Experimental and Numerical Study of Hydrodynamics with Heat

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Ind. Eng. Chem. Res. 2009, 48, 3177–3186

3177

Experimental and Numerical Study of Hydrodynamics with Heat Transfer in a Gas-Solid Fluidized-Bed Reactor at Different Particle Sizes Mahdi Hamzehei* and Hassan Rahimzadeh Department of Mechanical Engineering, Amirkabir UniVersity of Technology, Tehran 15875-4413, Iran

In this research, particle size effect on heat transfer and hydrodynamics of a nonreactive gas-solid fluidizedbed reactor were studied experimentally and computationally. A multifluid Eulerian model incorporating the kinetic theory for solid particles was applied to simulate the unsteady-state behavior of this reactor and momentum exchange coefficients were calculated by using the Syamlal-O’Brien drag functions. Simulation results were compared with the experimental data to validate the computational fluid dynamics (CFD) model. Pressure drops and temperature distribution predicted by the simulations at different particle sizes were in good agreement with experimental measurements at superficial gas velocity higher than the minimum fluidization velocity. Simulation results also indicated that small bubbles were produced at the bottom of the bed. These bubbles collided with each other as they moved upward forming larger bubbles. The influence of solid particles size on the gas temperature was studied. The results indicated that, for smaller particle size, due to a higher heat-transfer coefficient between the gas and solid phases, solid-phase temperature increases and mean gas temperature decrease, rapidly. Furthermore, this comparison showed that the model can predict hydrodynamic and heat-transfer behavior of gas-solid fluidized-bed reactors reasonably well. Introduction Fluidized-bed reactors are used in a wide range of applications in various industrial operations, including chemical, mechanical, petroleum, mineral, and pharmaceutical industries. Understanding the hydrodynamics of fluidized-bed reactors is essential for choosing the correct operating parameters for the appropriate fluidization regime. Computational fluid dynamics (CFD) offers an approach to understanding the complex phenomena that occur between the gas phase and the particles. The Eulerian-Lagrangian and Eulerian-Eulerian models have been applied to the CFD modeling of multiphase systems. For gas-solid flows modeling, usually, Eulerian-Lagrangian are called discrete particle models and Eulerian-Eulerian models are called granular flow models. Granular flow models (GFMs) are continuum-based and are more suitable for simulating large and complex industrial fluidized-bed reactors containing billions of solid particles. However, these models require information about solid-phase rheology and particle-particle interaction laws. In principle, discrete-phase models (DPMs) can supply such information. In turn, DPMs need closure laws to model fluid-particle interactions and particle-particle interaction parameters based on contact theory and material properties. In principle, it is possible to work our way upward from direct solution of Navier-Stokes equations. Lattice-Boltzmann models and contact theory to obtain all the necessary closure laws and other parameters required for granular flow models. However, with the present state of knowledge, complete a priori simulations are not possible. It is necessary to use these different models judiciously. Combined with key experiments, to obtain the desired engineering information about gas-solid flows in industrial equipment. Direct solution of Navier-Stokes equations or lattice-Boltzmann methods are too computation-intensive to simulate even thousands of solid particles. Rather than millions of particles, DPMs are usually used to gain an insight into various vexing issues such as bubble or cluster formations and their characteristics * To whom correspondence should be addressed. Fax: +98 21 66 41 97 36. E-mail address: [email protected].

or segregation phenomena. A few hundred thousand particles can be considered in such DPMs.1-3 Multiphase flow processes are key element of several important technologies. The presence of more than one phase raises several additional questions for the reactor engineer. Multiphase flow processes exhibit different flow regimes, depending on the operating conditions and the geometry of the process equipment. Multiphase flows can be divided into variety of different flows. One of these flows in gas-solid flows. Depending on the properties of the gas and solid phases, several different subregimes of dispersed two-phase flows may exist. With increasing gas flow rate, decreases in the bed density and the gas-solid contacting pattern may change from dense bed to turbulent bed, then to fast-fluidized mode and ultimately to pneumatic conveying mode. In all these flow regimes, the relative importance of gas-particle, particle-particle, and wall interaction is different. Therefore, it is necessary to identify these regimes to select an appropriate mathematical model. Apart from density and particle size as used in Geldart’s classification, several other solid properties, including angularity, surface roughness, and composition, may also significantly affect the quality of fluidization.4,5 At low gas velocity, solids rest on the gas distributor and the regime is a fixed-bed regime. The relationship between some flow regimes, the type of solid particles, and gas velocity is shown schematically in Figure 1. When superficial gas velocity increases, a point is reached beyond which the bed is fluidized. At this point, all the particles are just suspended by upward flowing gas. If gas velocity increases beyond a minimum fluidization velocity, homogeneous fluidization may exist for the case of fine solids, up to a certain velocity limit. At high gas velocities, the movement of solids becomes more vigorous. Such a bed is called a bubbling bed or a heterogeneous fluidized bed, in which gas bubbles generated at the distributor coalesce and grow as they rise through the bed. For deep beds of small diameter, these bubbles eventually become large enough to spread across the diameter of the vessel (slugging bed regime). In large-diameter columns, if the gas velocity increases yet further, then, instead of slugs, turbulent

