Numerical Computation and Experimental Verification of the Jet

characteristics is based on the volume-averaged mass conversion and Navier-Stokes equations. The model was tested in the two-dimensional bed with a ...
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Ind. Eng. Chem. Res. 2002, 41, 3696-3704

GENERAL RESEARCH Numerical Computation and Experimental Verification of the Jet Region in a Fluidized Bed Kai Zhang,*,†,‡ Hu Zhang,‡ Jonathon Lovick,‡ Jiyu Zhang,† and Bijiang Zhang† Institute of Coal Chemistry, Chinese Academy of Sciences, P.O. Box 165, Taiyuan 030001, China, and Department of Chemical Engineering, University College London, London WC1E 7JE, United Kingdom

The fluid dynamics in the jet region has been investigated numerically and experimentally by using a gas-solid jetting fluidized-bed cold model. A hydrodynamic model of gas-solid flow characteristics is based on the volume-averaged mass conversion and Navier-Stokes equations. The model was tested in the two-dimensional bed with a central jet and a cone-shaped distributor. The experimental data were obtained from the frame-by-frame analysis of films. The jet penetration height increases with an increase of the jet gas velocity and nozzle diameter. The jet frequency decreases with an increase in the jet gas velocity. Furthermore, a simple model to estimate the initial bubble diameter is presented using the assumption that the gas leakage velocity is equal to the minimum fluidization velocity. It has been verified that the hydrodynamic model shows a good agreement with experimental data, which makes this model a useful tool to study the jet characteristics in gas-solid fluidized beds. 1. Introduction A gas jet blown upward into a fluidized bed through the nozzle may form various patterns. These patterns are a chain of bubbles, a jet plume of a series of cavities, or a permanent cavity, which depend mainly on the solid property, bed configuration, and operating conditions.1 Grace and Lim2 suggested a simple criterion, i.e., do/dp e 25.4, as a necessary, but not sufficient, condition for a permanent jet formation. Whether it is a chain of bubbles, a pulsating jet, or a permanent jet, what is more important is that the gas with high velocity will result in a jet region between the nozzle and the jet top, where there exists extensive mixing and gas-solid contact. This region is also named the grid zone.3 The characteristics in the jet region are different from those in the bubbling region with respect to the high concentration of reactant gases and a complicated mechanism of gas-solid contact. For example, exothermic hot spots and particle agglomerations were found in the jet region of fluidized-bed coal gasifiers, such as the U-gas gasifier, the WRK gasifier, and the ICC gasifier.4 Tsukada and Horio3 divided the jet region into two areas: the bubbleforming area, where a void was formed and went up periodically, and the jet stem region, where a dilute spout existed stably. Computational fluid dynamics (CFD) from first principles has been regarded as a feasible tool in the area of fluidization engineering. This model can offer an * Corresponding author. Present address: Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, U.K. Tel: 0044-20-76793839. Fax: 0044-20-73832348. E-mail: [email protected]. † Chinese Academy of Sciences. ‡ University College London.

attractive alternative approach to obtain useful information about hydrodynamic characteristics in gas-solid fluidized beds, especially in those cases where the use of experimental methods is severely restricted by technical constraints. A representative example is that Barthod et al.5 improved the performance of a fluidized bed in the petroleum industry by means of CFD modeling. Gidaspow6 reviewed earlier studies on hydrodynamic computer models. Recently, many researchers7-11 gave their overviews of CFD applications related to gas-solid fluidized beds. The previous simulations were focused on the hydrodynamics in the overall bed. Gidaspow and Ettehadieh12 successfully carried out experiments and simulation on the porosity distribution in the fluidized bed. Bouillard et al.13 extended their work to the bed with an immersed obstacle symbolizing a heat exchanger. Kuipers et al.14-17 and Nieuwland et al.18 investigated the porosity distribution, local wallto-bed transfer coefficients, and detailed bubble behavior in either the two-dimensional bed or the semicircular bed. It is obvious that the majority of literature has been concerned with time-averaged quantities, but a few investigations of the transient jet behavior have been reported. We believe that an experimentally verified hydrodynamic model describing gas and solid dynamics in the jet region is helpful for scale-up and optimization of the fluidized bed. The objective of this study attempts to provide an experimental validation of the hydrodynamic model in a jet region. The model, derived from the first principles of momentum and mass conservation, can be solved by a modified SIMPLE (Semi-Implicit Method for PressureLinked Equation) algorithm. Based on this model, a CFD code CASICC (Computational Analysis for Significant Information of Coal Conversion) has been devel-

