Horizontal gas mixing in rectangular fluidized bed:A novel method for gas dispersion coefficients in various conditions and distributor designs

Asheesh Nautiyal,Chien-Song Chyang*,Pin-Wei Li,Hsin-Yung Hou

Department of Chemical Engineering,Chung Yuan Christian University,Taoyuan 320,Taiwan,China

1.Introduction

A gas- fluidized bed is a system ofrandomly spread macroscopic particles that are suspended by an upward flow of air.In a fluidized bed,widespread gas and solid mixing provide a large active surface area to enhance chemical reactions and heat transfer[1].Fluidized beds are used in industry,particularly for the combustion of various fuels,the bulk drying of materials,and some food processing techniques[2,3].Various studies discussed gas-solid mixing,dispersion in a fluidized bed and fluidized bed design to promote high levels of contact between gases and solids[4-6].

Numerous studies have been made to estimate only the dispersion coefficient(Dr)for a circularand rectangular fluidized bed,and itis generally constant at all levels in the bed(excluding the level near the bottom).Werther and Molerus[7]reported a detailed experimental study on the bubble behavior of fluidized beds with different diameters,sizes and densities.They suggested that near the distributor,a zone of maximum bubble growth exists in an annulus near the wall.With increasing bed height,this zone approaches the center of the bed.They stated that the presence of the wall prevents attraction to the inner bubbles from the outside;thus,the bubbles tend to coalesce,and the maximum bubble flow position approaches the central axis.If the wall is circular,the effectis circumferentially even,and the peak is atnearly the same radial location at a given height.However,Whitehead and Dent[8]suggested that if the wall is square,then the wall effect is not circumferentially even,and the peak position does not form a circular position but forms several spots arranged in a rectangle.

Some previous studies observed and discussed the dispersion behavior and dispersion coefficients of the tracer in the lateral and horizontal planes in rectangular and circular fluidized beds.Kunii and Levenspiel[9]reported lateral dispersion of fluidized solids,postulating thatsolid particles are displaced by the rising bubbles,and then the particles are drawn into the wake of the bubbles where they are mixed.Shi and Fan[10]measured the lateral dispersion coefficients of particles(Dsr)in a rectangular gas-solid fluidized bed.They developed an empirical,dimensionless correlation to estimate the lateral dispersion coefficient.Klinkenberget al.[11]presented a study of the concentration distributions caused by diffusion in a fluid moving in a cylindrical tube at uniform velocity.They obtained a solution for a general equation of nonisotropic diffusion using two-sided Laplace integrals with boundary conditions.Brenner[12]provided a method forthe solution ofthe diffusion model of homogeneous fluid displacement in beds of finite length.The results presented in Brenner's paper are given in dimensionless form for the instantaneous concentration of solute leaving the bed and the average solute concentration in the bed at any instant.

Using the study by Klinkenberget al.[11],Rowe and Evans[13]predicted the tracer dispersion at points within the bed using a singlephase model,primarily depending on the radial dispersion coefficient(Dr).In their study,the radial dispersion coefficient increases rapidly from a value near molecular diffusion with an increase in the excess gas flow(U-Umf.).Atimtay and Cakaloz[14]obtained the two-dimensional diffusion model.They measuredDrof a gas in a 10-cm-diameter fluidized bed charged with resin beads using a new strip staining technique with bromine as the tracer.Lin and Chyang[15]discussed the gas mixing in the radial direction within a cold model circular fluidized bed,which was studied using response surface methodology(RSM).Their results showed that the standard deviation of the time-averaged radial tracer concentration is well correlated with the operational and geometric parameters.?tefanica and Hrdli?ka[16]described the experimental apparatus for measurement ofDr,and the effects of gas velocity at various fluidization values and bed heights were investigated.The maximum value ofDron each level was found either in the center of the bed or at the wall.

Fuel mixing has a large impact on the overallperformance of a fluidized bed combustor(FBC).As the fuel's horizontal mixing improves,there is a more uniform local stoichiometric ratio in the reactor's cross section,and this lowers the occurrence of sites with unreacted fuel or oxygen.The effects of various operating parameters on the dispersion coefficient have been studied in previous works.Such studies primarily focus on the effects of the operating parameters,such as super ficial gas velocity and particle size,on dispersion behavior[17-19].