10.1021/ie801413q CCC: $40.75  2009 American Chemical Society Published on Web 02/02/2009

3178 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009

Figure 1. Progressive change in gas-solid contact (flow regimes) with change on gas velocity (from Kunni and Levenspiel2).

motion of solid clusters and gas voids of various size and shape are observed, Entrainment of solids becomes appreciable (turbulent fluidized-bed regime). With further increases in gas velocity, solids entrainment becomes very high (fast fluidization regime).3-7 The fundamental problem that is encountered in the modeling of these reactors is the motion of two phases where the interface is unknown and transient, and the interaction is understood only for a limited range of conditions. Also, a large number of independent variables such as particle density, size, and shape can influence hydrodynamic behavior.2,3,5 Taghipour et al.7 conducted experimental and computational studies of gas-solid fluidized-bed reactor hydrodynamics. The simulation results were compared to those obtained from the experiments. The model predicted bed expansion and gas-solid flow patterns reasonably well. Furthermore, the predicted instantaneous and time-average local voidage profiles showed similar trends with the experimental results. Kaneko et al.8 developed a DEM simulation code incorporated with the reaction and energy balances based on the soft sphere interaction at particle collision for a fluidized-bed olefin polymerization. Heat transfer from particles to the gas was estimated using the Ranz-Marshall equation. Their results were fairly realistic, with regard to particle and gas temperature behavior and also the bubbling behavior in a gas fluidized bed at elevated pressure. The formation of hot spots was observed on the distributor near the wall of the fluidizing column. Huilin et al.9 studied bubbling fluidized beds with binary mixtures by applying a multifluid Eulerian CFD model, according to the kinetic theory of granular flow. Their simulation results showed that hydrodynamics of gas bubbling fluidized bed are related with the distribution of particle sizes and the amount of dissipated energy in particle-particle interactions. Behjat et al.10 simulated a gas-solid fluidized bed, based on the Eulerian description of the phases and multiphase fluiddynamic model. They assumed that solid particles release a constant value of heat and that fine polymer particles have higher activity and generate more heat than coarse particles. Their results indicate that, considering two solid phases, particles with

Figure 2. (A) View of the experimental setup. (Legend: 1, digital camera; 2, digital video recorder; 3, Pyrex reactor; 4, pressure transducers; 5, thermocouples; 6, computer, A/D and DVR cards; 7, electrical heater; 8, rotameter; 9, blower; 10, filter; 11-14, cooling system; and 15, controller system.) (B) Geometry and meshing of the fluidized-bed reactor. (C) Positions of the pressure transducers and thermocouples.

smaller diameters have a lower volume fraction at the bottom of the bed and a higher volume fraction at the top of the bed. In addition, it was revealed that bed expansion was larger when a bimodal particle mixture was applied compared with the case of monodispersed particles. Also, gas temperature increases as it moves upward in the reactor, because of the heat of polymerization reaction, which leads to the higher temperatures at the top of the bed. Chiesa et al.11 have presented a computational study of the flow behavior of a laboratory-scale fluidized bed. They have also compared experiments results of a two-dimensional laboratory-scale bubbling fluidized bed with their computational results. Their results showed that, when Eulerian and Lagrangian approaches were applied, the numerical simulations led to a pattern that was rather similar to the experimental data. These simulation results were much closer to the experimental results

Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3179 12

than the results of the continuous model. Gobin et al. numerically simulated a fluidized bed using a two-phase flow method. In their research, time-dependent simulations have been performed for industrial and pilot reactor operating conditions. The numerical predictions are in good qualitative agreement with the observed behavior, in terms of bed height, pressure drop, and mean flow regimes. Wachem et al.13 experimentally verified Eulerian-Eulerian gas-solid model simulations of bubbling fluidized beds with existing correlations for bubble size or bubble velocity. A CFD model for a free bubbling fluidized bed was implemented in the commercial code CFX. This CFD model was based on a two-fluid model, including the kinetic theory of granular flow. CFD simulation of a fluidized-bed reactor has also been conducted by Fan et al.,14 who focused on the chemical kinetic aspects and took into account the intraparticle heat- and masstransfer rates, polydisperse particle distributions, and multiphase fluid dynamics. Gas-solid heat and mass transfer, polymerization chemistry, and population balance equations were developed and solved in a multifluid code (MFIX) to describe particle growth. Also, Zhong et al.15 studied the maximum spoutable bed heights of a spout-fluid bed packed with six types of Geldart group D particles, where they obtained the effects of particle size, spout nozzle size, and fluidizing gas flow rate on the maximum spoutable bed height. Their results shown that the maximum spoutable bed height of a spout-fluid bed decreases as the particle size and spout nozzle size each increase, which seem to exhibit the same trend as that of spouted beds. The increase in the fluidizing gas flow rate leads to a sharply decrease in the maximum spoutable bed height. Lettieri et al.16 used the Eulerian-Eulerian granular kinetic model available within the CFX-4 code to simulate the transition from bubbling to slugging fluidization for a typical Group B material at four fluidizing velocities. Results from simulations were analyzed in terms of voidage profiles and bubble size, which showed typical features of a slugging bed, and also good agreement between the simulated and predicted transition velocity. Mansoori et al.17 performed a simulation of gas-solid turbulent upward flow in a vertical pipe using k-ε turbulence modeling and a Eulerian-Lagrangian approach. Particle-particle and particle-wall collisions were simulated based on deterministic approach. The influence of particle collisions on the particle concentration, mean temperature, and fluctuating velocities was investigated. The profiles of particle concentration, mean velocity, and temperature were shown to be flatter by considering interparticle collisions. It was demonstrated that the effect of interparticle collisions had a dramatic influence on the particle fluctuation velocity. Despite many studies on the modeling and model evaluation of fluidized-bed hydrodynamics, only a few works have been published on the CFD modeling and model validation of combined reactor hydrodynamics and heat transfer. Also, only a limited number of works has been reported on the successful CFD modeling of fluidized-bed hydrodynamics with heat transfer. In this study, heat transfer and hydrodynamics of twodimensional nonreactive gas-solid fluidized bed reactor were studied experimentally and computationally. Attention was given to the influence of gas temperature and particles size effect on gas-solid heat transfer and hydrodynamics. A multifluid Eulerian model that incorporated the kinetic theory for solid particles was applied to simulate the gas-solid flow at different superficial gas velocities with different particle sizes. Momentum exchange coefficients were calculated using the Syamlal-O’Brien drag functions. Also, it is assumed that the inlet gas is hot and

Table 1. Values of Model Parameters That Used in the Simulations and Experiments description

value

comment

solids density, Fs gas density, Fg mean particle diameter, ds coefficient of restitution, ess maximum solids packing, Rmax angle of internal friction, φ

2500 kg/m 1.225 kg/m3 175, 275, 375 µm 0.9 0.61 30°

initial solid packing diameter of fluidized bed height of fluidized bed initial static bed height superficial gas velocity initial temperature of solid inlet gas temperature inlet boundary conditions outlet boundary conditions time steps maximum number of iterations convergence criteria

0.5 28 cm 100 cm 44 cm 38 cm/s 300 K 473 K velocity outward flow 0.001 s 20 10-3

glass beads air density uniform distribution fixed value from Syamlal et al.18,19 from Johnson and Jackson27 fixed value fixed value fixed value fixed value fixed value fixed value fixed value superficial gas velocity fully developed flow specified specified specified

the initial solid particle is at ambient temperature. Simulation results were compared with the experimental data for model validation. Experimental Setup Experiments were conducted in a Pyrex cylinder with a height of 100 cm and diameter of 28 cm (see Figure 2). The distributor consisted of a perforated plate with an open-area ratio of 0.8%. Spherical particles with diameters of 175, 275, and 375 µm and a density of 2500 kg/m3 were fluidized with air at a temperature of 473 K and pressure of 1 kPa. The static bed height was 44 cm, with a solids volume fraction of 0.5. Figure 2A shows a view of experimental setup and its equipment. The distributor consisted of a perforated plate with an open-area ratio of 0.8%. A blower supplied the fluidizing gas. A pressure-reducing valve was installed to avoid pressure oscillations and achieve a steady gas flow. The gas flow rates were measured by an air rotameter. The initial temperature of the solid particles was 300 K. An electrical heater increases the inlet gas temperature from ambient temperature to 473 K. A cooling system was used to decrease the temperature of the gas that exits from the reactor, to apply a closed cycle. Pressure fluctuations in the bed were obtained using three pressure transducers. Also, to control and monitor the fluidized-bed operation process, A/D, DVR cards, and other electronic controller systems were applied. The ratio of the distributor pressure drop to the bed pressure drop exceeded 11% for all operating conditions investigated. Seven thermocouples (Type K) were placed at different positions to measure the variation in gas temperature. Figure 2B shows the positions of the pressure transducers and thermocouples. The overall pressure drop and bed expansion were monitored at different gas velocities. A digital camera (Canon 5000) and a digital video recorder (SAMSUNG, SDC-415) were used to photograph the flow regimes and bubble formation through the transparent wall during the experiments. Numerical Procedures Governing Equations. The governing equations of the system include the conservation of mass, momentum, and energy. Equations of solid and gas phases were developed based on Eulerian-Eulerian model, using the averaging approach. By definition, the volume fractions of the phases must sum to 1:

3180 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009

Rg + Rs ) 1

(1)

(The terms Rg and Rs represent the volume fractions of gas and solid, respectively.) The respective continuity equations for the gas and solid phases without mass transfer between the phases are ∂ f (R F ) + ∇ · (RgFg ν g) ) 0 ∂t g g

(2)

∂ f (R F ) + ∇ · (RsFs ν s) ) 0 ∂t s s

(3)

and

The conservation of momentum equations for the gas phase and the solid phase are described by ∂ f f f f (R F ν ) + ∇ · (RgFg ν g ν g) ) -Rg ∇ Pg + ∇ · σ cg + RgFg g + ∂t g g g f f βgs(ν s - ν g) (4) and ∂ f ff f (R F ν ) + ∇ · (RsFs ν s ν s) ) -Rs ∇ Pg + ∇ · σ cs + RsFs g + ∂t s s s f f βgs(ν g - ν s) (5) where σ c is the Reynolds stress tensor, g the gravitational constant, and the expression [-Rs∇p + βgs(ν bg - b νs)] is an interaction force (drag and buoyancy forces) that represents the momentum transfer between the gas phase and the solid phase.5,7 Several drag models exist for the gas-solid interphase exchange coefficient βgs. In this research, the Syamlal-O’Brien drag model18,19 was used. The Syamlal-O’Brien drag law is based on the measurements of the terminal velocities of particles in fluidized or settling beds. The interphase exchange coefficient is expressed as

( )( )

Res f f 3 RsRgFg CD |ν - ν g| βgs ) 4 ν2 d νr,s s r,s s

(6)

where the drag coefficient (CD) is given by

(

CD ) 0.63 +

4.8

√Res ⁄ νr,s

)

2

(7)

and υr,s, which is a terminal velocity correlation, is expressed as νr,s ) 0.5{A - 0.06Res + [(0.06Res)2 + 0.12Res(2B - A) + with

and B)

{

(9)

0.8Rg1.28 if Rg e 0.85 if Rg > 0.85 Rg2.65

where Pg is the pressure. The viscous stress tensor (τcg) is assumed to have the Newtonian form c + R λ tr(D c )Ic cτg ) 2RgµgD (13) g g g g c c where I is the identity tensor and Dg is the strain rate tensor for the gas phase, which is given by f f T c ) 1 [∇ν (14) D g + (∇ν g) ] g 2 Granular flows can be classified into two distinct flow regimes: a viscous or rapidly shearing regime, in which stresses arise because of collisional or translational transfer of momentum, and a plastic or slowly shearing regime, in which stresses arise because of coulombic friction between grains in enduring contact. In this work, a combination of viscous and plastic regimes was applied to investigate the behavior of granular flow.

cs ) σ

A2]1⁄2} (8)

A ) Rg4.14

Figure 3. Simplified flowchart of the simulation procedure.

{

* -PspIc + cτsp (if Rg e Rg) * -PsvIc + τsv (if Rg > Rg)

(15) where Ps is the pressure and cτs is the viscous stress in the solid phase. The superscript p denotes plastic regime and the subscript v denotes viscous regime. In fluidized-bed simulations, εg* is usually set to the void fraction at minimum fluidization.15 The granular pressure is given by Psv ) K1Rs2Θs

(10)

The granular stress is given by c + λv tr(D c )Ic cτsv ) 2µsvD s s s

The solids Reynolds number (Res) is calculated as Fgds|ν g - ν s| µg f

Res )

f

(11)

λsv ) K2Rs√Θs (12)

(17)

where the second coefficient of viscosity for the solid phase (λvs ) is given by

The stress tensor for the gas phase is given by σ cg ) -PgIc + cτg

(16)