10.1021/ie010962u CCC: $22.00 © 2002 American Chemical Society Published on Web 06/19/2002

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oped. The gas and solid dynamics have been analyzed, including gas- and solid-phase velocities, pressure distribution, and voidage profile in the jet region. To validate this model, the jet development, jet penetration height, jet frequency, and initial bubble diameter were experimentally investigated in a two-dimensional gas fluidized bed with a cone-shaped distributor, which represented roughly a cross section of the ash-agglomerating coal gasifiers, such as KRW, U-gas, and ICC fluidized-bed gasifiers.4 The result indicates that the simulated data agree satisfactorily with the experimental data.

[

hh ) 150 -∇‚P

(φsdp)

g

β ) 150

∂(sFs) + ∇‚(sFsb us) ) 0 ∂t

(2)

MOMENTUM EQUATIONS

Solid Phase u s) ∂(sFsb + ∇‚(sFsb usb us) ) -s∇P - β(u bs - b ug) + ∂t g - ∇Ps (4) ∇‚sjjτs + sFsb where  represents the volume fraction (g + s ) 1), Ps the solid-phase pressure, β the gas-solid friction coefficient, F the density, and jτ the viscous stress tensor. The subscripts g and s indicate gas and solid phases, respectively. To solve the equations stated above, the basic variables including volume fraction , pressure P, gas-phase velocity vector b ug, and solid-phase velocity vector b us should be specified. According to the closure principles of governing equations, it is necessary that all other variables in the equations need to be derived from these basic variables. The major parameters of the other variables, i.e., gas-solid friction coefficient, β, solidphase pressure, Ps, and viscous stress tensor, jτ, are given in the following. Gas-Solid Friction Coefficient. When the voidage, g, is less than 0.8, the gas-solid friction coefficient, β, can be obtained from the well-known Ergun equation,19 which describes the relationship between pressure drop and superficial fluid velocity in the packed bed of fluid through porous media, given as follows:

b uo b uo (5)

(6)

s p

When the voidage, g, is greater than 0.80, the gassolid fraction coefficient, β, can be estimated by an empirical correlation proposed by Wen and Yu:20

g(1 - g) b -b us|f(g) F |u (φsdp) g g

(8)

In eq 8, f(g) ) g-2.65, which accounts for the presence of other particles in the fluid and indicates the correction for the drag coefficient of a single particle. The drag coefficient of a single particle, Cd, is related to the particle Reynolds number, Rep:

Cd ) 24[1 + 0.15Rep0.687]/Rep Cd ) 0.44

Gas Phase u g) ∂(gFgb + ∇‚(gFgb ugb ug) ) -g∇P - β(u bg - b u s) + ∂t g (3) ∇‚gjjτg + gFgb

g3(φsdp)

(1 - g)2 µg (1 - g)Fg + 1.75 |u bg - b us| (7) 2 g φsdp (φ d )

β ) 0.75Cd

Solid Phase

| |]

(1 - g)Fg

By comparison of eq 5 with eq 6, the gas-solid friction coefficient, β, becomes

Gas Phase (1)

+ 1.75

bg - b us) -g∇‚P ) β(u

CONTINUITY EQUATIONS ∂(gFg) u g) ) 0 + ∇‚(gFgb ∂t

µg 2

3

In the case of fluidized beds, b uo ) g(u bg - b us). If transient, convective, solid stress, and gravity terms are neglected, the gas momentum equation becomes

2. Theoretical Model Hydrodynamic models of fluidization adopt the principles of conservation of mass, momentum, and energy. The mass conversion and Navier-Stokes equations, describing gas and solid flow in the cold model of fluidized beds, are given in the vector form as

(1 - g)2

Rep < 1000 (9)

Rep g 1000

(10)

gFg|u bg - b us|(φsdp) µg

(11)

where

Rep )

Solid-Phase Pressure. The solid-phase pressure, ps, is related to the interaction among particles and plays an important role from both a physical and a numerical viewpoint.6 It prevents the solid fraction from reaching impossibly high values and helps to make the system numerically stable by converting imaginary characteristics into real ones.12 Bouillard et al.13 and Kuipers et al.14-16 regarded that the solid-phase pressure, ps, can be replaced by a particle-particle interaction elastic modulus, G(g), through the following relationship:

G(g) ) dps/dg

(12)

When the chain rule is applied, the gradient of the solidphase pressure ∇ps is written as follows:

∇ps ) G(g)∇g

(13)

Bouillard et al.13 and Kuipers et al.14-16 presented a generalized formula of G(g) in gas-solid fluidized beds as

G(g)/G0 ) exp[-c(g - 0)]

(14)

where c (called the compaction modulus) is the slope of ln(G) vs g, 0 the compaction voidage, and G0 the normalizing units factor. This method is based on the

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theory of powder compaction. The details can be found in Orr’s21 work. Viscous Stress Tensor. By using a suspended torsion pendulum, Hagyard and Sacerdote22 investigated the rheology of fluidized powders and found it could be roughly described by Newtonian behavior, that is

jτi ) - 2µi∇‚u bi + µ′i∇‚u bi E + µi[(∇u bi) + (∇u bi)T] (15) 3

(

)

where µi is the effective viscosity of each phase, µ′i the bulk viscosity of each phase, and E the unit tensor. The superscript T denotes transpose. Generally, µi >> µ′i; therefore, as a first approximation, eq 15, related only to the motion, can be written as follows:

jτi ) - 2µi∇‚u bi E + µi[(∇u bi) + (∇u bi)T] 3

(

)

(16)

3. Numerical Method The set of nonlinear coupling partial differential conservation equations cannot be resolved by an analytical method, even supplied with additional constitutive equations and the initial and boundary conditions. However, an approximate solution can be obtained from a numerical approach. It is very important to select an appropriate numerical scheme because it affects not only the solution accuracy but also the efficient use of the computer resources. Multidimensional analyses of single fluid flow and homogeneous two-phase flow appear to originate from one of two major generic sources: the ICE procedure developed at LASL and the SIMPLE procedure developed at Imperial College. Both procedures are, in fact, quite similar in concept. They use staggered grid and pressure equation concepts and assign the velocities at control volume boundaries. Their main differences lie in the exact form of the pressure equation, the manner in which the equations are formulated over the control volume, the degree of implicitness used in the time advancement, and the solution sequence. However, for multidimensional multifluid, it is in its infancy. The numerical analysis can be evolved from the same ancestors because it is quite a simple matter, at least in principle. The SIMPLE method23 is tried to extend the dense phase fluidization condition in this investigation. Figure 1 shows the staggered grid system used in this investigation. Solid and dashed lines present the main grid line and the control volume surface, respectively. The small circles are the main grid joints. The locations for x- and y-direction velocities are placed on the respective control volume faces, irrespective of whether the latter happen to be midway between the grid points. Figure 2 is the control volume for the x-direction velocity components, and Figure 3 is the control volume for the scalars. In Figure 2, the control volume for the xdirection velocity component is staggered in relation to the normal control volume. The pressure force acting on the control volume for the x-direction velocity components can be calculated by the difference PP -PE. This is one of the advantages of the staggered grid. The scalars are located on the jointing points of the main grid lines (see Figure 3). The features of this procedure are (1) a nonuniform staggered grid to avoid regarding a zigzag pressure field as a uniform profile by the momentum equations, (2) a control volume approach to

Figure 1. Staggered grid system.

Figure 2. Control volume for the x-direction velocity component.

Figure 3. Control volume for scalars.

discrete the partial differential equations, (3) the gas continuity equation as a pressure-correction formula, (4) the solid continuity equation to calculate each phase volume fraction, (5) source-term linearization, (6) difference schemes including upwind, PLS (Power-Law Scheme), or QUICK (Quadratic Upstream Interpolation for Convective Kinematics), (7) TDMA (TridiagonalMatrix Algorithm) and an under-relaxation approach to solve the algebraic equations, and (8) a changeable relaxation factor, R. It is selected as the bigger one between the difference of s,max - s and ω when s is close to s,max. Here ω is a very small value. This new method is quite effective in overcoming difficulty in the solution of each phase volume fraction and avoiding divergence in the iterative solution of strongly nonlinear multiphase flow equations. According to the model theory and numerical method mentioned above, a comprehensive CFD code, the CASICC software

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002 3699 Table 2. Range of Experiment Conditions experimental parameter

value

diameter of central jet (m) initial height of the bed (m) ratio of annular gas velocity to minimum fluidization velocity jet gas velocity (m/s)

