecatalysts. Rapid deactivation of the solids can then be handled, and excellenttemperature controlled allows large scale operations. In general, fluidized bedsare of special interest when a high degree of gas to solid contact coupled withlarge throughput of gas at fairly low pressure drop is needed.

Comparing the fluidized bed and fixed bed reactor investment costs, phys-ical characteristics and operation performance the important advantages anddisadvantages of fluidized beds relative to fixed beds can be summarized asfollows.

10.7 Chemical Reactor Modeling 893

Advantages of fluidized beds:

• The ability to withdraw and reintroduce solids continuously.• Possibility of continuous regeneration of the catalyst particles. This is par-

ticularly useful for chemical processes where the catalyst is rapidly deac-tivated.

• The rapid mixing of solids leads to close to isothermal conditions through-out the reactor. Low risk of hot spots, runaway and thermal instability.The fluidized bed is well suited for exothermic reactions.

• Low impact of internal and external diffusion phenomena because of thesmall particle size.

• Efficient gas-solid contacting. Heat and mass transfer rates between gasand particles are high when compared with fixed bed reactors.

• The convective heat transfer coefficients at the surfaces immersed in thebed are high. This property indicates that internal heat exchangers requirerelatively small surface areas.

Disadvantages of fluidized beds:

• For the same weight of catalyst, expansion of the bed requires an increasein reactor volume.

• The random movement of the particles causing back-mixing result in anoverall reactor behavior that is closer to a CSTR than a plug flow reactor.In many chemical processes, this leads to an increase in the reaction volumeand a loss of selectivity.

• The entrainment of solid particles necessitates the installation of a device(like a cyclone) for separating and recycling fines.

• Friable solids are pulverized and entrained by the gas and must be replaced.• Erosion of internals, pipes, and vessels from abrasion by particles can be

serious.• Broad residence time distributions of solids due to intense mixing, erosion

of the bed internals, and attrition of the catalyst particles.• Broad residence time distributions of the gas due to dispersion and gas-

bypass in the form of bubbles, especially when operated in the bubblingbed regime.

• Reactor hydrodynamics and modeling are complex. Scale-up and designthus presents serious challenges which limits the use of these reactors toapplications that can justify the significant research and development ef-forts involved.

10.7 Chemical Reactor Modeling

Although many of the earliest fluidized beds were operated at high gas veloc-ities, technical difficulties led to a decrease in superficial gas velocities in theearly years [56]. At the same time, the academic research focused mainly on


the bubbles and slugs in fluidized beds. Call to mind that the characteristicnature of basic science guides academic researchers to proceed analyzing anyproblem in a systematic manner trying to understand the low velocity phe-nomena before the more complex high velocity flow patterns were studied.This naturally led to significant advances in the understanding of processescarried out at relatively low gas velocities, high-velocity processes and hydro-dynamics were all simply ignored.

In recent years renewed interest in fluidized beds operated at high gas ve-locities (1.5 m/s or more) in hydrodynamic regimes beyond the bubbling andslugging regimes arise, firstly because some industrial fluid bed reactor pro-cesses were always operated at such high gas velocities. Secondly, the interesthas been increased by the development of new circulating fluidized bed (5-10m/s) and substantial external recycle of entrained solids. Larger units wereconstructed and operated to increase the profit and competitive ability.

10.7.1 Conventional Models for Bubbling Bed Reactors

In order to predict the performance of a chemical reactor information on thereaction kinetics, thermodynamics, heat and mass transfer, and flow patternsare generally required. For bubbling fluidized bed reactors in particular theflow and the phase interaction phenomena are the most challenging model-ing tasks. In the early days, ideal flow models such as plug flow, continuousstirred tank or mixed flow, dispersion, and the tank in series approaches wereassessed. Then, led by Toomey and Johnstone [130] a number of two-regionmodels were proposed. The basic advantage of these models was that they en-able the investigators to account for the observed non-homogeneity of densefluidized beds, identifying the dilute bubble and the dense emulsion phases.The word phase in this context refers to a region which may include both gasand solid particles. These regions are distinguished from one another in termsof the volume fraction of solids, by physical appearance, and through theirflow characteristics [54].

Two major discoveries in the understanding of the gas/solid interactionphenomena in bubbling fluidized beds were obtained in the 1960s [81]. Firstly,the Davidson and Harrison [29] analysis of the flow of gas within and in thevicinity of rising gas bubbles. Secondly, the Rowe and Partridge [115] andRowe [116] finding that a rising bubble was accompanied by a wake of solidsand that this was the main mechanism causing solids circulation in a densefluidized bed.

These developments led to a novel type of reactor models, the hydrody-namic models, in which the bed behavior was based on the characteristics ofthe rising bubbles. Several models of this kind were derived for the industrialrelevant fine particle suspensions in which the rising bubbles are surroundedby very thin clouds of circulating gas. Various combinations of assumptionshave been used to represent the phase interaction phenomena in this region[81]:


• Two- or three phase models. In the three phase case the bubble, cloud, andemulsion phases are treated as separate regions. The three-phase model canbe reduced to a two-phase model either by treating the bubble and cloudphases as a single region, or by treating the cloud and emulsion phases asa single region.

• The bubbles are spherical or non-spherical.• The bubble wake region is accounted for or ignored.• The simplest model versions consider one mean bubble size in the whole

bed. The simple model can be extended so that bubbles are allowed togrow as they rise in the bed but are of the same size over the cross sectionof the bed at any level in the bed. The most advanced models are similarto the previous group of extended models but consider also a bubble sizedistribution at any level in the bed.

• The solids in the bubbles are considered or ignored.

A few of the early bubbling bed modeling approaches are assessed in thesubsequent sub-sections. Reliable engineering models, at this simple level ofcomplexity, can only be derived based on appropriate empirical informationcharacterizing the important bed properties. The theory and typical param-eterizations used to determine the relevant behavior of the gas and solids inthe bubble, cloud, emulsion and wake regions are outlined.

Pressure Drop and Minimum Fluidization Velocity

Increasing the gas flow just passing the point of minimum fluidization condi-tions the onset of particle fluidization will occur when the drag forces actingon the particles due to the upward moving gas are balanced by the weight ofthe solid particles [30].

The cross sectional average global pressure drop over a fluidized bed op-erated at minimum fluidization conditions is normally calculated by an ex-trapolation of the Ergun [42] equation (6.13) for fixed packed beds, the flowregime that is prevailing until the minimum fluidization flow rate has beenreached, as described in chap 6:

−Δpt

Lmf= f

ρg(Usmf )2

dp(10.1)

where f denotes an appropriate friction factor, as for example (6.14), and Lmf

represents the height of the fixed bed at minimum fluidization conditions.The weight of the cross sectional averaged bed at minimum fluidization is:

−Δpt

Lmf= (1 − εmf )(ρp − ρg)g (10.2)

where εmf denotes the gas holdup (overall bed void fraction) in the bed atminimum fluidization (generally a measured characteristic of the bed packing).


The superficial gas velocity at minimum fluidization conditions Usmf is

then found by combining the two relations for the pressure drop:

Usmf =

√

dp(1 − εmf )(ρp − ρg)gρg f

(10.3)

Volumetric Gas Flow Rate and Bubble Rise Velocity

The volumetric gas flow rate to the bubble phase in a fluidized bed Qb isdefined as the average rate at which the bubble volume crosses any level inthe bed [22]. However, the flow of gas in the bubble phase is generally greaterthan the volumetric gas flow rate in bubble because of the so-called bulkthroughflow of gas into and out of each bubble (i.e., convective interfacialmass transfer).

A first estimate for Qb is given by the two-phase theory of fluidization,proposed by Toomey and Johnstone [130] and developed by Davidson andHarrison [29, 30]. In this theory a bubbling fluidized bed consists of two zonesor phases, referred to as the bubble phase consisting of pure gas and theemulsion phase consisting of uniformly distributed particles in a supportinggas steam. The emulsion phase is assumed to be operating at minimum flu-idization conditions Us

e ≈ Usmf , while the bubble phase carries the remaining

gas flow Usb ≈ Us

in − Usmf .

This simple model can then be used to estimate the volumetric gas flowflow rate in bubble phase as the excess gas flow above that required for mini-mum fluidization:

Qb ≈ (Usin − Us

mf )A (m3/s) (10.4)

where A is the cross sectional area of the bed (m2).Experimental investigations indicate that the actual Qb is 10−50% smaller

than the value given by the simple two-phase model [22]. Nevertheless, thetwo-phase theory has been the basis for much work in fluidization technology.For example, knowing the bubble phase volumetric gas flow rate the averagefraction of the bed area occupied by bubbles can be approximated as:

εb ≈Qb

Aub≈

Usin − Us

mf

ub(−) (10.5)

where ub is the average rise velocity of bubbles in the bubbling bed.Moreover, the simple two-phase theory results (10.4) and (10.5) can be

adopted determining a first estimate of the bed expansion ΔL = L − Lmf

due to the bubble formation in dense beds. The expansion of the bed aboveits depth at minimum fluidization is obtained from the bubble phase volumebalance:

AΔL = A

∫ L

0

(1 − εb)dz −A

∫ Lmf

0

(1 − εb)dz ≈∫ L

Lmf

Qb

ubdz (m3) (10.6)


provided that the average rise velocity of bubbles is known (since the Lmf andUs

in parameters are normally specified and Usmf is calculated from suitable

semi-empirical models).The rise velocity of a single bubble rising in an emulsion phase unaffected

by other bubbles and the walls is frequently calculated from an expressionderived for gas liquid flow. Bubbles in a fluidized bed behave in many wayslike bubbles in a low viscosity liquid [116, 82]. A small bubble is spherical, buta bubble becomes deformed with increasing size and the larger ones have aspherical capped shape. The smallest bubbles rise slowly. The larger bubblesgenerally rise faster. Bubbles flowing in series may coalescence to producelarger bubbles. Wall effects usually decreases the bubble rise velocity. How-ever, in contrast to gas-liquid flow there is a high degree of gas interchangebetween the bubbles and the gas in the dense emulsion phase. Davies and Tay-lor [31] studied the rate of rise of large spherical cap bubbles of air throughnitrobenzene or water. An approximate calculation showed that the velocityof rise of a single bubble is ubl,rise = 2

3 (gRn)1/2. In this formula Rn is theradius of curvature at the nose of the gas bubble. Clift et al [23] (p 206) sum-marized the experimental rate of rise data for gas bubbles in liquids found inmore recent publications.

For particular bubble Reynolds and Eotvos numbers, Reb = ρgdeubl,rise/μg > 150 and Eo = gΔρd2

e/σI ≥ 40, the data on spherical cap bubblesindicate that a semi-empirical relation for the terminal velocity expressed interms of the volume equivalent bubble diameter is appropriate [54]:

ubl,rise ≈[

23

√

Rn

de

]

(gde)1/2 ≈ 0.711(gde)1/2 (m/s) (10.7)

where ubl,rise denotes the rise velocity of a spherical cap bubble in liquid(m/s), and de is the equivalent bubble diameter (m). The term in the bracketis generally a weak function of Reb, but for Reb > 100 the pre-factor is aboutconstant. For gas-liquid flows the modified coefficient value is confirmed fairlywell in several experimental investigations[28, 116, 138, 31].

Even though (10.7) is strictly valid in liquids, the formula is widely usedfor calculations of the ideal rise velocity of single bubbles in fluidized beds,when the ratio of bubble to bed diameters is db/dt < 0.125 [29, 82]:

ubr,0 ≈ 0.711(gdb)1/2 (10.8)

However, experimental data on rise velocities in fluidized beds consisting ofgroup A and B particles indicates that the pre-factor in (10.8) is about 2/3as for a single spherical cap bubble in liquid. For group D particles the givenpre-factor of 0.711 seems appropriate.

Wall effects are found to retard the single bubbles when db/dt > 0.125.Kunii and Levenspiel [82] proposed that for 0.125 < db/dt < 0.6 the risevelocity of a single bubble can be estimated from:


ubr,0 ≈ [0.711(gdb)1/2] × 1.2 × exp(

− 1.49db

dt

)

(10.9)

For db/dt > 0.6, the bed operates in the slugging regime which is not veryinteresting from a chemical reactor engineering point of view.

In reactor modeling we need to deal with the behavior of the bubblingbed as a whole rather than single rising bubbles. In extending the simple two-phase theory, Davidson and Harrison [29] proposed that the average velocityof bubbles in a freely bubbling bed can be approximated by 4:

ub = Usb + ubr,0 ≈ Us

in − Usmf + ubr,0 (10.10)

where ubr,0 is the ideal rise velocity of an isolated bubble of the same size(10.8).

This relation is not generally valid but considered to give a fair approxima-tion of the average flow rate of bubbles when Us

in is close to Usmf for all types

of particles, and for all velocities of interest considering larger type B and typeD particles. However, in large diameter beds consisting of fine particles andoperated at higher gas velocities the real bubble rise velocity is several timesthe velocity predicted by (10.10). This deviation is primarily caused by theexistence of preferred emulsion flow patterns [81, 82]. For small B and fine Aparticles the emulsion phase gas and solids are not really stagnant but developdistinct flow patterns, frequently referred to as gulf streaming , induced by theuneven rise or channeling of gas bubbles. In these gulf streaming patterns thebubbles rise in bubble rich regions with less friction thus a substantially highermean bubble rise velocity occur in these zones. In small and narrow beds thisflow pattern might be reduced or prevented by the wall friction. Several mod-ified relations for the average bubble rise velocity in dense fluidized beds havebeen reported over the years, each of them valid for particular Geldart solids,bed designs and operating conditions. Kunii and Levenspiel [81], for example,proposed a correlation for Geldart A solids with dt ≤ 1 (m):

ub = 1.55{

(Usin − Us

mf ) + 14.1(db + 0.005)}

d0.32t + ubr,0 (m/s) (10.11)

A similar correlation was given for Geldart B solids with dt ≤ 1 (m):

ub = 1.6{

(Usin − Us

mf ) + 1.13d0.5b

}

d1.35t + ubr,0 (m/s) (10.12)

The Davidson-Harrison Model for Gas Flow Around Bubbles

The first model for the movement of both gas and solids and the pressuredistribution around single rising bubbles was given by Davidson and Harrison4 This expression and the arguments for its use were first presented by Nicklin [102]

for gas-liquid systems determining the rise velocity of a single bubble in a cloudof gas bubbles rising through a stagnant liquid, and later used by Davidson andHarrison [29] (p 28) for bubbles in fluidized beds.