The factor phase, is

µvs ,

(18)

which represents the shear viscosity for the solid

Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3181

√Θs

µsv ) K3Rs c s) is The strain rate tensor (D

(19)

f f T c ) 1 [∇ν D (20) s + (∇ν s) ] s 2 Similar to the functions typically used in plastic flow theories, an arbitrary function that allows a certain amount of compressibility in the solid phase represents the solid-pressure term for the plastic flow regime:

Psp ) RsP*

c s is the solids strain-rate tensor, and the following Here, D abbreviated terms are used: K1 ) 2(1 + ess)Fsg0,ss

K2 ) K3 )

(21)

{

( )

4 2 dsFs(1 + ess)Rsg0,ss - K3 3 3√π

P* ) 10

(R*g - Rg)10

5√π

c cτsp ) 2µspD s

(23)

where µsp )

P* sin φ 2√I2D

1 [(D - Ds22)2 + (Ds22 - Ds33)2 + (Ds33 - Ds11)2] + 6 s11 Ds122 + Ds232 + Ds312 (25)

The granular temperature (Θ) of solid phase, as an order of solid fluctuation, is defined as one-third of the mean square velocity of the random motion of the particles. The granular temperature is different from the solid phase temperature and is proportional to the granular energy. Therefore, granular energy is defined as the specific energy of the solid particles fluctuation. The following solid-phase granular energy equation must be solved to calculate the granular temperature Θ.21 3∂ 3 f f (F R Θ ) + ∇ · (FsRs ν sΘs) ) σ cs : ∇ · ν s + 2 ∂t s s s 2 ∇ · (kΘs ∇ Θs) - γΘs + φgs (26) υs is the generation of energy by the solid stress where σ cs : ∇ · b tensor, kΘs∇Θs is the diffusion of energy (where kΘs is the diffusion coefficient). The collisional dissipation of energy (γΘs) represents the rate of energy dissipation within the solid phase due to inelastic particle collisions, which was derived by Lun et al.22,23 γΘs )

[

12(1 - ess )g0,ss 2

ds√π

]

FsRs2Θs3 ⁄ 2

(27)

The transfer of kinetic energy due to random fluctuations in particle velocity (φgs) is expressed as φgs ) -3βgsΘs

(28)

In this work, the following algebraic granular temperature equation was used, with the assumptions that (i) the granular energy is dissipated locally, (ii) the convection and diffusion contributions are negligible, and (iii) only the generation and dissipation terms are retained:10,13 c )+ c ) + {K2 tr2(D c )R2 + 4K R [K tr2(D Θs ) [(-K1Rs tr(D s s 1 s s 4 s 2 c2)]}1 ⁄ 2) ⁄ (2R K )]2 (29) 2Ks tr(D s s 4

(31)

}

2 12(1 - ess )Fsg0,ss

(32)

(33)

ds√π

When using this algebraic equation instead of solving the balance for the granular temperature, much faster convergence is obtained during simulations.12,13 The term g0,ss is the radial distribution function that, for one solid phase,7,22 has the form

[ ( )]

(24)

Here, φ is the angle of internal friction, and I2D is the second invariant of the deviatoric stress tensor.14,20 I2D )

K4 )

(22)

A solids stress tensor based on the critical state theory is given as

()

dsFs √π [1 + 0.4(1 + ess)(3ess - 1)Rsg0,ss] + 2 3(3 - ess) 8Rsg0,ss(1 + ess)

where P* is represented by an empirical power law: 25

(30)

Rs

g0,ss ) 1 -

1 ⁄ 3 -1

(34)

Rs,max

where ess is the coefficient of restitution for particle collisions. This coefficient is a measure of the elasticity of the collision between two particles and is related to how much of the kinetic energy of the colliding particles that is present before the collision remains after the collision. The coefficient of restitution is defined as the ratio of the difference in the velocities of the two colliding particles after the collision, relative to the difference in their velocities before the collision.22 For a perfectly elastic collision, the coefficient of restitution is given as ess ) 1, whereas, for a perfectly plastic (or inelastic) collision, the coefficient of restitution is given as ess ) 0. The term kΘs is the diffusion coefficient for granular energy. The Syamlal-O’Brien7,18,19 model is expressed as kΘs )

15dsFsRs√πΘs 12 2 1+ η (4η - 3)Rsg0,ss + 4(41 - 33η) 5 16 (41 - 33η)ηRsg0,ss (35) 15π

[ ( )

]

with η ) 1/2(1 + ess). The internal energy balance for the gas phase can be written in terms of the gas temperature as follows: RgFgCpg