0.008-0.014 0.1-0.4 1.0-3.0 5.773-22.09

A two-dimensional bed with a central nozzle and a cone-shaped gas distributor, constructed from transparent Plexiglass, was employed in this investigation as shown in Figure 4. It was 2.05 m high and had an inside rectangular cross section of 0.051 m × 0.300 m. A jet tube in the center was attached to the inner bed surface. The compressed air volumetric flows were controlled by gate valves and measured by rotameters independently. All experiments were carried out at ambient temperature and atmospheric pressure. Resin and sand were used as solid materials, whose physical properties are listed in Table 1. In a typical run, the solid was first loaded into the bed to a specified initial bed height. The gate valves were then opened to introduce air into the bed. The volumetric gas flow rate was calculated and distributed to each rotameter to maintain the annular gas velocity between umf and 3umf. Finally, the second gas with high velocity was injected through the nozzle. The dynamic behavior in the jet region was recorded by video camera (National M-7). The main experimental condition is listed in Table 2.

while the smaller the grid spacing is, the more computation time will be spent and the more random computation error will occur. The simulated part became 0.15 m × 2.05 m when we adopted an axisymmetrical concept. The actual computational region was 0.15 m × 0.60 m in order to save the calculating time because there were no solid particles in the upper region of the bed. The grid spacing was determined by refining the grid until fluid flow properties, such as the jet contour and bubble size, changed by less than 5%. Total cell numbers were 30 × 60. Here, 30 was in the x direction and 60 in the y direction. The grids were dense near the jet nozzle, but those were dilute away from the jet nozzle, especially in the upper freeboard. The front and back wall effects could be neglected as the simulation was carried out in the two-dimensional rectangular bed. The left wall was treated as a symmetry plane, and the right wall was treated as an impermeable, rigid wall. Gas inflow was specified at the bottom of the bed, which was made impenetrable for the solid particles. The boundary condition at the top of the bed was a so-called pressure boundary; i.e., the pressure at the top of the bed was fixed at ambient atmosphere. It is important that the freeboard of the fluidized bed is high enough, so that a fully developed flow can be physically expected and the solid concentration at the top of the bed can be neglected. The boundary conditions (B.C.) were imposed as follows: B.C. 1. At y ) 0, ug,y ) us,x ) us,y ) 0, g ) g,min, keeping the gas inlet velocity at the minimum fluidization velocity, except for a jet nozzle in the center of the bed, at which a relative high inlet velocity was specified. B.C. 2. At y ) 0.60 m, P ) atmospheric, ∂u bg/∂y ) 0,u bs ) 0. B. C. 3. At x ) 0, us,x ) ug,x ) 0, due to the symmetry about the jet center. B.C 4. At y ) 0.15 m, b ug ) 0, i.e., a no-slip boundary for the gas phase but a slip boundary for the solid phase. Initially, the lower part of the fluidized bed was filled with particles at rest with a uniform solid volume fraction. The upper part of the bed was freeboard, which was high enough to make sure that the particle concentration at the top of the bed is negligible. The initial pressure distribution was calculated from the hydrostatic bed height. This procedure simulated closely the actual experimental procedures. In the freeboard, the solid volume fraction was zero, which might result in unrealistic values for particle velocity field and poor convergence. For this reason, a solid volume fraction of 10-8 was set in the freeboard. It is essential that a very small number of particles filled within the whole freeboard can provide more realistic results for the particle velocity in this region but would not have any influence on the fluid dynamic in the fluidized bed.

5. Numerical Computation Section

6. Results and Discussion

The grid size influences both convergence of each iteration and computational time. The larger the grid spacing is, the more numerical diffusion will take place,

A chain of bubbles forms above the nozzle at low gas velocities, but a permanent jet forms at high gas velocities for a given system. Many experimental mea-

Figure 4. Schematic diagram of the experimental apparatus: (A) two-dimensional bed; (B) central nozzle; (C) conical distributor plates; (D) rotameters; (E) valves. Table 1. Physical Properties of Solid Particles resin sand

dp (m)

Fs (kg/m3)

g,mf

umf (m/s)

744 × 10-6 1770 × 10-6

1474 2550

0.424 0.422

0.220 0.940

package, including numerical computation and graphic process codes, has been successfully developed. The numerical program written in FORTRAN language provides a series of significant computational results of the porosity, the pressure, and the gas- and solid-phase velocity fields in either two-dimensional Cartesian or axisymmetric cylindrical coordinate under either a steady- or unsteady-state condition. Numerous data are stored into the text files for the postprocessing, such as contour and vector plots, data reduction, menu, and animation. 4. Experimental Section