[29] (Chap 4). Two versions of the model were developed, considering two-and three-dimensional fluidized beds. The theory is based on the followingassumptions:

• There is no solids in a gas bubble. A three-dimensional bubble is spherical,whereas a two-dimensional bubble is cylindrical.

• As a bubble rises, the particles move aside, as would an incompressibleinviscid fluid having the same bulk density as the whole bed at incipientfluidization ρs(1 − εmf ) + ρgεmf .

• The gas flows in the emulsion phase as an incompressible viscous fluid.The relative velocity between the gas and the solid particles must satisfyDarcy’s law:

vg − vs = −K∇p (10.13)

where K is a permeability constant characteristic of the particles and thefluidizing fluid.

From potential flow theory, invoking the given model assumptions, the pres-sure distribution in the bed can be found by solving a Laplace equation. Theboundary conditions used express that far from a single bubble an undis-turbed pressure gradient exist as given by (10.2), and that the pressure in thebubble is constant. Then, after the pressure distribution is known the flowpattern for the solids and the gas in the vicinity of a rising bubble can becalculated. The solution shows that the pressure in the lower part of the bub-ble is lower than that in the surrounding bed, whereas in the upper part it ishigher. For this reason the gas flows into the bubble from below and leavesat the top. A distinct difference in the gas flow pattern has been identified,depending on whether the bubble rises faster or slower than the emulsion gas.The gas flow pattern is thus classified dependent on the relative velocity be-tween a single bubble ubr,0 and the emulsion gas ue = Us

mf/εmf far from thebubble.

For slow bubbles (ubr,0 < ue) in beds consisting of large particles theemulsion gas rises faster than the bubble, hence the faster rising emulsion gasshortcuts through the rising bubble on its way through the bed. The emulsionphase gas enters the bottom of the bubble and leaves at the top. Within thebubble an annular ring of gas is forced to circulate as it moves upwards withthe bubble. For a fast bubble (ubr,0 > ue) in beds consisting of small particlesthe emulsion gas rises slower than the bubble, but still the emulsion phasegas enters the bottom of the bubble and leaves at the top. However, sincethe bubble is rising faster than the emulsion gas, the gas leaving the top ofthe bubble is swept around and returns to the base of the bubble. The regionaround the bubble penetrated by this circulating gas is called the cloud. Therest of the gas in the bed does not mix with the recirculating gas but movesaside as the fast bubble with its cloud passes by. The cloud phase mightbe considered infinite in thickness at ubr,0 = ue, but thins with increasingbubble velocity. For very fast bubbles, ubr,0/ue > 5, the cloud is very thin


and most of the gas stays within the bubble. In general slow cloudless bubblesoccurs in beds of coarse particles (group D, or B) while fast bubbles withthin clouds are typically observed for group A and B particles at high gasvelocities.For a two-dimensional bed the cloud size is given by [29, 30]:

R2c

R2b

=ubr,0 + ue

ubr,0 − ue(10.14)

For a three-dimensional bed the cloud size is given by [29, 30]:

R3c

R3b

=ubr,0 + 2ue

ubr,0 − ue(10.15)

For a two-dimensional bed the ratio of cloud to bubble size is defined as [82]:

fc =R2

c

R2b

≈ ubr,0 + ue

ubr,0 − ue≈ 2ue

ubr,0 − ue≈

2Usmf/εmf

ubr,0 − Usmf/εmf

(10.16)

For a three-dimensional bed the ratio of cloud to bubble volume is defined as[82]:

fc =R3

c

R3b

=ubr,0 + 2ue

ubr,0 − ue≈ 3ue

ubr,0 − ue≈

3Usmf/εmf

ubr,0 − Usmf/εmf

(10.17)

These estimates for fc are obtained considering the particular case whenubr,0 ≈ ue.

The simple theory also provides estimates of the gas exchange betweenthe bubble and the emulsion phase. For a two-dimensional bed of unit thick-ness the flow of gas through the bubble is v = 4ueεmfrb, and for a three-dimensional bed the flow of gas through the bubble is v = 3ueεmfπr

2b .

Rowe [116] and Davidson et al [30] summarized the experimental inves-tigations of the properties of bubbles and clouds, explained the elementarytheories using nice illustrations and provided pictures of the physical phe-nomena observed.

The Wake Region

In an early study Rowe and Partridge [115] (see also [116]) did show thatgas fluidized beds are characterized by the formation of bubbles which risethrough denser bed zones of the bed and determine a gross scale gas and solidflow pattern.

It was observed that like a single bubble in liquid, a rising bubble in afluidized bed drags a wake of material consisting of a gas-solid mixture up thebed behinds it. Close to the bottom of the bed, just above the gas distributor,solids are entrained by the rising bubbles to form the bubble wake. There is


also a continuous interchange of solids between the wake and emulsion as thebubble rise. At the top of the bed the wake solids rejoin the emulsion to movedown the bed.

The bubble wake fraction is given by the volume ratio of the wake to thebubble:

fw =Vw

Vb(10.18)

The wake volume Vw is defined as the volume occupied by the wake withinthe sphere that circumscribes the bubble.

The wake fraction is normally determined from experimental analysis [82].

Bubble Size

In bubbling fluidized beds there are several mechanisms affecting the bubblesproperties [115, 116, 30, 82]. The main mechanisms determining the meanbubble size are bubble growth, bubble coalescence and bubble breakage. Theprimary bubble growth mechanism for a single bubble unaffected by otherbubbles is gas transfer from the emulsion phase into the bubble. In a freelybubbling bed the main mechanism for bubble growth is bubble-bubble co-alescence. Binary breakage of bubbles frequently occur as a result of smalldisturbances, often proceeded by a slight flattening of the bubble, initiatednear the top of the roof of the bubble and these rapidly grow in a cuttingedge of particles that may divide the parent bubble into two equal or unequaldaughter bubbles. Later these daughter bubbles commonly re-combine eitherby gas leakage from one bubble to the other or because a small bubble fallingbehind the larger one is caught in its wake and coalescence.

Experiments have shown that in beds consisting of B type particles thebubbles increase steadily, whereas in beds consisting of type A particles thebubbles grow rapidly until they reach a stable size determined by a state ofequilibrium between the coalescence and breakage processes [82].

When performing experimental investigations of bubble properties two di-mensional beds have proved to be useful [55]. These two-dimensional columnsare of rectangular cross section, the width being considerably greater thanthe thickness. The fluidized particles are contained in the gap between twotransparent faces, separated by 2-3 cm. The bubbles span the bed thicknessand are thus viewed. However, while these two-dimensional columns are usefulfor qualitative purposes, there are important quantitative differences betweenthe two- and three dimensional fluid bed flow behavior. These differences arisefrom quantitative differences in rise velocities of isolated bubbles, bubble co-alescence properties, bubble shape and wake characteristics, reduced solidsmixing, etc. To characterize the bubble properties in three dimensional densebeds intrusive probes are normally used [129]. For modeling purposes it isimportant to distinguish between the relations obtained characterizing two-and three dimensional beds.


The equivalent bubble diameter de, defined as the diameter of a sphericalbubble with a volume equal to the average bubble volume, is often used as ameasure of bubble size. A representative relation for the equivalent bubble di-ameter in a three-dimensional bed with B particles supported by a perforatedplate distributor was given by [134]:

de = de,0[1 + 27.2(Usin − Us

mf )]1/3(1 + 6.84z)1.21 (m) (10.19)

where de,0 denotes the initial bubble size at the distributor, and z is theposition above the distributor.

A more general relation for the equivalent bubble diameter in three-dimensional beds was derived by Darton et al [27]:

de =0.54g1/5

(Usin − Us

mf )2/5(z + 4√

A0)4/5 (m) (10.20)

where A0 is the catchment area for the bubble stream at the distributor plate,which characterize different distributors, and is usually the area of plate perorifice. For a porous plate distributor a typical value A0 ≈ 5.6 × 10−5(m2)was proposed.

To determine the growth of circular bubbles in two-dimensional fluidizedbeds Lim et al [89] modified the relation of Darton et al [27] and proposed:

de =[8(Us

in − Usmf )(23/4 − 1)

πλg1/2z + d

3/2e,0

]2/3

(m) (10.21)

where the initial equivalent bubble diameter just above the distributor is givenby:

de,0 =[8(Us

in − Usmf )A0

πλg1/2

]2/3

(m) (10.22)

The dimensionless proportionality constant λ ∼ 2 is related to the distance abubble travels in a stream before coalescing with the adjacent stream to forma single stream of larger bubbles.

The Basic Two-Phase Model

A number of fluidized bed reactor model versions are based on the crosssectional averaged two-phase transport equations as presented in sect 3.4.7.Due to the vigorous particle flow the fluidized beds are essentially isothermal,so no energy balance is generally required5. In addition, the necessary speciesmass (mole) balance can be deduced from (3.498). The solids are considered

5 In particular cases, considering extremely exothermic or endothermic processes,a global CSTR heat balance may be employed to determine a uniform operatingtemperature and the necessary heating/cooling capacity.


to be completely mixed in the dense phase and is essentially stagnant, so onlyan overall mass balance is needed for the solid phase.

For the particular case of an irreversible gas solid catalyzed reaction withno accompanying volume change, the mass (mole) balance for a species A inthe interstitial gas phase moving through the emulsion phase is frequentlysimplified assuming axially constant transport parameters (i.e., fe, ue, De,and kbe) [141, 142, 58]:

fe∂CAe

∂t+ feue

∂CAe

∂z= feDe

∂2CAe

∂z2+ fekbe(CAb − CAe) + rAeρefe (10.23)

where CAe represents the cross sectional average mole concentration of A inthe emulsion phase gas (kmol/m3), Us

e = feue the superficial velocity of theemulsion phase gas (m3/m2 s), ue the interstitial emulsion gas velocity (m/s),De an effective diffusivity for the emulsion gas (m2/s), kbe an interchange masstransfer rate coefficient per unit volume of bubble gas (m3/m3 s), rAe reactionrate in emulsion phase (kmol/kg s), fe is the fraction of the bed gas volumetaken by the emulsion gas (m3/m3), and ρe the mass concentration of thecatalyst particles in the emulsion phase (kg/m3).

The corresponding mass (mole) balance for species A in the bubble phaseis:

fb∂CAb

∂t+ fbub

∂CAb

∂z= fbDb

∂2CAe

∂z2− fbkbe(CAb − CAe) + rAbρbfb (10.24)

where CAb represents the cross sectional average mole concentration of A inthe bubble phase (kmol/m3), Us

b = fbub the superficial velocity of the bubblephase gas (m3/m2 s), Db an effective diffusivity for the bubble gas (m2/s),rAb reaction rate in bubble phase (kmol/kg s), fb the volume fraction of thebed gas taken by the bubble gas (m3/m3), and ρb the mass concentration ofthe catalyst particles in the bubble phase (kg/m3).

The Davidson-Harrison Two-Phase Model

The Davidson and Harrison [29, 30] bubbling bed reactor model representsone of the first modeling attempts that was based on bubble dynamics. Themodel rest on the following assumptions:

• The reactor operates at steady state, thus the transient terms in (10.23)and (10.24) disappear.

• The gas bubble are evenly distributed throughout the bed and are of equalsize.

• The bubble phase gas flow can be described by a plug flow model, hencethe bubble phase dispersion term in (10.24) vanishes.

• No solids in the bubbles, thus no catalytic reaction takes place in thebubble phase gas.


• The emulsion phase gas flows with a superficial velocity Usmf and is either

considered completely mixed or plug flow.• Gas is exchanged between the non-reactive bubble and the reactive emul-

sion phases by a combined molecular and convective transport flux.

Species mass balances were developed for the two phases and solved ana-lytically for first order reactions. Thus, for the case of a completely mixedemulsion phase the concentration of a reactant leaving this phase is given by:

Ce =Cin(1 − fbe

−X)

1 − fbe−X + kLmf

Us

(10.25)

where X = KbeL/Usb is the interphase exchange parameter, L represents the

bed height and Usb the bubble velocity, and fb = 1 − Us

mf/Us.

For the bubble phase operated in plug flow:

Cb = Ce + (Cin − Ce)e−X (10.26)

The concentration of unconverted reactant Cout leaving the bed as a wholewas then found from:

Cout = fbCb + (1 − fb)Ce (10.27)

The overall fraction of unconverted reactant becomes:

Cout

Cin= fbe

−X +(1 − fbe

−X)2

1 − fbe−X + kLmf

Us

(10.28)

For the case when the emulsion phase gas is in plug flow, the overallfraction unconverted reactant leaving the bed becomes:

Cout

Cin=

1m1 −m2

[

m1e−m2L(1 −

m2LUsmf

XUs) −m2e

−m1L(1 −m1LU

smf

XUs)]

(10.29)where m1 and m2 are obtained from:

2L(1− fb)m = (X − kLmf

Us)± [(X − kLmf

Us)2 − 4

kLmf

UsX(1− fb)]1/2 (10.30)

In the latter expression m = m1 with the positive sign and m = m2 with thenegative sign.