Most studies related to dispersion in the horizontal plane use smallscale circular and rectangular beds.Van Deemter[20]investigated mixing in small and large fluidized beds and demonstrated the importance of the size effect.The author concluded that the differences and similarities between a small and large bed could be attributed to the flow regime.Zhanget al.[21]suggested that the particle concentration is evenly distributed across the bed core but rapidly increases at the boundary layer towards the wall surfaces in circulating fluidized bed boilers.They mentioned that the cross-sectional average solid volume fraction was low,even as low as 0.003.The high particle concentration at the wall in such a dilute phase might be attributed to the membrane-tube wall con figuration and to the small bed aspect ratio,which is an order of magnitude of 10.The effects of various operating parameters on the dispersion coefficient are considered a wellunderstood phenomenon;yet,the effect of the side walls on the dispersion coefficient has not been discussed considerably.Therefore,it is necessary to investigate the effect of the chamber walls on dispersion behavior.

Using the radial dispersion coefficient(Dr)for a circular fluidized bed is acceptable,but switching to a rectangular bed,Drneeds to be decomposed into Cartesian form,DxandDy,in the horizontal plane.This study uses a new analytic solution to estimateDxandDy.

Rectangular fluidized-bed combustors have been widely adopted for commercial application.For most of the commercial fluidized beds,tuyere nozzles or bubbling caps are used as the distributor instead of perforated plates.Fora perforated plate,the gas jetfrom the distributoris in the vertical direction.Rather,for a fluidized bed equipped with tuyeretype distributor,the gas jetfromthe nozzle is in the horizontalplane.So,it becomes necessary to develop a method for estimating dispersion coefficients,DxandDy,in a rectangular fluidized bed.Additionally,in a rectangular fluidized bed,with a relatively low length-to-width ratio,back-mixing of solids does not substantially occur.

In this study,a novel analytical solution,in terms ofDxandDy,was established for a governing equation of the tracer dispersion in the steady state.However,the solutions are valid for specialized situations,such as tracer injection from a point source with the physically relevant boundary conditions[15].Further,a novelsurface fitting ofthe obtained analytical solution to the experimental data is presented in this study.The tracer concentration data points along theXandYaxes at a certain height(Z-axis)were collected using controllable probes with CO2detection tips.In the fluidized bed,the dispersion coefficients are discussed within a Cartesian coordinate system of a three-dimensional dispersion model.In addition,the obtained results were compared with the conventional model[13],which shows that the values ofDxandDyremain in the same range.

2.Experimental

All ofthe experiments are conducted in a lab-scale rectangular fluidized bed,fabricated with a transparentacrylic column.The cross section ofthe bed is 0.2 m×0.4 m,and itis 1.5 min height.Aschematic diagram of the apparatus is shown in Fig.1.The experimental setup is similar to thatofour previously published papers[15,17].The fluidizing air is supplied by a 15 hp.Roots blower,and the super ficial gas velocities are set to be multiples of minimum fluidization velocity(Umf),that is,3.5Umf,5Umf,and 6.5Umf.Glass beads with a mean density of 2500 kg·m-3and a mean size of 385 μm are used as the bed material.The minimum fluidizing velocity is 0.115 m·s-1.The static bed height(Hs)is 0.2 m.At the bottom,an acrylic perforated plate with 343 holes of 1.85 mm ID is employed as the gas distributor.In this experiment,carbon dioxide is used as the tracer gas.The glass bead used in this study is smooth and the absorption of CO2by glass beads can be neglected.The injection rate of the tracer gas is 0.20%of the fluidizing air flow rate.The tracer gas was continuously injected from the center while maintaining the same distributor levelviaa 6.4 mm ID stainless-steel tube into the bed.After reaching a steady state of the tracer gas in the bed,using a stainless-steel probe of 6.35 mm I.D,the downstream tracer gas is sampled at 5 and 10 cm(sampling height(Z)is the distance between tracer gas injector and sampling point)above the distributor in a horizontal plane at fixed distances in 2D coordinate geometry,as shown in Fig.2.To observe the effect of the side walls,the tracer position shifted from the center to near the wall of the chamber along the long side,i.e.,theX-axis.To examine the effect of the bed height,two static bed heights of 15 cm and 20 cm were prepared.

To study the effect of distributor design on the dispersion coefficients,the perforated plate was replaced with a multi-horizontal nozzle distributor.The super ficial gas velocities are 3.5Umfand 5Umf.The bed materials are the same and the static bed height is 25 cm.The height of tracer injection is 10 cm and the sampling height(Z)is 10 cm above the distributor for the multi-horizontal nozzle distributor.