(

)

∂Tg + Vg · ∇ Tg ) - ∇ · (Rgkg ∇ Tg) - Hgs (36) ∂t

where kg is the thermal conductivity of the gas. The solid heat conductivity includes direct conduction through the fractional contact area and indirect conduction through a wedge of the gas that is trapped between the particles. The thermal energy balance for the solid phases is given by

(

RsFsCps

)

∂Ts + Vs · ∇ Ts ) ∇ · (Rsks ∇ Ts) + Hgs ∂t

(37)

The heat transfer between the gas and the solid is a function of temperature difference between the gas and solid phases.10 0 Hgs ) -γgs (Ts - Tg)

(38)

The heat-transfer coefficient is related to the particle Nusselt number (Nus), using the following equation:

3182 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 0 γgs )

6kgRsNus 2

ds

(39)

Here, kg is, again, the thermal conductivity of the gas phase. The particle Nusselt number (Nus) is determined by applying the following correlation for a porosity range of 0.35-1.0: Nus ) (7 - 10Rg + 5Rg2)(1 + 0.7Res0.2Pr1 ⁄ 3) + (1.33 - 2.4Rg + 1.2Rg2)Res0.7Pr1 ⁄ 3 (40) where Res is the relative solid Reynolds number and Pr is the gas Prandtl number.12,13 Cpg µg Pr ) kg

velocities on the wall were set to zero (no slip condition). The normal velocity of the particles was also set at zero. The following boundary equations apply for the tangential velocity and granular temperature of particles at the wall:24-28 ν s,w ) -

f

Initial and Boundary Conditions. The initial values of the variables for all the fields (Rg,Rs,Vg,Vs) are specified for the entire computational domain. Initially, solid particle velocity was set at zero (in minimum fluidization), and gas velocity was assumed to have the same value everywhere in the bed. The temperature of solid phase was 300 K, and, for the gas phase, the temperature was set to 473 K. At the inlet, all velocities and volume fraction of all phases were specified. The outlet boundary condition was outward flow and was assumed to be fully developed flow. Table 1 shows the values of the model parameters that were used in the simulations. The other variables were subject to the Newmann boundary condition. The gas tangential normal

)

f

∂ ν s,w √3πFsRsg0,ss√Θs ∂n

(42)

The general boundary condition for granular temperature at the wall takes the form Θs,w ) -

(41)

(

6µsRs,max

( )

ksΘs ∂Θs,w √3πFsRsνs2g0Θs3⁄2 + ess,w ∂n 6Rs,maxess,w

(43)

Here, b Vs,w is the particle slip velocity, ess,w the coefficient of restitution at the wall, and Rs,max the volume fraction of particles at maximum packing.10,15,28 The boundary conditions for the energy equation are set such that the walls are adiabatic. Computational Fluid Dynamics (CFD) Simulation and Computation Solution. The governing equations were solved using the finite volume approach. The two-dimensional (2D) computational domain was discretized by 18 600 rectangular cells. A time step of 0.001 s with 20 iterations per time step and a convergence criterion of 10-3 was chosen. This iteration was adequate to achieve convergence for the majority of time steps. The calculation domain is divided into a finite number of control volumes. Volume fraction, density, and turbulent kinetic energy are stored at the main grid points that are placed in the center of each control volume. A staggered grid arrangement is used, and the velocity components are solved at the control volume surfaces. To accelerate the simulation, the semi-implicit method for pressure-linked equations (SIMPLE) scheme and automatic time-step adjustment are used. For all simulations reported here, a second-order spatial discretization method was used to improve the accuracy of the simulations.3,4,27,28 The simplified flowchart of the simulation algorithm is shown in Figure 3. Results and Discussion

Figure 4. Comparison of experimental and simulation bed pressure drop (P1 - P3) data at different solid particle sizes.

Figure 5. Comparison of experimental and simulated bed pressure drop at different superficial gas velocities (ds ) 0.275 mm).