The details of the model derivation and the analytical solution can befound in [29, 141, 142, 141, 117]. It is noted that this model was developedbefore the importance of bubble clouds and wakes were realized.

Although this model is derived from first principles and uses a minimum ofempirical information, the model predictions have been found to agree fairlywell with experimental data in a number of cases and may thus be adequatefor design purposes. However, the empirical relationships are derived fromexperiments with laboratory scale equipment, and this has caused the validityof their application to large industrial units to be questioned.


The van Deemter Two-Phase Reactor Model

This model represents one of the first modeling approaches used to describelarge industrial beds that has been fully documented and discussed in the openliterature. The model was adopted by the Shell company designing a fluidizedbed reactor for their solid catalyzed hydrogen chloride oxygen process. How-ever, the model has many of the same limitations as the Davidson-Harrisonmodel because is was developed before the importance of bubble clouds andwakes were realized.

The fairly general transport equations constituting the basic two-phasemodel, given in (10.23) and (10.24), were simplified making the van Deemtermodel specific assumptions [132, 142, 47]:

• The reactor operates at steady state, thus the transient terms disappear.• The bubble gas flow can be described by a plug flow model, hence the

bubble phase dispersion term vanishes.• No solids in the bubbles so no catalytic reaction takes place in the bubble

phase, thus the reaction term disappears in the bubble gas mole balance.

For a single reaction, a generic cross sectional average species mole balancefor component A in the bubble phase gas can thus be written as:

fbubdCAb

dz+ fbkbe(CAb − CAe) = 0

(

kmol

m3s

)

(10.31)

The corresponding cross sectional average species mole balance for componentA in the emulsion phase gas is written as:

feuedCAe

dz−fbkbe(CAb−CAe)−feDe

d2CAe

dz2+rAeρefe = 0

(

kmol

m3s

)

(10.32)

The overall outlet concentration CA,out is calculated as:

UsinCA,out = fbubCAb + feueCAe (10.33)

in which Usin denotes the overall superficial gas velocity (m3/m2s).

The model equations are normally solved applying the following initial andboundary conditions :

Bubble phase: CAb = CAb|z=0

Emulsion phase [26]: −DedCAe

dz

∣

∣

∣

∣

z=0

= ue(CAe|z=0 − CAe)

dCAe

dz

∣

∣

∣

∣

z=L

= 0

(10.34)

The undetermined model parameters like Usb , ub, kbe, De, fe, fb, ue and Us

e

were approximated from empirical correlations determined by experimentalmeasurements. Us

in and εmf are generally measured or specified, thus Usmf

can be determined from (10.3).


A typical correlation for kbe is given by [117, 47]:

Usin

fbkbe≈ (1.8 − 1.06

d0.25t

)(3.5 − 2.5Z1.4

) (m) (10.35)

where Z is the total reactor height (m).The effective diffusivity in the emulsion phase may be approximated by [6]:

De ≈ 0.35(gUsin)1/3d

4/3t (m2/s) (10.36)

In the basic two-phase model a prescribed fraction of the total gas flow ratethrough the bed, which is coinciding with the minimum fluidization operatingconditions, is assumed to move through the emulsion phase [130, 29]. Therelative velocity between the interstitial gas and the solids in the emulsionphase is thus ue = Us

mf/εmf . The rest of the gas constitutes the bubblephase. The superficial bubble gas velocity is thus generally approximated as:

Usb = fbub ≈ Us

in − Usmf (m3/m2s) (10.37)

Likewise, the superficial emulsion gas velocity is approximated as:

Use = feue ≈ Us

mf = ueεmf (m3/m2s) (10.38)

The average bubble rise velocity (10.10) is approximated by the bubble risevelocity among a swarm of bubbles [134]:

ub = Use − Us

mf + ubr,0 ≈ ubr,0 ≈ K√

gdb (10.39)

where K = 0.64 when dt < 0.1 (m), K = 1.6d0.4t when 0.1 < dt < 1.0 (m),

and K = 1.6 when dt > 1.0 (m).The flow pattern for solids is generally downward near the wall and upward

in the central core. This particle movement also affects the gas flow in theemulsion phase. In this model the net rise velocity of solids is neglected, henceus = 0.

For a perforated plate distributor the bubble size might be approximatedby (10.19).

The Kunii-Levenspiel Three-Phase Model

The celebrated Kunii-Levenspiel [78, 79, 81, 82] (p 289) reactor model pre-sented in this section was designed for the particular case of fast bubblesin vigorously bubbling beds which is relevant for industrial applications withGeldart A and AB solids. In such beds there are definite gross mixing patternsfor the solid, downward near the wall and upward in the central core. Thishas a marked effect on the gas flow in the emulsion phase, which is also forceddownward near the wall. However, based on experimental data analysis, Kunii


and Levenspiel found that at gas flow rates above Usin/U

smf ∼ 6 − 11 there

are practically no net gas flow through the emulsion phase.To take these observations into account a three phase model was proposed

in which many ideas from the Davidson bubble theory, the Davidson-Harrisonand the van Deemter reactor models were adapted. The Rowe cloud and wakeobservations were also considered. The basic model assumptions were:

• The bed is assumed to consist of three phases, the bubble, cloud andemulsion, with the wake region considered to be a part of the cloud phase.

• For industrial relevant operating conditions well above minimum fluidiza-tion, Us

in � Usmf and practically all the moving gas is transported in the

bubble phase. In this fast bubbling regime, for which ub > 5Usmf/εmf , the

clouds are very thin.• Interchange mass transfer coefficients are used to account for the mass

transfer between the phases.• The rising bubbles contain no solid.• The bubble phase operates under plug flow conditions.• The cloud and emulsion phases are stagnant (ue ≈ uc ≈ 0).• Spherical bubbles of uniform size accompanied by wakes throughout the

bed.• First-order reaction is assumed. The catalytic reaction rate is given in

accordance with modern literature [47].

For a single catalytic reaction, a generic cross sectional average speciesmole balance for component A in the bubble phase gas can be written as:

ubdCAb

dz+ kbc(CAb − CAc) + rAbρb = 0

(

kmol

m3s

)

(10.40)

where kbc denotes the interchange coefficient determining the mass transferbetween the bubble and cloud gas phases, referred to unit bubble gas volume(m3/m3s), and CAc the molar concentration of species A in the cloud phasegas (kmol/m3).

For a single reaction a generic species mole balance for component A inthe stagnant cloud phase gas is written as:

kbc(CAb − CAc) = rAcρcVc

Vb+ kce(CAc − CAe)

(

kmol

m3s

)

(10.41)

where Vc denotes the volume of the cloud phase (m3), Vb the volume of thebubble phase (m3), ρc the mass concentration of the catalyst particles in thecloud phase (kg/m3), and kce denotes the interchange coefficient determiningthe mass transfer between the cloud and emulsion phases, referred to unitbubble volume (m3/m3s).

For a single reaction a generic species mole balance for component A inthe stagnant emulsion phase gas is written as:


kce(CAc − CAe) = rAeρefe

fb≈ rAeρe

1 − fb

fb

(

kmol

m3s

)

(10.42)

In general, the model can be solved numerically with appropriate initialconditions for the plug flow ODE.

Bubble phase:CAb = CAb|z=0 (10.43)

The model can be solved analytically for first order reactions. Eliminat-ing the unknown CAc and CAe concentrations from the model equations, weobtain:

fbubdCAb

dz= KrCAb

(

kmol

m3s

)

(10.44)

where the overall rate coefficient Kr contains all the interfacial transfer resis-tances as well as the reaction resistance terms.

The overall rate coefficient Kr can be defined as [47]:

Kr = k

⎡

⎢

⎢

⎢

⎢

⎣

ρb +1

kkbc

+ 1ρc

VcVb

+ 1k

kce+ 1

ρe(1−fb)

fb

⎤

⎥

⎥

⎥

⎥

⎦

(1/s) (10.45)

Integration over the whole reactor bed from z = 0 to z = L gives:

CAb = CA,in exp(−KrL

Usb

)(

kmol

m3

)

(10.46)

The celebrated K-L model did gain its popularity because it is very simple,yet developed for large scale industrially relevant flows and considers most ofthe pertinent bubbling bed phenomena.

Over the years the range of uses of the K-L model has been extended tochemical processes that can not be described by first order kinetics. For theseproblems no analytical solution can be obtained so the resulting set of DAEequations are solved numerically. Gascon et al [48], for example, investigatedthe behavior of a two zone fluidized bed reactor for the propane dehydro-genation and n-butane partial oxidation processes employing the K-L modelframework.

In any case the undetermined model parameters are determined from ap-propriate empirical correlations [78, 79, 81, 82]. The mass transfer coefficients,all based on unit bubble volume, can be obtained from:

kbc = 4.5us

mf

db+ 5.85(

D1/2Amg1/4

d5/4b

) (m3/m2s) (10.47)


kce = 6.78(εmfDAmub

d3b

)1′/2 (m3/m2s) (10.48)

where DAm is the diffusivity for species A in the mixture (m2/s).In the two-phase view the solids in the bubble phase is ignored (εb ≈ 1),

thus the overall void fraction in the bubbling bed is:

εov = fbεb + (1 − fb)εe ≈ fb + (1 − fb)εe (10.49)

If εov and fb are known from experiments, εe can be determined by thisequation.

However, in many cases εov and fb are unknown, so εe cannot be deter-mined in this way. For modeling purposes Kunii and Levenspiel [81] proposedsome rough approximations. For beds of Geldart A solids, εe ≈ εmb, whereasfor beds of Geldart B and D solids, εe ≈ εmf .

In the latter case, where εe ≈ εmf , the volume balance over the bed canbe written as:

1 − εov ≈ (1 − fb)(1 − εmf ) (10.50)

The volume fraction of the bed consisting of bubbles fb is determined by theprevailing flow regime. For fast bubbles in vigorously bubbling beds, whereub > 5Us

mf/εmf and Usin � Us

mf , the clouds are thin and one may use thefollowing approximation [81]:

fb =Vb

Vr≈

Usin − Us

mf

ub − Usmf

≈ Usin

ub(10.51)

The volume fraction of bed gas comprising the emulsion phase gas is:

fe = 1 − fb =ub − Us

in

ub − Usmf

(10.52)

The distribution of solids in the three regions were defined by:

γb =Vbs

Vb, γc =

Vcs

Vb, γe =

Ves

Vb(10.53)

A material balance for the solids and (10.50) then relates these γ-values:

fb(γb + γc + γe) = 1 − εov = (1 − εmf )(1 − fb) (10.54)

hence

γe =(1 − εmf )(1 − fb)

fb− γb − γc (10.55)

Values of γc have been estimated considering a spherical bubble and account-ing for the solids in both the cloud and wake regions:

γc = (1−εmf )(Vc

Vb+Vw

Vb) ≈ (1−εmf )(

3Usmf/εmf

0.711(gdb)1/2 − Usmf/εmf

+fw) (10.56)


where the volume of the cloud surrounding each of the fast rising bubbles isgiven by:

Vc

Vb≈

3Usmf/εmf

0.711(gdb)1/2 − Usmf

(10.57)

Like bubbles in liquids, it might be expected that every rising bubble in flu-idized beds has an associated wake of material rising behind it. The ratio ofwake to bubble volume fw = Vw/Vb has to be determined by experiments, butthe void fraction of the wake is frequently assumed to be that of the emulsionphase.

Moreover, experiments in beds of uniformly sized bubbles indicates thatγb ≈ 0.001 ∼ 0.01.

One may imagine that just above the distributor solid is entrained by therising bubbles to form the bubble wake. This solid is carried up the bed withthe bubbles at velocity ub and is continually exchanged with fresh emulsionsolid. At the top of the bed the wake solids rejoin the emulsion to movedown the bed at velocity us. The upward velocity of gas flowing through theemulsion is thus:

ue =umf

εmf− us (10.58)

Under particular operating conditions the circulation of solids may then gethigh enough so that the gas flow is directed downward in the emulsion. How-ever, in vigorous bubbling beds the rise velocity of solids is about zero:

us ≈ 0 (10.59)

The distribution of solids γ used in the original Kunii and Levenspiel model[82] can be linked to the bulk solid density in the following manner [47]. Thebulk density of solids in the bubble phase yields:

ρb = γbρs, (10.60)

where ρs is the mass density of the solid catalyst material.The bulk density of solids in the cloud and wake phases is given by:

ρc = γcρsVb

Vc(10.61)

in which Vc is the volume of the cloud phase, and Vb is the volume of thebubble consisting of pure gas.

The bulk density of solids in the emulsion phase is obtained from:

ρe = γeρsVb

Ve= γeρs

fb

fe≈ γeρs

fb

1 − fb(10.62)

In the latter relation the volume of the thin cloud phase is ignored.


10.7.2 Turbulent Fluidized Beds

Models for turbulent fluidized bed reactors are normally based on one di-mensional gas flow, despite the significant radial density gradients observedexperimentally [56, 57].