The concentrations of the tracer gas in the bed are analyzed using a nondispersive infrared gas analyzer(ETG IMA 3000B).The dimensionless ratio,C/Co,represents the concentration of the tracer gas in the study,whereCis the concentration of the tracer gas in the bed where the gas is mixed with the bed particles,andCois the concentration of the tracer gas within the freeboard.The tracer gas and fluidizing air are assumed to be well mixed.In this study,Cois 2350×10-6.Fig.3 shows the con figuration of the 3D lab-scale rectangular fluidized bed with the tracer gas injection and the data sampling locations in the horizontal(X-Y)plane and in the vertical(Z)direction.Our lab scale fluidized bed setup is a small-scale replica of commercially used fluidized bed technology,so the results from this study can be applied more accurately to a large size fluidized bed.

2.1.Modeling

In this study,a governing equation for tracer dispersion in a rectangular fluidized bed is formulated.An analytical solution is obtained for the governing equation of tracer dispersion with the physically relevant boundary conditions.

The governing equation for tracer dispersion in the rectangular bed is as follows:

whereCis the tracer concentration(kg·cm-3);Vx,Vy,andVzare the super ficial gas velocity vectors(cm·s-1)in the respective directions;andDx,Dy,andDzare the dispersion coefficients(cm2·s-1)in the respective directions.In a fluidized bed, flow through the bed is commonly represented by dispersed plug flow,where the mechanismofmixing involves the effective dispersion coefficients,DyandDr,known as axial and radial dispersion,respectively.Axial dispersion(Dy)can be offset by radial dispersion(Dr),which means that radial dispersion in fluences the plug flow behavior.For a steady state,plug flow,and negligible vertical dispersion(compared with the convective transport)from a point source in the rectangular bed,Eq.(1)can be simplified,as follows:

The tracer progresses as a thin slab(dZ)in a rectangular bed where a massM(kg)oftracer is released atX=Y=Z=0.Rearranging Eq.(2)in the moving frame of reference with the direction of flow,thenZ/U= τ and ?C/?(Z/U)=?C/?τ and Eq.(2)transforms into the following equation:

Fig.3.Continuous release of the tracer at mid-width W(X=0),(Y=0)and(Z=0).(a)The lab-scale fluidized bed.(b)Tracer point at mid-width.

The solution is found by collecting thefandηterms on separate sides of the equation,as follows:

As assumed in the beginning,τ=Z/U,we return to the stationary frame.Further,the solution transforms into the following equation:

Dividing Eq.(6)by dZ(the tracer moving as a thin slab of heightdZ),the following equation is obtained:

whereMis?m·(dZ/U)in which?mis the mass flow rate(kg·s-1,i.e.,Co·U·A),Cois the tracer gas concentration(kg·cm-3)atZ= ∞,andAis the cross-sectional area of theXYplane.Eq.(7)can be written as follows:

The obtained experimental data are analyzed in OriginPro 8,which includes conversion of the worksheet data into the matrix for the 3D surface plot in theX-Y-Zdimensions.Origin's 3D surface fitting is performed by the nonlinear least squares fitter(NLFit)tool using a “user de fined built-in function”based on the compiled language(Origin C)of Eq.(8)[24,25].The solution,i.e.,Eq.(8),is written in the“Z-Script function form”,whereZis the dependent variable,andXandYare independentvariables.The other parameters in the“Z-Scriptfunction form”for surface fitting of Eq.(8)are de fined as follows:Z0,U1,D1,andD2 are named for the parametersA,U,DxandDy,respectively.Next,the“Z-Script function form”is compiled,and the parameter initialization is performed with constant and initial guess values.The nonlinear fitting always starts with an initial guess of the parameter to ensure greater convergence.Finally,auto-iterations are performed to achieve the best fit of Eq.(8)to the experimental data.

Previously,Rowe and Evans[13]have discussed the solution and estimation ofDrusing a single-phase modelwith remote boundary conditions,as shown in Eq.(9).The results from Eq.(9)are used for comparison with the proposed model,i.e.,Eq.(8).

3.Results and Discussion

3.1.Effect of super ficial gas velocities(U)on the gas dispersion coef ficients(Dx and Dy)

Fig.4.The experimental 3D surface plot of tracer gas concentrations with various super ficial gas velocities.(H s=20 cm;Z=5 cm).(a)U=3.5U mf;(b)U=5U mf;(c)U=6.5U mf.