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Figure 5. Computed instantaneous voidage in the bed.

surements were employed to determine the jet (or bubble) boundary. Rowe et al.24 observed that a bubble plume, rather than a permanent jet, was a more typical mode of gas discharge at an orifice. A permanent jet may appear at an orifice when particles around the orifice are not well fluidized or the flow of solids toward the orifice is hindered by surfaces. A jet can be stabilized by increasing the particle size, gas velocity through the orifice, and gas to solid density ratio or decreasing the ratio of the bed height to orifice diameter.25 Yang et al.26 and Luo,27 using a pitot-tube technique, measured and analyzed the gas velocity field in a semicylindrical fluidized bed and a two-dimensional fluidized bed, respectively. Their results indicated that gas-phase velocities changed considerably at the jet boundary. The

simulated result from CASICC code indicated that the gas velocity, solid velocity, and pressure profile changed significantly when the voidage was 0.8. The boundary of jet (or bubble) is defined as the line having a voidage of 0.8. This result confirms the arbitrary definition by Gidaspow and Ettehdieh.12 Figure 5 is a typical example of the simulation results using resin as the bed material. 6.1. Jet Development. A jet originally appears vertical and periodically breaks up into bubbles. Figure 6 provides the contours of the jet development process at different times in the bed. A gas jet is entered in the bed at t ) 0. A jet appears at t ) 0.05 s and begins to grow at a later time. At t ) 0.175 s, the first jet has collapsed from the top of the nozzle and a new bubble is formed; then a new cycle starts. A jet region can be

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Figure 6. Process of a jet development. Figure 8. Comparison of the experimental, simulated, and predicted jet penetration heights (resin).

Figure 7. Comparison of the experimental, simulated, and predicted jet penetration heights (sand).

divided into two zones: the lower zone and the upper zone. At the lower zone, the jet boundary is conical and quite stable. Particles are entrained into the jet and propelled upward by the high-velocity gas stream. In this zone, the axial gas momentum flux drops rapidly with height. The radial gas momentum flux also falls very steeply with the radial distance from the jet center. The upper zone is the bubble-forming region, the boundary becomes unstable, the axial gas momentum flux profiles fall gradually, and the particles with relatively high momentum flux are thrown back to the emulsion phase, which eventually results in the bubble detaching from the top of the jet. Tsukada and Horio3 also got a similar conclusion. 6.2. Jet Penetration Height. As gas enters a fluidized bed, it penetrates some vertical distance before breaking up into bubbles. Various methods were employed to measure the jet penetration height experimentally. They were mainly visual observation, highspeed photography, and probe techniques. Different expressions for the jet penetration height were also proposed. However, discrepancies among these expressions existed mostly because of the different definitions and the limited experimental data. Knowlton and Hirsan28 suggested three different definitions of the jet penetration height, i.e., LB, LMAX, and LMIN. LB was the deepest penetration of bubbles into the bed before they lost their momentum. LMAX was the penetration depth of a series of interpenetrating cavities. LMIN was the minimum penetration depth. LMAX was adopted as the jet penetration height in this paper. The experimental jet penetration height was obtained from frame-byframe analysis of the video data. Figures 7 and 8 provide the comparison of experimental and simulated jet penetration heights with the predicted results from various correlations.27,29-31 The

Figure 9. Effect of the orifice diameter on the jet penetration heights (resin).

jet penetration height increases with an increase in the jet gas velocity for a given nozzle diameter. Generally speaking, the simulated and experimental values are closer to the correlation of Luo27 than to that by Blake et al.,29 Basov et al.,30 and Yang and Keairns.31 It should be noted that we used the same experimental setup as Luo.27 The influences of the orifice diameter on the jet penetration height are shown in Figure 9 under constant jet gas velocities. The results indicate that the jet penetration height increases with increasing nozzle diameter because of an increase in the momentum of gas introduced from the nozzle. It is clear that the jet penetration height obtained from the hydrodynamic model, without using any fitted parameters, exhibits much smaller errors than those from empirical correlations using dimensionless analyses. 6.3. Jet Frequency. The jet frequency is another important parameter to describe the jet behavior. Nguyen and Leung32 reported that the jet frequency was 10-25 Hz, while Huang and Chyang25 stated that it was 2-7 Hz. The jet frequency was obtained by counting the total number of jets per second. Figure 10 plots the comparisons of experimental and simulated jet frequencies employing resin as the solid material. The jet frequency decreases with an increase in the jet gas velocity. The simulated result shows that the jet frequency decreases with an increase in the particle diameter when the other physical properties are kept constant. The jet frequency is dependent upon operation conditions and solid materials. 6.4. Initial Bubble Diameter. A bubble is formed when the jet breaks up. Experimental initial bubble