For reactors operated in the turbulent regime the following reactor mod-eling approaches have been proposed:

• The first modeling attempts employed one dimensional pseudo- homoge-neous plug flow models for the gas phase. For first order reactions themodel can be written as:

Usg

dC

dz+ k(1 − ε)C = 0

(

kmol

m3s

)

(10.63)

The customary initial condition is:

C = Cin (10.64)

• For non-ideal flows the modeling approach has been extended adaptingone dimensional pseudo-homogeneous axial dispersion models for the gasphase.The latter model can be written as:

d

dz(Dg,ax

dC

dz) + Us

g

dC

dz+ k(1 − ε)C = 0

(

kmol

m3s

)

(10.65)

The boundary conditions are [26]:

CinUsg,in =CUs

g −Dg,axdC

dzat z = 0 (10.66)

dC

dz= 0 at z = L (10.67)

The dispersion coefficient for the turbulent regime has been determinedfrom correlations on the form:

Dg,ax ≈ 0.84ε−4.445ov (m/s) (10.68)

where εov is the overall void fraction in the turbulent regime.

10.7.3 Circulating Fluidized Beds

For reactors operated in the fast fluidization and pneumatic conveying regimesthe following modeling approaches have been used [57, 96, 13]:

• One dimensional pseudo-homogeneous plug flow models for the species inthe gas phase.


For pneumatic conveying all the particles are evenly dispersed in the gas.This makes contacting ideal or close to ideal. The plug flow model is thuswell suited for the dilute transport reactors, but has also been used for thedenser fast fluidization regime neglecting gradients in the solids distribu-tion. For first order reactions the model can be written as:

Usg

dC

dz+ k(1 − ε)C = 0

(

kmol

m3s

)

(10.69)

The initial condition is:C = Cin (10.70)

• One dimensional pseudo-homogeneous axial dispersion model for thespecies in the gas phase.This model has been used for denser transport reactors and reactors op-erated in a dilute fast fluidizartion regime intending to account for thenon-ideal flow behavior. For first order reactions the model can be writtenas:

d

dz(Dax

dC

dz) + Us

g

dC

dz+ k(1 − ε)C = 0

(

kmol

m3s

)

(10.71)

The boundary conditions are [26]:

CinUsg,in =CUs

g −DaxdC

dzat z = 0 (10.72)

dC

dz= 0 at z = L (10.73)

The dispersion coefficient was determined by correlations on the form:

Dg,ax ≈ 0.1953 × ε−4.1197ov (m/s) (10.74)

where εov is the overall void-age in the reactor. This correlation is valid forall the regimes ranging from turbulent fluidization to pneumatic transport.

• Core/annulus models for the gas phase.Most state of the art CFB reactor models operated in the fast fluidizationregime make use of the core/annulus approach, which dates back to thework of [17]. These models are based on the experimental data interpre-tation that two zones exist in the riser at every axial location, an upwardmoving dilute core phase and a dense annulus phase with high solids con-centration. The gas was assumed to pass upward in plug flow through thecentral core. Some of this gas was exchanged with the outer annular regionwhere the gas is stationary. Each of the the two-regions was assumed tobe radially uniform. An inter-region mass transfer coefficient and the ra-dius of the core, both assumed to be independent of height, were obtainedby fitting the models to experimental axial gas mixing data. The catalystparticles were assumed to be small enough so that both intra particle dif-fusion resistances and external particle mass transfer do not need to be


considered. Applying this model to a first order reaction with steady stateoperation leads to molar balances of the generic form [73]:Core region:

Usg (R2

r2c

)dCc

dz+

2Krc

(Cc − Ca) + k(1 − εcs)Cc = 0(

kmol

m3s

)

(10.75)

Annulus region:

(r2c

R2 − r2c

)2Krc

(Cc − Ca) + k(1 − εas)Ca = 0(

kmol

m3s

)

(10.76)

where K and k are the cross flow mass transfer coefficient and first orderkinetic rate constant, respectively. Furthermore, Kagawa et al [73] em-ployed the following model parameters: kc = k(1 − εc), ka = k(1 − εa),rc/R = 0.85, εc = 0.6〈εs〉A, εa = 2.0〈εs〉A, and Uc = U/0.852.A common initial boundary condition is:

Cc = Cin at z=0 (10.77)

It is recognized that this model is formally analogous to the two-phasebubbling bed model, with the annulus replacing the stagnant dense phaseand the dilute core replacing the bubble phase.

As the interest in high velocity fluidized beds operating in the turbulent,fast fluidization and pneumatic conveying regimes has grown, several attemptshave been made to provide appropriate reactor models, often by extending themodels originally devised for the bubbling bed reactors.

Kunii and Levenspiel [81] presented models for bubbling beds for all typesof particles considering the lean phase freeboard and for fast fluidized beds. Afreeboard entrainment reactor model was developed to account for the extraconversion taking place above dense bubbling beds. For fast fluidized bedreactors the vessel was divided into two zones, a lower dense zone and an upperfree-board zone. Later, Kunii and Levenspiel [83, 84] deduced engineeringmodels for determining the performance behavior of a CFB reactor extendingthe bubbling bed model by adjusting the parameters to the turbulent, fastfluidization and pneumatic conveying regimes.

The Overall Pressure Balance Around a CFB Loop

In the CFB loop the pressure drop over the riser must be balanced by thatimposed by the flow through its accompanying components such as the down-comer and the recirculation device. The pressure drops across the downcomer,the solids circulation and control device, and the riser are the major elementsin the pressure balance around the CFB loop. The pressure drop balance isthus such that the sum of the control device, fluidized bed riser and cyclone


(piping) pressure drops must equal the downcomer and hopper pressure drops.The hopper pressure drop is usually negligible so that:

ΔpR + ΔpC + ΔpCD ≈ ΔpD (10.78)

A balance of the pressure around the CFB loop requires quantitative infor-mation for the pressure drops in each component. Typical design formulas forthe pressure drop associated with the major components in a CFB loop canbe outlined as follows [44, 20, 114, 1].

• Pressure Drop Across the Riser, ΔpR:For the gas flow in the riser, kinetic energy in the gas phase is partiallytransferred to the solids phase through gas-particle interactions and is par-tially dissipated as a result of friction. Under most operating conditions influidized beds, gravitational effects dominate the overall gas phase energyconsumption. Thus, neglecting the particle acceleration effects, the pres-sure drop in the riser can be approximated by the weight of the particles:

ΔpR =∫ L

0

ρpεpgdz = ρpεpgL (10.79)

where L is the riser hight.• Pressure Drop Through the Cyclone, ΔpC :

Cyclone pressure drop is essentially a consequence of the vortex energy,the solid loading and the gas-wall friction. The main contribution is thesuspension vortex energy. Generally, the pressure drop through a cycloneis proportional to the velocity head and approximated by [111]:

ΔpC = kρg(Usg,in)2 (10.80)

The value for the coefficient k may vary from 1-20, depending on thecyclone design [44].

• Pressure Drop Across the Downcomer, ΔpD:The solids flow in the downcomer is either in a dense fluidized state or ina moving packed state.The maximum pressure drop through in the downcomer is establishedwhen particles are fluidized, a state that can be expressed in terms of thepressure drop under the minimum fluidization condition as [44]:

Δp|D,max = LD(1 − αg,mf )ρpg (10.81)

where LD is the height of the solids in the downcomer.• Pressure Drop Through the Solids Flow Control Devices, ΔpCD:

If the particles in the downcomer are fluidized, the pressure drop throughthe mechanical solids flow control device can be expressed as [72]:

ΔpCD =1

2ρp(1 − αg,mf )

(

Ws

C0A0

)2

(10.82)


where Ws is the solids feeding rate and A0 is the opening area for themechanical valve. The coefficient C0 ∼ 0.7 − 0.8 over a variety of systemsand control device opening configurations.The solids flow rate can be controlled by non-mechanical valves such asan L-valve with a long horizontal leg. The overall pressure drop across anL-valve is normally calculated as a linear sum of two terms accounting forthe pressure drop through the elbow and the pressure drop caused by thegas-solid flow in the horizontal leg [140]:

ΔpCD = ΔpCD,elbow + ΔpCD,leg =1

2ρp(1 − αg,mf )

(

Ws

C0A0

)2

(10.83)

where C0 = 0.5.

10.7.4 Simulating Bubbling Bed Combustors UsingTwo-Fluid Models

In this section the application of multiphase flow theory to model the perfor-mance of fluidized bed reactors is outlined. A number of models for fluidizedbed reactor flows have been established based on solving the average funda-mental continuity, momentum and turbulent kinetic energy equations. Theconventional granular flow theory for dense beds has been reviewed in chap 4.However, the majority of the papers published on this topic still focus onpure gas-particle flows, intending to develop closures that are able to predictthe important flow phenomena observed analyzing experimental data. Veryfew attempts have been made to predict the performance of chemical reactiveprocesses using this type of model.

Alternative Two-Fluid Model Closures

According to Enwald & Almstedt [40], the existing ensemble averaged two-fluid model closures for bubbling beds, developed by Simonin and co-workers(e.g., [123, 122, 33, 11, 64, 65, 7, 8, 126, 9, 100]), Drew [35], Drew and Lahey[36], and the group at Chalmers University of Technology (e.g., [39, 108, 109,110, 40, 41]), are frequently divided into four different model classes.

With increasing model complexity, these model versions are:

• Constant Particle Viscosity (CPV) models.• Particle Turbulence (PT) models.• Particle and Gas Turbulence (PGT) models.• Particle and Gas Turbulence with Drift Velocity (PGTDV) models.

The continuity and momentum equations that are common for these modelversions are listed below. The model equations adopted for non-reactive mix-tures can be deduced from the more general formulations (3.293) and (3.296),respectively.


The continuity equation used is expressed as:

∂

∂t

(

αk〈ρk〉Xk

)

+ ∇ ·(

αk〈ρk〉Xk〈vk〉Xkρk

)

= 0 (10.84)

The momentum equation employed is given by:

∂

∂t

(

αk〈ρk〉Xk〈vk〉Xkρk

)

+ ∇ ·(

αk〈ρk〉Xk〈vk〉Xkρk〈vk〉Xkρk

)

= −∇ ·(

αk〈Tk〉Xk + αkTRe,Xk

k

)

+ αk〈ρk〉Xkg + 〈MkI〉(10.85)

In order to separate the average of products into products of average, weightedaveraged values are commonly introduced. The phasic - and mass averageshave been defined by (3.277) and (3.278), respectively. Hence, the instanta-neous velocity is decomposed into a weighted mean component and a fluctu-ation component in accordance with (3.279).

The Reynolds stress tensor of phase k is given by:

TRe,Xk

k = 〈ρk〉Xk〈v′′kv

′′k〉Xkρk (10.86)

The total stress tensor is conventionally decomposed into a pressure termand a viscous stress term. The average total stress term in the momentumequations may thus be re-written as:

−∇ · (αk〈Tk〉Xk) = −∇ · [αk(〈pk〉Xke + 〈σk〉Xk)]

= −∇(αk〈pk〉Xk) −∇ · (αk〈σk〉Xk)(10.87)

The viscous stress tensor of both phases can be modeled using the rigorousNewtonian strain-stress relation:

〈σk〉Xk = −μB,k∇ · 〈vk〉Xkρke − 2μk(〈Sk〉Xkρk − 13∇ · 〈vk〉Xkρke) (10.88)

where μB,k represents the bulk viscosity of phase k (kg/ms).The average strain rate tensor is defined by:

〈Sk〉Xkρk =12(∇〈vk〉Xkρk + (∇〈vk〉Xkρk)T ) (10.89)

It was explained in chap 4 that the particulate phase pressure, 〈pp〉Xp , con-sists of three effects, one kinetic contribution corresponding to momentumtransport caused by particle velocity fluctuation correlations, 〈pp,kin〉Xp , onecollisional contribution caused by particle interaction, 〈pp,coll〉Xp , and one be-ing a contribution from the gas phase pressure, 〈pg〉Xp . The pressure in theparticulate phase is thus given by:

αp〈pp〉Xp = αp〈pp,kin〉Xp + αp〈pp,coll〉Xp + αp〈pg〉Xp (10.90)


The particulate phase total stress tensor can then be written as:

−∇ · (αp〈Tp〉Xp) = −∇(αp〈pp,kin〉Xp) −∇(αp〈pp,coll〉Xp)

− 〈pg〉Xp∇αp − αp∇〈pg〉Xp −∇ · (αp〈σp〉Xp)(10.91)

The interfacial momentum transfer to phase k is defined by [36, 39]:

〈MkI〉 = 〈Tk · ∇Xk〉 (10.92)

This relation can be reformulated adopting one out of several possible mod-eling approaches. The conventional continuum mechanical approach for re-writing the interfacial momentum transfer terms for dispersed flows was out-lined in sect 3.4.3. Hence, an alternative approach for calculating the inter-facial momentum transfer terms based on kinetic or probabilistic theories, asproposed by Simonin and co-workers, is examined in this section.

The gradient of the phase indicator function, which appears in (10.92),was defined by (3.288). The expression for MkI then becomes [36, 39]:

〈MkI〉 = 〈Tk · ∇Xk〉 = −〈Tk · nkδk〉 (10.93)

as we recall that ∇Xk = (∂Xk/∂n)nk, where (∂Xk/∂n) = −δk.He & Simonin [65] argued that to find a relation for the drag force acting

on a single sphere in a suspension, the velocity field of the undisturbed flow isneeded. They derived a momentum equation for an undisturbed flow based onprobabilistic arguments. Based on the momentum equations for the disturbedand undisturbed flow, they derived an expression for the interfacial momentumtransfer. The interfacial momentum transfer term was thus decomposed asfollows:

〈Tp · npδp〉 = −〈Tg · ngδg〉 = − 〈˜Tg · ngδg〉 − 〈δTg · ngδg〉=〈˜Tg · ∇Xg〉 − 〈δTg · ngδg〉≈〈pge · ∇Xg〉 − 〈δTg · ngδg〉=〈pg∇Xg〉 − 〈δTg · ngδg〉=〈pg〉∇αg + 〈p′g∇Xg〉 − 〈δTg · ngδg〉≈〈pg〉∇αg + Fg

(10.94)

where ˜Tg denotes the total stress tensor of the undisturbed flow. The undis-turbed pressure is decomposed in accordance with the Reynolds procedurepg = 〈pg〉 + p′g.