Simulation results were compared with the experimental data, to validate the model. The pressure drop was measured experimentally for three sizes of solid particle and compared with those predicted by CFD simulation. As indicated in Figure 4, the overall pressure drop in the bed decreased significantly at the beginning of fluidization and then fluctuated around an almost-steady-state value after ∼3.5 s. Pressure drop fluctuations are expected as bubbles continuously split and coalesce in a transient manner in the fluidized bed. The results show that the pressure drop increases as the particle size increases. Comparison of the model predictions, using the Syamlal-O’Brien drag functions, and experimental measurements on the pressure drop show good agreement for most operating conditions. These results (for ds ) 0.275 mm) are the same as those reported by Tagipour et al.7 and Behjat et al.10 A comparison of the time-average bed pressure drop, using the Syamlal-O’Brien drag functions, against superficial gas velocity is plotted in Figure 5. The simulation and experimental results show better agreement at velocities above Umf The discrepancy for U < Umf may be attributed to the solids not being fluidized, thus being dominated by interparticle frictional forces, which are not predicted by the multifluid model for simulating gas-solid phases. The results show that, with increasing gas velocity, the pressure drop (P1 - P2 and P1 P3) increases but the rate of increase for (P1 - P3) is larger

Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3183

Figure 6. Simulation void fraction profile of a two-dimensional (2D) bed (Ug ) 38 cm/s, ds ) 0.175 mm).

Figure 7. Comparison of the experimental and simulated void fraction and bubbles for three particle sizes and three different times.

than for (P1 - P2). Comparison of the model predictions and experimental measurements on pressure drop (for both cases) show good agreement at different gas velocities. Figure 6 shows simulation results for the void fraction contour plot, based on gas volume fraction, for Ug ) 38 cm/s and ds ) 0.175 mm. The increase in bed expansion and variation of the fluid-bed voidage can be observed. At the start of the simulation, waves of voidage are created, which travel through the bed and subsequently break to form bubbles as the simulation progresses. Initially, the bed height increased with bubble formation until it leveled off at a steady-state bed height. The observed axisymmetry gave way to chaotic transient generation of bubble formation after 1.5 s. The bubbles coalesce as they move upward producing bigger bubbles. The bubbles become stretched as a result of bed wall effects and interactions with other bubbles. The contour plots of the solids fraction shown in Figure 7 indicate similarities between the experimental and simulations for three particle sizes and at three different times. The results show that the bubbles at the bottom of the bed are relatively small. The experiments indicated the presence of small bubbles near the bottom of the bed; the bubbles grow as they rise to the top surface with coalescence. The elongation of the bubbles is due to wall effects and interactions with other bubbles. The Syamlal-O’Brien drag model provided similar qualitative flow patterns. Generally, the size of the bubbles predicted by the CFD models are similar to those observed experimentally. Any discrepancy could be due to the effect of the gas distributor, which was not considered in

the CFD modeling of the fluid bed. In practice, jet penetration and hydrodynamics near the distributor are significantly affected by the distributor design. The increase in bed expansion and the greater variation of the fluid-bed voidage can be observed in Figures 6 and 7 for particles with ds ) 0.175 mm. According to experimental evidence, this type of solid particle should exhibit a bubbling behavior as soon as the gas velocity exceeds minimum fluidization conditions. It is also worth noting that the computed bubbles show regions of voidage distribution at the bubble edge, as experimentally observed. In Figure 7, symmetry of the void fraction is observed at three different particle sizes. The slight asymmetry in the void fraction profile may result from the development of a certain flow pattern in the bed. Similar asymmetry has been observed in other CFD modeling of fluidized beds.5,7 The void fraction profile for large particles is flatter near the center of the bed. Simulation results for void fraction profile are show in Figure 8. In this figure, the symmetry of the void fraction is observed at three different particle sizes. The slight asymmetry in the void fraction profile may result form the development of a certain flow pattern in the bed. Similar asymmetry has been observed in other CFD modeling of fluidized beds.5,7 Void fraction profile for large particle is flatter near the center of the bed. The simulation results of the time-averaged cross-sectional void fraction at different solid particles diameter is shown in Figure 9 for Ug ) 38 cm/s. This figure shows that, with increasing solid particles

3184 Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009

Figure 8. Simulated void fraction at different solid particles diameter (at z ) 20 cm, Ug ) 38 cm/s, t ) 5.0 s).

Figure 9. Simulation results of void fraction at different solid particles diameter (Ug ) 38 cm/s, t ) 5.0 s).

Figure 11. Simulation results of the effect of particle size on solid particle temperature (t ) 5.0 s).