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Figure 10. Comparison of experimental and simulated jet frequencies (resin).

diameters were the average values of up to 50 bubbles, which were obtained from the frame-by-frame analysis. The simulated bubble diameter is defined as the diameter of a circle having the same surface as the void in the emulsion phase during the simulation process. This void is the boundary of the gas volume fraction as 80%. The model for bubble formation in a fluidized bed was first proposed by Davidson and Harrison.33 They assumed that the fluidized bed resembled an inviscid liquid and implied that there was no net exchange of gas between the bubble and its surrounding emulsion phase. The validity of this model was later questioned by Nguyen and Leung32 and Yang et al.26 Nguyen and Leung32 measured the actual bubble volume photographically in a two-dimensional bed at 1.5umf. They found that a net loss of gas from the bubble to the surrounding emulsion phase was 47%. Yang et al.26 investigated the gas leakage from the bubble to the emulsion phase in a large-scale semicircular fluidized bed by using frame-by-frame analysis of filming. They also suspected substantial gas leakage from the bubble to its surrounding emulsion phase and proposed a simple model for the hemispherical bubble. A model is presented in this investigation to describe the gas leakage from the bubble to its surrounding emulsion phase. The velocity of leakage is equal to the minimum fluidization velocity, which is similar to the assumptions by Zene34 and Yang et al.26 The primary assumptions for the leakage model are as follows: (1) the emulsion phase of the fluidized bed is regarded as an inviscid fluid; (2) the bubble is cylindrical in form; (3) the bubble forms after a jet collapse when the time is tB; and (4) the net gas-leaking velocity to its surrounding emulsion phase is equal to the minimum fluidization velocity. Therefore, the balance of the gas volume discharging from the nozzle is

V T ) VB + VL

(17)

where VT is the total gas volume, VB is the bubble volume, and VL is the leakage volume to the surrounding emulsion phase. Therefore,

VT ) QtB ) u0d0wtB

(18)

VB ) πDB2w/4

(19)

VL ) πDBwumftB

(20)

and

Figure 11. Comparison of the gas leakage model with experimental and simulated initial bubble diameters (resin).

Figure 12. Comparison of the gas leakage model with experimental and simulated initial bubble diameters (sand).

thus

DB2 + 4umftBDB - 4u0d0tB/π ) 0

(21)

Figures 11 and 12 show the comparison of experimental, simulated, and predicted initial bubble diameters. It is clear that the predicted bubble diameters, using the gas leakage model, are close the experimental and simulated results, while the predicted bubble diameters, assuming no gas leakage, are considerably larger than the other data. 7. Conclusion Based on the first principles of mass and momentum conversion, a hydrodynamic model describing gas-solid flow characteristics in the fluidized bed has been presented. This model was experimentally verified by using the frame-by-frame analysis of video. The jet development, jet penetration height, jet frequency, and initial bubble diameter were investigated in a twodimensional bed with a cone-shaped distributor. The results showed that the numerical calculation is in a good agreement with the experimental measurement. A void (jet or bubble) boundary can be determined at the contour of the gas volume fraction as 0.8, where basic flow field variables (gas velocity, solid velocity, and pressure profile) have distinct changes. This result further confirms the definition by Gidaspow and Ettehdieh.12 A jet region can be divided into two zones: the