The interfacial momentum transfer terms for the disturbed flow are ap-proximated by the steady drag force:

Fg ≈ −〈δTg · ngδg〉 ≈ 〈Xpρpvr/τgp〉 ≈ 〈Xpρp〉〈vr〉Xpρp/〈τgp〉Xpρp

≈ αp〈ρp〉Xp〈vr〉Xpρp/〈τgp〉Xpρp(10.95)

where τgp is the particle relaxation time.


We reiterate that for a dispersed flow Fp the macroscopic generalized dragforce normally contains numerous contributions, as outlined in chap 5. How-ever, for gas-solid flows the lift force fL, the virtual mass force fV , and theBesset history force fB components are usually neglected [39]. The conven-tional generalized drag force given by (5.27) thus reduces to:

Fp ≈ Np(fD + fL + fV + fB) ≈ NpfD (10.96)

where the forces in the brackets on the right hand side are the forces actingon a single particle in a suspension and Np is the number of particles per unitvolume.

The generalized drag force is then expressed as:

Fg = −Fp = −Xpρp

τgp(vg − vp) (10.97)

in which the particle relaxation time τgp is defined by:

1τgp

=3

4dp

ρg

ρpCD|vg − vp| (10.98)

The averaged drag force was approximated by [8]:

〈Fg〉 = −〈Fp〉 =〈−Xpρp

τgp(vg − vp)〉

≈ − 34dp

αp〈CD〉Xpρp〈ρg〉Xp〈|vr|〉Xpρp〈vr〉Xpρp

(10.99)

The average drag coefficient used is [51]:

〈CD〉Xpρp =(

17.3〈Rep〉Xpρp

+ 0.336)

α−1.8g (10.100)

An alternative parameterization for the drag coefficient is given by [34, 50].The particle Reynolds number is given by:

〈Rep〉Xpρp =〈ρg〉Xp〈|vr|〉Xpρpdp

μg(10.101)

The average particle relaxation time is given by:

〈τgp〉Xpρp =43

〈ρp〉Xpdp

〈ρg〉Xg 〈CD〉Xpρp〈|vr|〉Xpρp(10.102)

The resulting decomposition of the interfacial momentum transfer term isequivalent to the conventional closure outlined in sect 3.4.3, and adopted byseveral investigations on gas solids flow [64, 65, 39, 108]. Nevertheless, as forthe conventional formulation, several simplifying assumptions are invoked in


this model closure as well. Most important, the viscous terms in the undis-turbed flow were neglected, the average undesturbed pressure is approximatedby the mean pressure of the bulk phase, and the terms involving correlationsof the pressure fluctuations are negligible in gas-solid flows. Moreover, addi-tional closures are needed for the stress tensors, the fluctuating terms and themean relative velocity.

According to Bel F’dhila & Simonin [11], the average of the relative velocitybetween each particle and the surrounding fluid vr can be expressed as afunction of the mean relative velocity and a drift velocity due to the correlationbetween the instantaneous distribution of the particles and the large scaleturbulent fluid motion with respect to the particle diameter:

〈Xpρpvr〉 = 〈Xpρpvp〉 − 〈Xpρpvg〉 (10.103)

Introducing weighted velocity variables in the first and second term in thisequation, while decomposing the instantaneous velocity in the third term intoits weighted and fluctuating components, we obtain:

〈Xpρp〉〈vr〉Xpρp = 〈Xpρp〉〈vp〉Xpρp − 〈Xpρp〉〈vg〉Xpρp − 〈Xpρpv′′g 〉 (10.104)

Dividing all terms by the factor 〈Xpρp〉, we get:

〈vr〉Xpρp = 〈vp〉Xpρp − 〈vg〉Xpρp − 〈Xpρpv′′g 〉/〈Xpρp〉 (10.105)

The last term on the right hand side of (10.105) is defined as the driftvelocity:

vdrift = 〈v′′g 〉Xpρp = 〈Xpρpv′′

g 〉/〈Xpρp〉 (10.106)

where v′′g is the gas fluctuating velocity.

The term 〈|vr|〉Xpρp is the average relative velocity length that is approx-imated by [64, 65]:

〈|vr|〉Xpρp ≈√

〈vr〉Xpρp · 〈vr〉Xpρp + 〈v′′r · v′′

r 〉Xpρp (10.107)

where v′′r is the fluctuating relative velocity.

The term 〈v′′r · v′′

r 〉Xpρp is determined by:

〈v′′r · v′′

r 〉Xpρp = 2(kp + kg − kgp) (10.108)

where kg represents the turbulent kinetic energy of the gas phase (m2/s2),kgp the gas-particle fluctuation covariance (m2/s2), kp = 3θp/2 the turbulentkinetic energy analogue of the particulate phase (m2/s2), and θp the granulartemperature of the particle phase (m2/s2).


The Constant Particle Viscosity (CPV) model

The first attempts at describing the gas-particle flows in fluidized beds wereperformed using rather simple models neglecting both the Reynolds stresses,TRe,Xk

k in (10.86), and the kinetic pressure-gradient term, αp〈pp,kin〉Xp , in(10.91). No turbulence models are thus used for any of the phases.

The momentum equation for the gas phase is thus given by:

∂

∂t

(

αg〈ρg〉Xg 〈vg〉Xgρg

)

+ ∇ ·(

αg〈ρg〉Xg 〈vg〉Xgρg 〈vg〉Xgρg

)

=

− αg∇〈pg〉Xg −∇ ·(

αg〈σg〉Xg

)

+ αg〈ρg〉Xgg + 〈Fg〉(10.109)

The viscous stress tensor of the gas phase is modeled using a reduced form ofthe Newtonian strain-stress relation (10.88):

〈σg〉Xg = −2μg

(

〈Sg〉Xgρg − 13∇ · 〈vg〉Xgρge

)

(10.110)

The bulk viscosity is set to zero for the continuous gas phase, in line withwhat is common practice for single phase flows.

The momentum equation for the particulate phase is written as:

∂

∂t

(

αp〈ρp〉Xp〈vp〉Xpρp

)

+ ∇ ·(

αp〈ρp〉Xp〈vp〉Xpρk〈vp〉Xpρp

)

=

−∇ ·(

αg〈Tp〉Xp

)

+ αp〈ρp〉Xpg + 〈MpI〉 =

−∇(

αp〈pp〉Xp

)

−∇ ·(

αp〈σp〉Xp

)

+ αp〈ρp〉Xpg + 〈MpI〉 =

−∇(

αp〈pp,coll〉Xp

)

− αp∇〈pg〉Xp −∇ ·(

αp〈σp〉Xp

)

+ αp〈ρp〉Xpg + 〈Fp〉

(10.111)

where 〈Fp〉 is given by (10.99) to (10.108). Since no turbulence closure isemployed in the CPV model, (10.107) is approximated as:

〈|vr|〉Xpρp ≈√

〈vr〉Xpρp · 〈vr〉Xpρp , (10.112)

and the average relative velocity vector (10.105) is approximated by:

〈vr〉Xpρp = 〈vp〉Xpρp − 〈vg〉Xpρp (10.113)

The particle collisional pressure-gradient term in (10.91) is approximatedby [50]:

∇(αp〈pp,coll〉Xp) ≈ −G(αg)∇αg (10.114)


This term is often referred to as a particle-particle interaction force and hasthe effect of keeping the particles apart above a maximum possible parti-cle packing. The particle-particle interaction coefficient G(αg) is named themodulus of elasticity. A survey of different particle-particle interaction forcemodels are given by Massoudiet et al [98].

Enwald and Almstedt [40] adopted a relation for the particle-particle in-teraction force proposed by Bouillard et al [16]:

∇(αp〈pp,coll〉Xp) ≈ −G0 exp(−c(αg − α∗))∇αg (10.115)

where G0, c and α∗ are empirical constants. Enwald and Almstedt [40] setα∗ = 0.46 to limit the void-age from decreasing below this value. The othertwo constants were chosen as G0 = 1.0 (kg/ms2) and c = 500.

In the CPV model the viscous stress tensor for the particulate phase ismodeled using a simplified version of the Newtonian strain-stress relation(10.88), similar to that employed for the gas phase:

〈σp〉Xp = −2μp

(

〈Sp〉Xpρp − 13∇ · 〈vp〉Xpρpe

)

(10.116)

In this particular model version, the bulk viscosity is set to zero for the par-ticulate phase. The particle viscosity variable, μp, is set to a constant value.Enwald and Almstedt [40] used a particle viscosity value of μp ≈ 1.0 (kg/ms),being representative for the experimental data presented by [24](p 77).

The Particle Turbulence (PT) models

Over the years the CPV model has been shown not to be appropriate to repre-sent important details of certain gas-particle flows. For this reason more rigor-ous closures have been developed for the total stress tensor of the particulatephase, intending to obtain better representations of the physical phenomenainvolved.

The PT model represents an extension of the basic CPV model and con-tains extended closures for the particle collisional pressure and the particle-particle velocity correlation terms, as well as simple attempts to account forsome of the gas-particle interaction phenomena. For the gas phase, on theother hand, the same set of transport equations as for the CPV model areemployed. The particulate phase continuity equation is also the same, but themomentum equation for the particulate phase is modified.

To model the particle velocity fluctuation covariances caused by particle-particle collisions and particle interactions with the interstitial gas phase, theconcept of kinetic theory of granular flows is adapted (see chap 4). This theoryis based on an analogy between the particles and the molecules of dense gases.The particulate phase is thus represented as a population of identical, smoothand inelastic spheres. In order to predict the form of the transport equationsfor a granular material the classical framework from the kinetic theory of


dense gases is used [21]. However, as explained by Peirano and Leckner [109],to derive the closure laws for the fluxes that occur in these equations themethod of Grad [59] was preferred to that of Chapman-Enskog [21]. It is thusemphasized that the work of Simonin and co-workers is based on the resultsof Jenkins and Richman [69] that derived the necessary closure laws using theGrad’s 13 moment system for a dense gas of inelastic spheres. It follows thatthe transport equations discussed in this section are derived from the classicalresults of the kinetic theory for dense gases [21], in combination with Grad’stheory [59]. Bear in mind that in chap 4 the Chapman-Enskog method wasused, so the closure laws obtained by Simonin and co-workers are similar butnot completely identical to those given earlier.

Moreover, He & Simonin [64, 65] considered the early models developedby Jenkins and Richman [69] appropriate for granular flows in vacuum, butinaccurate in the dilute zones of the bed where the interstitial gas phasefluctuations may affect the particles. He & Simonin [64, 65] thus extendedthe kinetic theory of granular materials in vacuum to take into account theinfluence of the interstitial gas.

In the PT model the extended momentum balance for the particle phaseyields:

∂

∂t

(

αp〈ρp〉Xp〈vp〉Xpρp

)

+ ∇ ·(

αp〈ρp〉Xp〈vp〉Xpρp〈vp〉Xpρp

)

=

− αp∇(

〈pg〉Xp

)

−∇(

αp[〈pp,kin〉Xp + 〈pp,coll〉Xp ])

−∇ ·(

αp〈σp〉Xp + αpTRe,Xpp

)

+ αp〈ρp〉Xpg + 〈Fp〉

(10.117)

where 〈Fp〉 is given by (10.99) to (10.108). However, in accordance with theturbulence closure employed in the PT model, the relative velocity covarianceterm (10.108) therein is approximated by:

〈v′′r · v′′

r 〉Xpρp = 2kp (10.118)

Moreover, the average relative velocity vector (10.105) is approximated by(10.113), as for the CPV model, because the drift velocity is neglected.

The effective stress tensor of the particulate phase can be expressed byan analogy to Newton’s law of viscosity for viscous fluids (10.88), adoptingthe well known gradient and Boussinesq hypotheses modeling the Reynoldsstresses (10.86):

−αp〈σp〉Xp − αpTRe,Xpp =αpμB,p∇ · 〈vp〉Xpρpe

+ 2αpμp,eff

(

〈Sp〉Xpρp − 13∇ · 〈vp〉Xpρpe

)

(10.119)


In the PT model the particle pressure terms in (10.91) are modeled by:

−∇(

αp[〈pp,kin〉Xp + 〈pp,coll〉Xp ])

= −∇(

αp〈ρp〉Xp(1 + 2αpg0(1 + e))θp

)

(10.120)The transport equation for the granular temperature θp, written in terms

of the turbulent kinetic energy analogue of the particulate phase kp, is givenby [64, 65, 109]:

∂

∂t

(

αp〈ρp〉Xpkp

)

+ ∇ ·(

αp〈ρp〉Xp〈vp〉Xpρpkp

)

=

∇ ·(

αp〈ρp〉Xp(Kcollp + Kt

p)∇kp

)

−(

αp〈σp〉Xp + αp〈TRep 〉Xp

)

: ∇〈vp〉Xpρp

− αp〈ρp〉Xp

〈τgp〉Xpρp(2kp − kgp) + αp〈ρp〉Xp

e2 − 13τ c

p

kp

(10.121)

where kp = 32θp represents the turbulent kinetic energy analogue of the par-

ticulate phase (m2/s2), Kcollp the collisional diffusion coefficient (m2/s), Kt

p

the turbulent diffusion coefficient (m2/s), e the restitution coefficient, and τ cp

the particle-particle collision time (s). Furthermore, it is emphasized that thegas-particle covariance kgp is set to zero in the PT-model.