a lower gas temperature. Because the static initial bed height is constant (44 cm), for small particles, the particle content in the bed is greater than that for large particles, so the surface contact between the hot gas and the cold particles increase. With increasing heat transfer between the gas phase and the solid phase, the solid phase temperature increases (see Figure 11) and mean gas temperature decreases (see Figure 10). The effects of solid particle diameter on solid phase temperature are shown in Figure 11. This figure indicates that a decrease in particle size causes a higher heat-transfer coefficient between the gas phase and the solid phase (resulting in a higher contact surface between the gas phase and the solid phase), which, in turn, leads to an increase in solid particle temperature. In addition, the temperature gradient between the solid phase and the gas phase is greater at the top of the bed, which leads to a larger heat-transfer rate, compared to that at the bottom of the bed. The effect of numerical implementation, including the time step, discretization schemes, mesh size, and convergence criterion on the results, was studied using sensitivity analysis. The simulated results of the solids volume fraction profile from the second-order discretization schemes at a time step of 0.001 s with a convergence criterion of 10-3 (the general numerical procedure in this study) were compared to those at the firstand second-order discretization schemes, with a time step of 0.0005 s and a convergence criterion of 10-4, with 100 iterations at each time step (higher-quality numerical procedures require more computational time). The results show no significant difference in overall hydrodynamic behavior and bubble shapes among these simulations, indicating that the implemented numerical simulation was adequate for a measure of the degree to which these simulations capture the correct hydrodynamic characteristics of the bed. Conclusion

Figure 10. Comparison of simulation and experimental results of the effect of particle size on gas temperature (t ) 5.0 s).

diameter, the void fraction and bed height increase and steady-state conditions arrive rapidly. The influence of the size of the solid particles on the gas temperature is shown in Figure 10. This figure indicates that a decrease in particle size that is due to a higher heat-transfer coefficient between the gas phase and the solid phase leads to

In this research, unsteady-state heat transfer and hydrodynamics in a gas-solid fluidized-bed reactor has been investigated. Preliminary investigation of multiphase flow models revealed that the Eulerian-Eulerian model is suitable for the modeling of industrial fluidized-bed reactors. The model includes continuity equations, momentum and energy equations for both phases, and the equations for the granular temperature of solid particles. A suitable numerical method that uses a finite volume method has been applied to discritize the equations. Simulation results also indicated that small

Ind. Eng. Chem. Res., Vol. 48, No. 6, 2009 3185

bubbles were produced at the bottom of the bed. These bubbles collided with each other as they moved upward, forming larger bubbles. The influence of the size of the solid particles on the gas temperature was studied. The results indicated that, for smaller particle sizes, because of a higher heat-transfer coefficient between the gas phase and the solid phase, the solid phase temperature rapidly increases and the mean gas temperature rapidly decreases. To validate the model, predicted pressure drops and gas temperature variation were compared to corresponding values of experimental data. The modeling predictions compared reasonably well with experimental data. Furthermore, this comparison showed that the model can predict the hydrodynamic and heat-transfer behavior of gas-solid fluidized beds reasonably well.

Nomenclature CD ) drag coefficient Ds ) solids strain rate tensor (s-1) Cp ) specific heat (J/(kg K)) ds ) solid diameter (mm) ess ) coefficient of restitution of particle ess,w ) restitution coefficient at the wall Hgs ) heat transfer between the gas and the solid (J/(m3 s)) g ) gravitational constant; g ) 9.81 m/s2 g0,ss ) radial distribution function H0 ) static bed height (cm) I ) identity tensor I2D ) second invariant of the deviatoric stress tensor (s-2) k ) thermal conductivity (J/(m K s)) K1 ) granular stress constant (kg/m3) K2 ) granular stress constant (kg/m2) K3 ) granular stress constant (kg/m2) K4 ) granular stress constant (kg/m4) kΘs ) diffusion coefficient for granular energy (kg/(s m)) Nus ) particle Nusselt number P ) pressure (Pa) Pr ) gas Prandtl number Re ) Reynolds number s ) stress tensor (Pa) T ) thermodynamic temperature (K) t ) time (s) Ug ) superficial gas velocity (m/s) V ) velocity (m/s) Vr,s ) terminal velocity correlation (m/s) Greek Symbols 0 γgs ) gas-solid heat-transfer coefficient (J/(m3 K s)) γΘs ) collision dissipation of energy (J/(m3 K s)) βgs ) gas-solid interphase exchange coefficient R ) volume fraction Rs,max ) volume fraction for the particles at maximum packing Rg* ) packed-bed void fraction Θs ) granular temperature (m2/s2) λ ) second coefficient of viscosity (kg/(m s)) µ ) viscosity (kg/(m s)) cτ ) Reynolds stress tensor (Pa) σ c ) stress tensor (Pa) F ) density (kg/m3) φ ) angle of internal friction φgs ) transfer of kinetic energy (kg/(s3 m))

Subscripts g ) gas mf ) minimum fluidization s ) solids Superscripts p ) plastic regime v ) viscous regime

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ReceiVed for reView September 19, 2008 ReVised manuscript receiVed December 10, 2008 Accepted December 29, 2008 IE801413Q