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lower zone and the upper zone. The jet boundary is conical and quite stable at the lower zone. The upper zone is the bubble-forming region, the boundary becomes unstable, and the axial gas momentum flux profiles fall gradually. The jet penetration height increases with an increase of the jet gas velocity for a given nozzle diameter and with an increase of the nozzle diameter for a given jet gas velocity because of an increase in the momentum of gas introduced from the nozzle. The jet frequency is dependent upon the operation conditions and solid materials, which decreases with an increase in the jet gas velocity. A simple model considering gas leakage to its surrounding emulsion phase is provided. The velocity of leakage is assumed as the minimum fluidization velocity of the solid particle. The experimental and simulated initial bubble diameters indicate that the gas leakage model can be used to approximately calculate the initial bubble diameter after a jet breaks up. Acknowledgment This research was financially supported by NNSFC (National Natural Science Foundation of China) and Royal Society of London. Dr. K. Zhang acknowledges the kind hospitality of Professors Rex Thorpe and John Davidson as well as the Department of Chemical Engineering during the time he spent at the University of Cambridge conducting a Royal Society Royal Fellowship. The support from the Henry Lester Trust is gratefully acknowledged by H. Zhang. Nomenclature c ) compaction modulus Cd ) drag coefficient DB ) initial bubble diameter, m dp ) particle diameter, m do ) nozzle diameter, m g ) acceleration due to gravity, m/s2 G0 ) particle-particle interaction elastic modulus for s ) 0, Pa G(g) ) particle-particle interaction elastic modulus, Pa L ) jet penetration height, m P ) pressure, Pa Ps ) solid-phase pressure, Pa Q ) volumetric flow rate issuing from the nozzle, m3/s Rep ) particle Reynolds number t ) time, s tB ) time of bubble foramtion, s b u ) velocity vector, m/s uo ) jet gas velocity, m/s umf ) minimum fluidization velocity, m/s VB ) bubble volume, m3 VL ) volume leaked to the surrounding emulsion phase, m3 VT ) total gas volume through the jet m3 w ) thickness of the bed, m Greek Letters R ) relaxation factor β ) gas-solid friction coefficient, kg/(m3‚s) ω ) very small value  ) volume fraction * ) compaction volume fraction jτ ) viscous stress tensor, Pa µ ) shear viscosity, Pa‚s F ) density, kg/m3 φ ) sphericity

Subscripts g ) gas phase max ) maximum value mf ) minimum fluidization state s ) solid phase x ) X direction y ) Y direction Operators ∇ ) gradient ∇‚ ) divergence

Literature Cited (1) Massimilla, L. Gas jets in fluidized beds. In Fluidization, 2nd ed.; Davidson, J. F., Clift, R., Harrison, D., Eds.; Academic Press: London, 1985; p 133. (2) Grace, J. R.; Lim, C. J. Permanent jet formation in beds of particle solids. Can. J. Chem. Eng. 1987, 65, 160. (3) Tsukada, M.; Horio, M. Gas motion and bubble formation at the distributor of a fluidized bed. Powder Technol. 1990, 63, 69. (4) Zhang, K. Assessment of coal and biomass gasification process available in the world and progress done in China. Final Report, Chinese Academy of Sciences, Taiyuan, China, 1997. (5) Barthod, T. B.; Del Pozo, M.; Mirgain, C. CFD-aided design improves FCC performance. Oil Gas J. 1999, 66. (6) Gidaspow, D. Hydrodynamics of fluidization and heat transfer: Supercomputer modeling. Appl. Mech. Rev. 1986, 39, 1. (7) Gidaspow, D. Multiphase Flow and Fluidization: Continuum and kinetic theory descriptions; Academic Press: New York, 1994. (8) Zhang, K. Hydrodynamic modelling for fluid dynamics in jetting fluidized beds. Ph.D. Dissertation, Institute of Coal Chemistry, Chinese Academy of Sciences (in Chinese), Taiyuan, China, 1996. (9) Kuipers, J. A. M.; van Swaaij, W. P. M. Computational fluid dynamics applied to chemical reaction engineering. Adv. Chem. Eng. 1998, 24, 227. (10) Hjertager, B. H.; Mathiesen, V.; Solberg, T. Computational analysis of some fluidized systems. In Advanced Technologies for fluid-particle system; Arastoopour, H., Ed.; AIChE Symposium Series 321; AIChE: New York, 1999; Vol. 95, pp 1-6. (11) Van Wachem, B. G.; Schouten, J. G.; van den Bleek, C. M.; Krishna, R.; Sinclair, J. L. Comparative analysis of CFD models of dense gas-solid systems. AIChE J. 2001, 47, 1035. (12) Gidaspow, D.; Ettehadieh, B. Fluidization in two-dimensional beds with a jet. Part II: Hydrodynamic modeling. Ind. Eng. Chem. Fundam. 1983, 22, 193. (13) Bouillard, J. X.; Lyczkowski, R. W.; Gidaspow, D. Porosity distributions in a fluidized bed with an immersed obstacle. AIChE J. 1989, 35, 908. (14) Kuipers, J. A. M.; Tammes, H.; Prins, W.; van Swaaij, W. P. M. Experimental and theoretical porosity profiles in twodimensional gas-fluidized bed with a central jet. Powder Technol. 1992, 71, 87. (15) Kuipers, J. A. M.; Prins, W.; van Swaaij, W. P. M. A numerical model of gas-fluidized beds. Chem. Eng. Sci. 1992, 47, 1913. (16) Kuipers, J. A. M.; Prins, W.; van Swaaij, W. P. M. Numerical calculation of wall-to-bed heat transfer coefficient in a two-dimensional gas-fluidized bed. AIChE J. 1992, 38, 1079. (17) Kuipers, J. A. M.; van Duin, K. J.; van Beckum, F. P. H.; van Swaaij, W. P. M. Computer simulation of the hydrodynamics of a two-dimensional gas-fluidized bed. Comput. Chem. Eng. 1993, 17, 839. (18) Nieuwland, J. J.; Veenendaal, M. L.; Kuipers, J. A. M.; van Swaaij, W. P. M. Bubble formation at a single orifice in gasfluidized beds. Chem. Eng. Sci. 1996, 51, 4087. (19) Ergun, S. Fluid flow through packed columns. Chem. Eng. Prog. 1952, 48, 89. (20) Wen, C. Y.; Yu, Y. H. Mechanics of fluidization. Chem. Eng. Prog. Symp. Ser. 1966, 62, 100. (21) Orr, C., Jr. Particulate Technology; Macmillan: New York, 1966; p 421. (22) Hagyard, T.; Sacerdote, A. M. Viscosity of suspension of gas-fluidized spheres. Ind. Eng. Chem. Fundam. 1966, 5, 500.