The bulk viscosity μB,p and the effective dynamic viscosity of the partic-ulate phase μp,eff are given by:

μB,p =43dpαp〈ρp〉Xpg0(1 + e)

√

θp

π(10.122)

μp,eff = 〈ρp〉Xp(νcollp + νt

p) (10.123)

in which νcollp and νt

p are the collisional and turbulent viscosities of the par-ticulate phase.

The collisional and turbulent viscosity values were calculated from [64, 65,109]:

νcollp =

45g0(1 + e)(νt

p + dp

√

θp

π) (10.124)

νtp =

(

23

τ tgp

〈τgp〉kgp + θp(1 + αpg0A)

)

/

(

2〈τgp〉Xpρp

+B

τ cp

)

(10.125)

where A = 2(1 + e)(3e− 1)/5 and B = (1 + e)(3− e)/5. The average particlerelaxation time 〈τgp〉Xpρp is obtained from (10.102). Moreover, it is emphasizedthat the gas-particle covariance kgp and the interaction time between theparticle motion and the gas phase velocity fluctuations τ t

gp are set to zero inthe PT-model.


In the formulation of the transport equations, several characteristic timescales are defined. In this framework these time scales are considered funda-mental in the classification and the understanding of the dominant mecha-nisms in the suspension flow. The particle relaxation time τgp was alreadydefined in (10.98). The particle-particle collision time τ c

p , is defined by:

τ cp =

dp

24αpg0

√

π

θp(10.126)

The radial distribution function g0 accounts for the probability of particlecontact. A possible parameterization is given by [94]:

g0 = (1 − αp/αp,max)−2.5αp,max (10.127)

where αp,max is the maximum packing of the particulate phase (≈ 0.64).Alternative parameterizations for g0 can be found in [34, 50, 95, 19, 104, 109].

The collisional and turbulent diffusion coefficients are modeled by[126, 109]:

Ktp =

θp(1 + αpg0C)(

95〈τgp〉Xpρp

+ Dτc

p

) (10.128)

Kcollp = αpg0(1 + e)(

65Kt

p +43dp

√

θp

π) (10.129)

where C = 3(1 + e)2(2e− 1)/5 and D = (1 + e)(49 − 33e)/100.

The Particle and Gas Turbulence (PGT) model

The PGT model represents an extension of the PT models in that the gasturbulence is taken into account by including the Reynolds stress tensor inthe momentum equation for the gas phase. The turbulence model used forthe gas phase is similar to the standard single phase k-ε turbulence modelpresented in sect 1.3.5, although additional generation and dissipation termsmay be added to consider the presence of particles. In the PGT model thedrift velocity is neglected.

In the PGT momentum equations the average drag force 〈Fp〉 is given by(10.99) to (10.108). Moreover, the average relative velocity vector (10.105) isapproximated by (10.113), as for the CPV and PT models, because the driftvelocity is neglected in the PGT model too.

In the momentum equation for the gas phase the Reynolds stress tensor isapproximated by the gradient and Boussinesq hypotheses and given by:

TRe,Xgg =〈ρg〉Xg 〈v′′

gv′′g 〉Xgρg

=23〈ρg〉Xgkge − 2μt

g

(

〈Sg〉 −13∇ · 〈vg〉Xgρge

) (10.130)


where μtg is the dynamic turbulent viscosity of the gas phase. The viscous

stress tensor used is given by (10.110).Simonin and Viollet [122] calculated the dynamic viscosity for the gas

phase from a modified k − ε model. The time scale of the large eddies of thegas phase flow was given by:

τ tg = 3Cμkg/(2εg) (10.131)

The dynamic turbulent viscosity of the gas phase flow was given by μtg =

2〈ρg〉Xgkgτtg/3, in accordance with the standard single phase turbulence the-

ory presented in sect 1.3.5.The transport equation that was used for the turbulent kinetic energy of

the gas phase is written as [122, 126, 8]:

∂

∂t

(

αg〈ρg〉Xgkg

)

+∇ ·(

αg〈ρg〉Xg 〈vg〉Xgρgkg

)

= ∇ ·(

αg

μtg

σk∇kg

)

− αgTRe,Xgg : ∇〈vg〉Xgρg − αg〈ρg〉Xgεg + Πkg

(10.132)

where Πkg represents the gas-particle interaction phenomena (kg/ms3). Thisinteraction term is modeled by:

Πkg =αp〈ρp〉Xp

〈τxgp〉Xpρp

(

− 2kg + kgp + vdrift · 〈vr〉Xpρp

)

(10.133)

However, in the PGT model the drift velocity vdrift is neglected and set tozero. The average particle relaxation time 〈τgp〉Xpρp is obtained from (10.102).

The transport equation for the dissipation rate of the gas-phase turbulentkinetic energy is given by [122, 126, 8]:

∂

∂t

(

αg〈ρg〉Xgεg

)

+ ∇ ·(

αg〈ρg〉Xg 〈vg〉Xgεg

)

= ∇ ·(

αg

μtg

σε∇εg

)

−

αgεgkg

(

Cε1αgTRe,Xgg : ∇〈vg〉Xgρg + Cε2〈ρg〉Xgεg

)

+ Πεg

(10.134)

where Πεg denotes the interaction term in the εg equation (kg/ms4). Thisinteraction term is modeled by:

Πεg = Cε3εgkg

Πkg (10.135)

The parameter values chosen in the gas phase turbulence model are the sameas those used for the standard single phase k-ε model (see sect 1.3.5). Theadditional interaction term parameter is set at a fixed value, Cε3 = 1.3, assuggested by Elghobashi and Abou-Arab [38].

For the particulate phase, the PT-model equations that were describedin sect 10.7.4 are used with minor extensions. That is, in the PGT-model


the transport equation for kp (10.121) contains the gas-particle fluctuationcovariance, kgp, to take into account the effect of the gas phase turbulence.

The effective particle phase viscosity is still obtained from (10.123). Inaddition, the turbulent viscosity of the particulate phase is calculated from(10.125) in which kgp is obtained from a separate balance equation. The in-teraction time between the particle motion and the gas velocity fluctuationsτ tgp, is modeled as suggested by Csanady [25]:

τ tgp =

τ tg

√

1 + 1.45(

3〈vr〉Xpρp · 〈vr〉Xpρp/2kg

)

(10.136)

The transport equation for the gas-particle fluctuation covariance is givenby [126]:

∂

∂t

(

αp〈ρp〉Xpkgp

)

+ ∇ ·(

αp〈ρp〉Xp〈vp〉Xpρpkgp

)

=

∇ ·(

αp〈ρp〉Xpνt

gp

σk∇kgp

)

− αp〈ρp〉Xp〈v′′gv

′′p〉 : ∇〈vp〉Xpρp

− αp〈ρp〉Xp〈v′gv

′p〉 : ∇〈vg〉Xgρg − αp〈ρp〉Xpεgp + Πgp

(10.137)

where νtgp denotes the gas-particle turbulent viscosity (m2/s), εgp the dissi-

pation rate of the gas-particle fluctuation covariance (m2/s3), and Πgp theinteraction term in the kgp model (kg/ms3).

The dissipation rate of the gas-particle fluctuation covariance εgp and thegas-particle turbulent viscosity νt

gp are defined by:

εgp = kgp/τtgp (10.138)

νtgp = kgpτ

tgp/3 (10.139)

The gas-particle fluctuation correlation tensor 〈v′′gv

′′p〉 is expressed by:

〈v′′gv

′′p〉 =

13kgpe − νt

gp

(

〈Sgp〉 −13tr(〈Sgp〉)e

)

(10.140)

The average gas-particle strain rate tensor is given by:

〈Sgp〉 =12

(

∇〈vg〉Xgρg + (∇〈vp〉Xpρp)T

)

(10.141)

The interaction term in (10.137) is modeled by:

Πgp = −αp〈ρp〉Xp

〈τxgp〉Xpρp

(

(1 +αp〈ρp〉Xp

αg〈ρg〉Xg)kgp − 2kg − 2

αp〈ρp〉Xp

αg〈ρg〉Xgkp

)

(10.142)


The Particle and gas turbulence model with drift velocity(PGTDV) Model

The PGTDV model consists of the same equations as the PGT model de-scribed in sect 10.7.4, the only difference being that the drift velocity is con-sidered in the PGTDV model. The drift velocity vdrift is included in (10.105)and (10.133).

The drift velocity takes into account the dispersion effect due to the par-ticle transport by the fluid turbulence. From the limiting case of particleswith diameter tending towards zero, for which the drift velocity reduces tosingle turbulence correlation between the volumetric fraction of the dispersedphase and the turbulent velocity fluctuations of the continuous phase. Thedrift velocity: vdrift is modeled as [33]:

vdrift = Dtgp(

1αg

∇αg − 1αp

∇αp) (10.143)

Based on semi-empirical analysis, the fluid-particle turbulent dispersion ten-sor, Dt

gp, is expressed in terms of the covariance between the turbulent veloc-ity fluctuations of the two phases and a fluid particle turbulent characteristictime:

Dtgp = τ t

gpkgp/3 (10.144)

The model assumes that the particles are suspended in a homogeneous fieldof gas turbulence.

It is mentioned, although not used in the model evaluation by Enwaldand Almstedt [40], that a much simpler closure for the binary turbulent dif-fusion coefficient Dt

gp has been derived by Simonin [123] by an extension ofTchen’s theory. This simple closure has been used by Simonin and Viollet[124], Simonin and Flour [125] and Mudde and Simonin [100] simulating sev-eral dispersed two-phase flows.

Initial and Boundary Conditions

To simulate a rectangular fluidized bed reactor the bed vessel dimensionshave to be specified first. The vessel used for validation has a rectangularcross section [40]. The bed vessel was 0.3 (m) wide, 2.22 (m) high and 0.2 (m)deep.

Proper boundary conditions are generally required for the primary vari-ables like the gas and particle velocities, pressures and volume fractions at allthe vessel boundaries as these model equations are elliptic. Moreover, bound-ary conditions for the granular temperature of the particulate phase is requiredfor the PT, PGT and PGTDV models. For the models including gas phaseturbulence, i.e., PGT and PGTDV, additional boundary conditions for theturbulent kinetic energy of the gas phase, as well as the dissipation rate ofthe gas phase and the gas-particle fluctuation covariance are required. The


boundary conditions for the primary variables are normally specified adopt-ing the standard single phase flow approaches. For some of the variables likethe turbulence properties and the volume fractions one has to use empiricalor semi-empirical information obtained from experiments to approximate theboundary values. The specification of the velocities at the inlet may requirespecial attention to consider the different geometries of the gas distributors.

The initial conditions are generally specified in correspondence with thestate of a fluidized bed operating at minimum fluidization conditions. Thebed height at minimum fluidization conditions is then set to Lmf , and thegas volume fraction is set to αmf at the bed levels below Lmf and unity inthe freeboard. The pressure profile in the bed is initialized using the Ergun[42] equation, whereas the pressure in the freeboard is set to the operationalpressure at the outlet. The horizontal velocity components of both phases andthe vertical particle velocity component are set to zero. The vertical interstitialgas velocity in the bed is normally initiated as Us

mf/αmf , and Usmf in the

freeboard.The gas density is initiated by use of the ideal gas law requiring that the gas

pressure, species composition and temperature are known. When turbulenceis considered, kg, kp and εg are frequently set to small but non-zero values.kgp is set to zero.

To obtain an asymmetrical flow, as observed for real cases, particular flowperturbations are generally introduced for a short time period as the flowdevelops in time from the start. Small jets at the bottom are often used forthis purpose.

Model Evaluations

Enwald and Almstedt [40] assessed the four different two-fluid model closuresgiven above to investigate the effect of the gas phase turbulence, drift veloc-ity and three dimensionality on the fluid dynamics of a bubbling fluidizedbed. A few characteristics features of the different models were observed. TheCPV model results generally deviated from those obtained by the more rigor-ous model versions. Nevertheless, the CPV model results were often in betteragreement with the experimental data than the other model predictions. Com-paring the PGT and PT model results it was observed that at atmosphericconditions the gas phase turbulence did not have any significant effect on thebed behavior. However, at higher pressures significant changes in the resultswere observed. Moreover, the drift velocity included in the most advancedmodel version PGTDV did not have any noticeable effect on the results atany pressure. Furthermore, strictly grid independent solutions were not ob-tained, and the three-dimensional effects were considered considerable.

10.7.5 Bubbling Bed Reactor Simulations Using Two-Fluid Models

The bubbling bed reactor flow investigation performed by Lindborg [91] andLindborg et al [92] is assessed in this section.


The kinetic theory closures applied in the PGT model employed by Lind-borg [91] are in many ways distinct from those examined by Enwald andAlmstedt [40]. The model closures derived from kinetic theory and adoptedby Lindborg [91] are consistent with the theory presented in chap 4. It is em-phasized that these closures are derived using the Chapman-Enskog method,whereas Simonin and co-workers derived their closures using a combinationof the Chapman-Enskog and Grad methods. Moreover, in the work by Lind-borg et al the granular temperature is not considered a particle turbulenceclosure but a property of the granular material. Furthermore, the solid phasestress closure is extended considering the impact of the long term particle-particle interactions such as sliding or rolling contacts. This modification maybe necessary for certain flow problems because the internal momentum trans-port closure derived from the kinetic theory of granular flows considers onlythe contributions from particle translation and short term particle-particleinteractions. In particular dense flows at low shear rates the stress genera-tion mechanism due to the long term particle-particle interactions in whichlarge amounts of energy is dissipated, may be significant. The stress ten-sor closure proposed by Srivastava and Sundaresan [127] for the long termparticle-particle interactions was adopted.