3704

Ind. Eng. Chem. Res., Vol. 41, No. 15, 2002

(23) Patankar, S. V. Numerical heat transfer and fluid flow; Hemisphere Publishing Corp.: Bristol, PA, 1980. (24) Rowe, P. N.; MacGielivray, H. J.; Cheesman, D. J. Gas discharge from an orifice into a gas fluidized bed. Trans. Inst. Chem. Eng. 1979, 57, 194. (25) Huang, C. C.; Chyang, C. S. Gas discharge modes at a single nozzle in two-dimensional fluidized beds. J. Chem. Eng. Jpn. 1991, 24, 633. (26) Yang, W. C.; et al. Cold flow scaleup facility experimental results 16-inch air tube test. Topical Report DOE/MC/19122-T1, 1983. (27) Luo, G. H. A study on particle segregation and jet flow behavior in gas fluidized beds. Ph.D. Dissertation, Institute of Coal Chemistry, Chinese Academy of Scicnces (in Chinese), Taiyuan, China, 1996. (28) Knowlton, T. M.; Hirsan, I. The Effect of Pressure on Jet Penetration Depth in Semi-Cylindrical Gas-Fluidized Beds. In Fluidization III; Grace, J. R., Matsen, J. M., Eds.; Plenum Press: New York, 1980; p 315. (29) Blake, T T. R.; Webb, H.; Sunderland, P. B. The nondimensionalization of equations describing fluidization with applica-

tion to the correlation of jet penetration height. Chem. Eng. Sci. 1990, 45, 365. (30) Basov, V. A.; Markhevka, V. I.; Melik-Akhnazarov, T. K.; Orochko, D. I. Investigation of the structure of nonuniform fluidized bed. Int. Chem. Eng. 1969, 9, 263. (31) Yang, W. C.; Keairns, D. L. Estimating the jet penetration depth of multiple vertical grid jets. Ind. Eng. Chem. Fundam. 1979, 18, 317. (32) Nguyen, X. T.; Leung, L. S. A note on bubble formation at an orifice in a fluidized bed. Chem. Eng. Sci. 1972, 27, 1748. (33) Davidson, J. F.; Harrison, D. Fluidized particles; Cambridge University Press: Cambridge, England, 1963. (34) Zene, F. Bubble formation and grid design. Inst. Chem. Eng. Symp. Ser. 1968, 30, 136.

Received for review November 29, 2001 Revised manuscript received May 13, 2002 Accepted May 17, 2002 IE010962U