Srivastava and Sundaresan [127] calculated the total stress as a linearsum of the kinetic, collisional, and frictional stress components, where each ofthe contributions are evaluated as if they were alone. The extended particlepressure and viscosity properties are calculated as:

pp = pp,kin + pp,coll + pp,fric (10.145)

μp = μp,kin + μp,coll + μp,fric (10.146)

This model is supposed to capture the two extreme limits of granular flow,which are designating the rapid shear and quasi-static flow regimes. In therapid shear flow regime the kinetic stress component dominates, whereas inthe quasi-static flow regime the friction stress component dominates [127].

The frictional stress closure derived by Srivastava and Sundaresan [127] isbased on the critical state theory of soil mechanics. Moreover, it was assumedthat the granular material is non-cohesive and has a rigid-plastic rheologicalbehavior. At the critical state the granular assembly deforms without volumechange, ∇ · vp = 0, and the frictional stress tensor equals the critical statefrictional stress tensor. This particular frictional stress closure, which strictlyspeaking is valid only at the critical state, is frequently used as a simplerepresentation of these stresses in the granular assembly even when ∇·vp = 0.In particular, based on a set of test simulations the simplified frictional stressclosure was found adequate for bubbling fluidized bed simulations.

The critical state pressure is given by [70]:

pp,crit =

{

F(αp−αp,min)r

(αp,max−αp)s if αp > αp,min

0 if αp ≤ αp,min

(10.147)


where F , r and s are empirical constants. The model asserts that the frictionalinteractions do not occur for particle volume fractions less than αp,min.

At the critical state, the solids frictional viscosity is given by:

μp,crit =pp,crit

√2 sinφ

2αp

√

Sp : Sp + Θp/d2p

(10.148)

The parameters adopted in the work of Lindborg et al [92] were taken fromOcone et al [105]. Johnson et al [70] used a similar set of parameters. Bothsets of parameter values are listed in table 10.4.

Table 10.4. Empirical parameters for frictional stresses.

Ocone et al [105] Johnson et al [70]

φ Angle of internal friction [◦] 28 28.5F Constant [N/m2] 0.5 0.5r Constant 2 2s Constant 3 5αp,min Lower volume fraction 0.5 0.5

limit for friction

The solids frictional pressure and viscosity are thus approximated by:

pp,fric ≈ pp,crit (10.149)

μp,fric ≈ μp,crit (10.150)

The term β〈v′g · Cp〉 in the granular temperature equation is normally

neglected due to a general lack of understanding of the physics representedby this term [34, 50]. In most cases, the closures found in the literature haveno significant effect on the solution.

In one of the proposed modeling approaches the production of granulartemperature represented by the gas-particle velocity covariance term is in-terpreted as a mechanism that breaks a homogeneous fluidized bed with noshearing motion into a non-homogeneous distribution. Koch [74] proposed aclosure for these gas-particle interactions for dilute suspensions. Koch andSagani [75] extended the closure of Koch [74] accounting for the dense sus-pensions effects. Lindborg et al [92] re-wrote the given closure in terms of theparticle relaxation time for Stokes flow τgp = ρpd

2p/18μg as:

β〈v′g · Cp〉 =

αpρpdp|vp − vg|2RsF2

4τ2gp

√

πθp

(10.151)

where Rs represents the energy source due to a specified mean force actingon the particles.


An expression for Rs obtained by a fit of lattice-Boltzmann simulationswas used:

Rs =1

g0(1 + 3.5√αp + 5.9αp)(10.152)

By applying the F definition of Benyahia et al [12], the closure can be ex-pressed as a function of the friction coefficient β instead of the dimensionlessdrag coefficient F [92]:

β〈v′g · Cp〉 =

β2dp|vp − vg|2Rs

4α4gαpρp

√

πθp

(10.153)

The Governing Equations

For reactive flows the governing equations used by Lindborg et al [92] resemblethose in sect 3.4.3, but the solid phase momentum equation contains severaladditional terms derived from kinetic theory and a frictional stress closure forslow quasi-static flow conditions based on concepts developed in soil mechan-ics. Moreover, to close the kinetic theory model the granular temperature iscalculated from a separate transport equation. To avoid misconception themodel equations are given below (in which the averaging symbols are disre-garded for convenience):

The continuity equation for phase k (= g, p) is:

∂

∂t(αkρk) + ∇ · (αkρkvk) = MwCO2

Rk,CO2 (10.154)

The momentum equation for the gas phase is expressed as:

∂

∂t(αgρgvg)+∇·(αgρgvgvg) = −αg∇pg −∇·(αgσg)+αgρgg+Mg (10.155)

The single phase gas phase viscous stress tensor (1.69), in which thebulk viscosity of the gas is set to zero, is used. The resulting viscousstress model coincides with (10.110):

σg = −μg

(

∇vg + (∇vg)T − 23(∇ · vg)e

)

The drag force coefficient used was taken from Gibilaro et al [51]. From(10.100) and (10.101) a derived coefficient was defined by:

β =(

17.3Rep

+ 0.336)

ρg|vp − vg|dp

αpα−1.8g (10.156)


The momentum equation for the solid phase can be written as:∂

∂t(αpρpvp) + ∇ · (αpρpvpvp) = −αp∇pg −∇ · pp + αpρpg + Mp (10.157)

The solids phase pressure tensor is modeled in accordance with(10.145). The pressure tensor is thus expressed as:

pp = −(

− pp +αdμB,d∇·vk

)

e−αdμd

(

∇vk +(∇vk)T − 23(∇·vk)e

)

(10.158)The solid phase pressure is calculated from (4.89) and (10.149):

pp =pp,kin + pp,coll + pp,fric

=αpρpΘp [1 + 2(1 − e)αpg0] + pp,crit

(10.159)

The radial distribution function can be approximated from [95]:

g0 =1 + 2.5αp + 4.5904α2

p + 4.515439α3p

[

1 −(

αp

αp,max

)3]0.67802 (10.160)

The solid phase bulk viscosity is calculated from (4.92):

μB,p =43αpρpdpg0(1 + e)

√

Θp

π

The particle phase viscosity is modeled in accordance with (10.146):

μp =μp,kin + μp,coll + μp,fric

=2μdilute

p

αpg0(1 + e)

[

1 +45αpg0(1 + e)

]2

+45αpρpg0(1 + e)

√

Θp

π+ μp,crit

(10.161)

The dilute particle viscosity is calculated from (4.93):

μdilutep =

596

ρpdp

√

πΘp

The granular temperature equation for the particle phase is written:

32

[

∂

∂t(αpρpΘp) + ∇ · (αpρpvpΘp)

]

= − (pp,kin + pp,coll + pp,fric) : ∇vp

−∇ · (−αpΓp∇θp)− 3βΘp + β〈v′

c · Cd〉 − γp

+32(MwCO2

Rp,CO2)Θp

(10.162)


The total heat flux is modeled in accordance with (4.95). Hence, theconductivity of the granular temperature is calculated from [50]:

αpΓp =αp(Γp,kin + Γp,coll)

=152

μdilutep

(1 + e)g0

[

1 +65αpg0(1 + e)

]2

+ 2α2pρpdpg0(1 + e)

√

Θp

π

The collisional energy dissipation term, as derived by [68], is given by(4.98):

γp = 3(1 − e2)α2pρpg0Θp

[

4dp

√

Θp

π−∇ · vp

]

The term β〈v′c ·Cd〉 is normally neglected due to a general lack of un-

derstanding of the physics represented by this term. However, to verifythis assumption Lindborg et al [92] performed numerous simulations tostudy the influence of these phenomena as represented by the closureproposed by [75].

The molecular temperature equation for phase k (= g, p) is defined by:

∑

c

αkρkωk,cCpk,c∂Tk

∂t+

∑

c

αkρkωcCpk,cvk · ∇Tk =

∇ · (kk∇Tk) +∑

j

(−ΔHr,k,j)Rk,j + Qk

(10.163)

For spherical particles the volumetric heat transfer coefficient is givenas the product of the specific surface area and the interfacial heattransfer coefficient (10.177). The volumetric interfacial heat transfercoefficient is modeled by (10.178)

.

The species mass balance for phase k (= g, p) yields:

∂

∂t(αkρkωk,c)+∇· (αkρkvkωk,c) = ∇· (αkρkDk,c∇ωk,c)+Mwc

Rk,c (10.164)

in which Dk,c represents an effective mass diffusion coefficient of speciesc in phase k.

Initial and Boundary Conditions

Initially, there is no gas flow through the reactor, and the true volume fractionof solids in the bed is about that of maximum packing. However, Lindborget al [92] adopted the customary approach of specifying the initial conditionsin correspondence with the state of a bubbling bed operating at minimumfluidization conditions. The bed height at minimum fluidization conditionswas thus set to an estimated value Lmf , and the gas volume fraction was setto αmf at the bed levels below Lmf and unity in the freeboard. The pressure


profile in the bed was initialized using the Ergun [42] equation, whereas thepressure in the freeboard is set to the operational pressure at the outlet.The horizontal velocity components of both phases and the vertical particlevelocity component are set to zero. The vertical interstitial gas velocity in thebed is initiated as Us

in/αmf , and Usmf in the freeboard.

The gas density was either initiated by a fixed value or calculated by useof the ideal gas law requiring that the gas pressure, species composition andtemperature are known. When turbulence is considered, kg, kp and εg arefrequently set to small but non-zero values. Typical levels of the turbulentkinetic energy and the dissipation rate are about 10−5 (m2s−2) and 10−5

(m2s−3), respectively. The granular temperature is set at about 10−5 (m2s−2).To obtain an asymmetrical flow, as observed for real cases, particular flow

perturbations are generally introduced for a short time period as the flowdevelops in time from the start. Heterogeneity was introduced by tilting thegravity vector by 1 % the first second if axi-symmetry is not assumed.

The governing equations are elliptic so boundary conditions are requiredat all boundaries. The normal velocity components for both phases are set tozero at the vertical boundaries. The wall boundary conditions for the verticalvelocity component, k and ε are specified in accordance with the standardwall function approach. The particle phase is allowed to slip along the wallfollowing the boundary condition given by (4.99).

Uniform flow is assumed at inlet. The gas pressure is set at outlet. Theparticles are not allowed to leave the reactor. For the scalar variables, exceptpressure, Dirichlet boundary conditions are used at inlet, whereas Neumannconditions are employed at the other boundaries [92].

During start up the system is very unstable. To simplify the conditions inthe reactor, reactants are gradually introduced after 5 seconds to avoid largevariations of density in the gas phase.

Cold Flow Reactor Simulations

In a bubbling bed flow assessment performed by Lindborg [91] and Lindborget al [92], the CPV model was used to simulate the bed behavior in thecylindrical vessel investigated experimentally by Lin et al [90]. The elasticitymodulus parameterization by Ettehadieh et al [43] was employed in thesesimulations:

G(αg) = −10−10.46αg+6.577 (Nm−2) (10.165)

A particular set of cold flow simulations were run with the physical propertiesand model parameters presented in table 10.5.

It was concluded that the CPV model is not able to reproduce the ex-perimentally determined flow pattern, as is easily recognized by comparingFig 10.8 and Fig 10.13, using a physical μp value. The flow patterns predictedare in many ways opposite to the measured ones. The simulations may give anupward flow in the center region of the tube and downwards close to the wall,


Table 10.5. Physical properties and model parameters [92] (case 1).

Cold Flow Simulations

poutg Outlet pressure (atm) 1.0

ρg Gas density (kg/m3) 1.21μg Gas viscosity (kg/ms3) 1.8 × 10−5

ρp Particle density (kg/m3) 2500dp Particle diameter (μm) 500e Restitution coefficient 0.997αp,max Maximum particle packing 0.62αp,mf Particle volume fraction 0.58

at minimum fluidizationup,mf Minimum fluidization velocity (m/s) 0.19Lmf Bed height at minimum fluidization (m) 0.113H Column height (m) 0.3164D Reactor diameter (m) 0.1380Δr Lateral resolution (mm) 3.450Δz Axial resolution (mm) 2.825

whereas the measured data for the same case shows that the flow should bedownwards in the center and upwards close to the wall and visa versa. Never-theless, by increasing the particle viscosity artificially by a factor of about 5the simulated flow patterns coincide much better with the observed ones. Thedeviation between the simulated and measured flow patterns may thus be ex-plained by the limitations of the CPV model closures. However, the influenceof the axi-symmetry boundary condition applied in the two-dimensional flowsimulations is strictly not verified yet.

From the study by Enwald and Almstedt [40] it may be concluded that aPT model can be sufficient simulating bubbling fluidized bed reactors. How-ever, it was shown by Lindborg et al [92] that the gas phase turbulence isimportant considering the species mixing within the gas phase. A PGT typeof model was thus recommended for the purpose of simulating reactive flows inbubbling bed reactors. Moreover, using the (10.153) closure, the simulationsperformed by Lindborg et al [92] confirmed that the β〈v′

c · Cd〉 term in thegranular temperature equation has no significant effect on the solution andcan be neglected.

Simulating the Lin et al data sets by use of a PGT model, Lindborg [91] andLindborg et al [92] obtained good predictions of the flow as shown comparingthe measured flow patterns Fig 10.8 and the predicted ones Fig 10.14. Thegas bubbles generally form close to the gas distributor near the wall, andmigrate towards the center as they rise. On their way up through the bedthey withdraw particles into the wake and thereby create a gulf streamingcirculation where the bubbles create a net upflow of solids in the low densityregions and a downflow of solids in the dense regions. On average, at thelowest gas flow rate, descending particles are observed at the center of thevessel while ascending closer to the wall (case (a)). With increasing gas flow


a)

0.1

0.10.1

0.10.2

0.2 0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.40.5

0.5

0.5

0.5

0.5

z [m

]

r [m]−0.05 0 0.05

0

0.03

0.06

0.09

0.12

0.15

0.18

Vector scale5 cm/s

b)

0.01

0.01

0.01

0.1 0.1 0.10.2

0.20.2

0.3

0.30.3

0.3

0.3

0.4

0.4

0.40.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.50.5

0.5

0.5

z [m

]r [m]

−0.05 0 0.050

0.03

0.06

0.09

0.12

0.15

0.18Vector scale

10 cm/s

c)

0.1

0.1

0.1

0.20.2

0.2

0.2

0.3

0.3

0.3

0.3

0.3

0.3 0.3

0.40.4

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.5

0.5

0.5

z [m

]

r [m]−0.05 0 0.05

0

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0.06

0.09

0.12

0.15

0.18

0.21

0.24

Vector scale25 cm/s

d)

0.10.1 0.1 0.1

0.2

0.2

0.2

0.2

0.20.3

0.3

0.3

0.3

0.3

0.3

0.3

0.4

0.4

0.40.4

0.4

0.4

0.4

0.4

0.5

0.50.

5

0.5 0.5

z [m

]

r [m]−0.05 0 0.05

0

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

Vector scale50 cm/s

Fig. 10.13. Particle circulation patterns at various fluidizing velocities for a gasfluidized bed consisting of 0.42−0.6 (mm) diameter glass beads [90]. Simulated flowpatterns obtained with the CPV model of Lindborg [91]. (a) Us

in = 32 (cm/s) andUs

in/Usmf = 1.65, (b) Us

in = 45.8 (cm/s) and Usin/Us

mf = 2.36, (c) Usin = 64.1 (cm/s)

and Usin/Us

mf = 3.31, (d) Usin = 89.2 (cm/s) and Us

in/Usmf = 4.6


a)

0.010.01

0.010.1 0.1 0.10.2 0.2 0.20.3 0.3 0.30.4 0.4 0.4

0.5 0.5 0.5

z [m

]

r [m]−0.05 0 0.05

0

0.03

0.06

0.09

0.12

0.15

0.18

Vector scale5 cm/s

b)

0.01

0.01

0.01

0.10.1

0.10.2

0.2 0.20.3 0.3 0.3

0.4 0.4 0.4

0.5

0.5

0.5

0.5

z [m

]r [m]

−0.05 0 0.050

0.03

0.06

0.09

0.12

0.15

0.18Vector scale

10 cm/s

c)

0.01

0.01 0.01

0.10.1

0.10.2

0.20.20.3 0.3 0.3

0.4 0.4 0.4

z [m

]

r [m]−0.05 0 0.05

0

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

Vector scale25 cm/s

d)

0.010.01

0.01

0.10.1 0.1

0.2 0.2 0.2

0.3 0.3 0.3

0.4

0.4

0.4

0.4

z [m

]

r [m]−0.05 0 0.05

0

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

Vector scale50 cm/s

Fig. 10.14. Particle circulation patterns at various fluidizing velocities for a gasfluidized bed consisting of 0.42−0.6 (mm) diameter glass beads [90]. Simulated flowpatterns obtained with the PGT model of Lindborg [91]. (a) Us

in = 32 (cm/s) andUs

in/Usmf = 1.65, (b) Us

in = 45.8 (cm/s) and Usin/Us

mf = 2.36, (c) Usin = 64.1 (cm/s)

and Usin/Us

mf = 3.31, (d) Usin = 89.2 (cm/s) and Us

in/Usmf = 4.6


rate the center-directed bubble migration is more pronounced, hence the solidscirculation pattern gradually turns so that particles eventually ascend at thecenter and descend near the wall (cases (b)-(d)). Although the transitionbetween the two main circulation patterns are not perfectly predicted, theparticle motion dependence on the superficial gas velocity is captured very wellsince both the particle velocity magnitudes and circulation patterns coincidewith the experimental data. In addition to the gas and solid motion in thebubbling bed, the bed expansion estimated from the same data correspondsquite well with the simulations. Moreover, the simulated mean bubble sizeand bubble rise velocity were in fair agreement with the frequently employedcorrelations given in the literature.

Simulation of a Sorption Enhanced Steam reforming Process

Jørgensen [71] simulated a sorption enhanced steam reforming process em-ploying an extended PGT model derived by Lindborg [91] and Lindborg et al[92]. The reactor configuration used operating the novel chemical process isdefined in table 10.6.

Table 10.6. Circular axi-symmetric reactor configuration used by [71].

Reactor configuration

Column height (m) 1.5Reactor diameter (m) 0.25Bed height at minimum fluidization (m) 0.5Particle volume fraction at minimum fluidization 0.61Outlet pressure (bar) 5Inlet gas velocity (m/s) 0.1Density dispersed phase (kg/m3) 1500Numerical resolution (m) 0.0125Grid resolution 122 × 12Particle diameter (μm) 500

The kinetic model used for the conventional steam reforming process wastaken from Xu and Froment [139]. The reaction takes place on a Ni/MgAl2O4

catalyst. The kinetics can be described by three equations, where two areindependent:

CH4 + H2O ←→ CO + 3H2 (10.166)

CH4 + 2H2O ←→ CO2 + 4H2 (10.167)

CO + H2O ←→ CO2 + H2 (10.168)

The rate equations are given by the following equations:

r1 =k1

p2.5H2

(pCH4pH2O − p3H2

pCO

K1)

DEN2(10.169)


r2 =k2

pH2

(pCOpH2O − pH2pCO2K2

)DEN2

(10.170)

r3 =k3

p3.5H2

(pCH4p2H2O − p4

H2pCO2

K3)

DEN2(10.171)

where

DEN = 1 + KCOpCO + KH2pH2 + KCH4pCH4 + KH2OpH2O/pH2 (10.172)

All equilibrium and kinetic constants are taken from Xu and Froment [139].The removal of CO2 from hot streams is considered a possible future tech-

nology for energy production. The sorption enhanced reaction process (SERP)has the potential to reduce the costs of hydrogen production by steam methanereforming, in addition to removing CO2 from the product stream. In this pro-cess, a CO2 acceptor is installed together with the catalyst for removal of CO2

from the gas phase, and hence pushing the equilibrium limits toward a higherH2 yield. The steam reforming process may thus be run at lower tempera-tures than conventional steam reforming (723-903 K), possibly reducing theinvestment and operational costs significantly [103].

The Lithium Silicate equilibrium reaction is:

Li4SiO4 + CO2 ←→ Li2CO3 + Li2SiO3 (10.173)

The mathematical model for the sorption reactions is given by [118]:

dx

dt= k1(PCO2 − PCO2,eq

)n1(1 − x)n2 (10.174)

where x is the fractional conversion of the reaction, defined as x = q/qmax.q is defined as the mass of CO2 adsorbed divided by mass adsorbent. qmax

refers to maximum amount of CO2 that can be adsorbed.The temperature dependence of k1 is given by:

k1 = k10e−Ea

R ( 1T − 1

T0) (10.175)

The reaction rate for the sorption is then given by:

rad =qmax

MwCO2

dx

dt(10.176)

The parameters in the Li4SiO4 sorption kinetics are given in table 10.7.A review of the heat transfer characteristics of fluidized beds has been given

by Yates [143]. It is generally accepted that the heat transfer between gas andparticles is very efficient in fluidized beds as a result of the high surface areaof the particle phase. The heat transfer fluxes between an immersed surfaceand the gas-fluidized bed material are more important from a practical design


Table 10.7. Parameters for the adsorption reaction kinetics [118].

Li4SiO4

k10 (s−1) 1.84e-4Ea (J/kmol) 1.1e8qmax (gCO2/gadsorbent) 0.20n1 0.26n2 2

point of view. Due to the efficient particle mixing in fluidized bed vessels, thesebeds are frequently assumed to be operated in an isotermal mode. However, forparticular processes, the effective conductivity of the bed material is requiredto predict possible temperature gradients within the bed.

The gas-particle interfacial heat transfer term in the temperature equa-tions can be modeled by:

Qg = −Qp = −6αpkg

d2p

Nup(Tp − Tg) = hv(Tp − Tg) (10.177)

A large number of empirical correlations is available for estimating the Nus-selt number in both packed- and fluidized beds. A Nusselt number correlationproposed by Gunn [60] was used. The Nusselt number parameterization rep-resents a functional fit to experimental data for Reynolds number up to 105

in the porosity range 0.35 − 1:

Nup = (7 − 10αg + 5α2g)(1 + 0.7Re0.2

p Pr0.3)

+ (1.33 − 2.4αg + 1.2α2g)Re

0.7p Pr0.3

(10.178)

where

Nup =hgpdp

kg, Rep =

αgρg|vg − vp|μg

and Pr =μgCp,g

kg(10.179)

According to Natarajan and Hunt [101] the effective thermal conductivityof the solids phase kp can be expressed as a linear sum of the kinetic- andmolecular thermal conductivities:

kp = kp,kin + kp,m (10.180)

The kinetic conductivity for a two-dimensional system is calculated from arelation given by Hunt [66]:

kp,kin = ρpCppdp

√

θpπ3/2

32αpg0(10.181)

For gas-particle systems the molecular conductivities of the different phasesare frequently calculated employing two semi-empirical relations deduced fromthe model of Zehner and Schlunder [145]. The original model of Zehner and


Schlunder was originally derived to describe the effective radial conductivityin fixed beds. By using a cell concept the heat is assumed to be transferred bymolecular conduction both in a pure gas phase with surface fraction 1−√

αp,and through a gas-solid bulk phase with the complimenting portion of thesurface fraction, √

αp. Deviations from sphericity and inter-particle point-contacts were taken into account by further dividing the gas-solid bulk phaseinto a surface fraction, φ, where heat is transferred through inter-particlecontact and 1 − φ for heat transfer through the remaining surface [10]. How-ever, in the two-fluid model framework it is necessary to separate the overallbulk thermal conductivity into individual conductivities for the gas and solidphases. Such a division has been proposed by Syamlal and Gidaspow [128].This two-fluid model correlation has later been adopted for calculating theeffective thermal conductivity of dense phase fluidized beds [15, 76, 107].

The effective thermal conductivity of the gas phase was given by:

kg,eff = kg

(1 −√αp)

αg(10.182)

The solid phase molecular conductivity was determined by:

km,p =kg√αp

(φA + (1 − φΛ)) (10.183)

where

Λ =2

(1 −B/A)

(

(A− 1)(1 −B/A)2

B

Aln

(

A

B

)

− B − 11 −B/A

− 12(B + 1)

)

(10.184)

The coefficients incorporated in these formulas are:

A =kp

kg, B = 1.25

(

αp

αg

)10/9

and φ = 7.26 · 10−3 (10.185)

The simulations were run with the feed gas composition presented in table10.8. This composition represents a steam to carbon-ratio of 6. The feed gasenters the column with a temperature of 848K.

Fig 10.15 shows the composition field (mole fractions under dry conditions)in the reactor after 100 seconds. A rapid decrease in the methane and CO2

mole fractions, and a steep increase in the H2 mole fraction are observed. Theendothermic steam methane reforming process reactions are fast, so most ofthe conversion is taking place immediately after the gas reactants enter thereactor. The CO2-sorption process is slower and takes place in the entirebed, removing CO2 from the gas and thus shifting the chemical equilibriumlimit for the SERP toward a higher H2 yield compared to the conventionalsteam reforming process. The exothermic sorption reaction also reduces thetemperature drop compared to the conventional steam methane reformingprocess. The temperature of the gas entering the vessel is 848 K.


In Fig 10.16 a) instantaneous fields of the solids volume fraction and ve-locity vectors after 50 seconds simulation time are shown. The black areasindicate locations where the solid fraction is below 0.2. It is seen that thesolids in the bottom of the vessel have a tendency to move toward the centerof the tube and rise at a radial position halfway between the wall and thecenter. The particles are moving down again away from the upflowing areaboth closer the wall and the center of the vessel.

In Fig 10.16 b) instantaneous fields of the dry H2 mole fraction and thegas velocity vectors after 10 seconds simulation time are shown. That is, thereactions were turned on 5 seconds after the flow was initiated. The hydrogenproduction is fast in the inlet zone and the hydrogen produced are transportedtoward the exit.

It is also seen that the gas bubbles created in the bottom of the vessel havea tendency to move toward the center of the tube and rise at a radial positionhalfway between the wall and the center. These bubbles carry some of thesolids in their wakes producing the solids circulation pattern seen in (a).

There are no experimental data available for this process yet, so no firmvalidation has been performed. Nevertheless, the flow pattern is deemed tobe reasonable and the chemical conversion is in fair agreement with thoseobtained in fixed bed simulations [118].

Table 10.8. Feed gas composition

Component Weight fraction Mole fraction

CH4 0.129 0.143H2O 0.871 0.857


Fig. 10.15. Simulation of a chemical reactive mixture. Cross sectional average drymole fraction profiles of CH4, CO2, H2O, CO, H2 and temperate profile after 100seconds. The results are averaged over the 2 last seconds to smooth the profiles.


Fig. 10.16. Simulation of a chemical reactive mixture. (a) Instantaneous fields ofthe solids volume fraction and the particle velocity vectors after 50 seconds. (b)Contour plot of an instantaneous dry H2 mole fraction field during start up of theprocess, 5 seconds after the reactants enter the column and 10 seconds after thestart up of the flow. The consistent gas velocity vector field is given in the sameplot.

a) b)

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