Fig.4(a)-(c)show the 3D surface plot of the tracer gas concentrations at the sampling height of 5 cm with super ficial gas velocities of 3.5Umf,5Umf,and 6.5Umf,respectively.TheXaxis andYaxis represent the length and width,respectively,of the lab-scale fluidized bed as shown in Fig.3.The unit is in centimeter.From Fig.4(a)-(c),it can be observed that as the super ficial gas velocity is increased,the tracer gas concentration spreads farther.

Subsequently,Fig.5(a)-(c)show the nonlinear least squares surface fitting,based on the analytical solution,on the 3D surface plot ofthe experimental data to estimateDxandDy.The values of Adj.R-square in all surface fittings are greater than 97%,which indicates the quality of the best fit.Table 1 shows the estimated dispersion coefficients,DxandDy,from the nonlinear surface fitting of the experimentaldata under different working conditions.

Fig.6(a)and(b)show the 2D form of the concentration profiles of the tracer gas along theXandYaxes with various super ficial gas velocities.The concentration profiles flatten as the super ficial gas velocity is increased,which indicates that the mixing extent at a higher super ficial gas velocity is improved.Fig.6(a)and(b)also show that the tracer gas concentration reaches a steady value,which is equivalent to the CO2concentration in the air(approximately 400×10-6).Because ofthe low sampling height(Z=5 cm),the jet effect is vigorous.Therefore,the maximum vertical tracer gas concentration could be overvalued.The maximum points are distantfrom the peak line.From Table 1,Dxis generally greater thanDy,which may be due to the effect of the side walls.The ratio ofDxtoDyaverages approximately 1.4-1.5,exceptundersome conditions,such aslowersampling height(Z=5 cm)and highersuperficial gas velocity(＞3.5Umf)where the jet effect is dominant.

Table 1Summary of the experimental conditions and results with tracer at center

Fig.6.Experimental 2D concentration profiles of tracer gas with various super ficial gas velocities(H s=20 cm;Z=5 cm).(a)C/C o vs.Location on X-axis;(b)C/C o vs.Location on Y-axis.

3.2.Effect of data sampling heights(Z)on the gas dispersion coef ficients(Dx and Dy)

Fig.7(a)-(c)show the 3D surface plot of the tracer gas concentrations at the sampling height of 10 cm with different super ficial gas velocities of 3.5Umf,5Umf,and 6.5Umf,respectively.It can be observed that as the super ficial gas velocity is increased,the tracer gas concentration spreads farther,and the trends are similar to Fig.4(a)-(c).However,the effectof the side walls on the tracer gas concentration profile in Fig.7(b)and(c)is noticeable.The walleffectatZ=5 cm(Fig.5)is not highly significant.

Fig.7.The experimental 3D surface plotof concentrations of the tracer gas with various super ficialgas velocities.(H s=20 cm;Z=10 cm).(a)U=3.5U mf;(b)U=5U mf;(c)U=6.5U mf.

Fig.8(a)-(c)show the nonlinear least squares surface fitting,based on the analytical solution,on the 3D surface plot of the experimental data to estimateDxandDy.The values of Adj.R-square in all surface fittings are greater than 99%,which indicates the quality of the best fit.The estimated dispersion coefficients,DxandDy,are also listed in Table 1.As the sampling height(Z=10 cm)is increased,the jet effect is no longer significant.The side walls along the length of the chamber act as an immediate barrier at a higher super ficial gas velocity;thus,the tracer concentration near the wall is detected as markedly higher.This may affect the estimation of the dispersion coefficients,DxandDy;thus,Dxis not equal toDy.If the bed is sufficiently large throughout its width,then the tracer gas can disperse farther without being blocked by the side walls.In this ideal case,Dxwill be equal toDy.

Fig.9.Experimental 2D concentration profiles of tracer gas with various super ficial gas velocities.(H s=20 cm;Z=10 cm).(a)C/C o vs.Location on X-axis;(b)C/C o vs.Location on Y-axis.

Fig.9(a)and(b)show the 2D form of the concentration profiles of the tracer gas along theXandYaxes with various super ficial gas velocities.The concentration profiles flatten as the super ficial gas velocity is increased,and the trends are similar to those in Fig.6(a)and(b).However,the jet effect in Fig.9(a)and(b)is not significant compared with that in Fig.6(a)and(b),demonstrating the significance of the sampling height in estimating the dispersion coefficients.

Fig.10(a)and(b)show the dispersion behavior in theX-Yplane with various fluidization values at sampling heights of 5 cm and 10 cm,respectively.As can be observed in Fig.10(a)and(b),the dispersion coefficientsDxandDyare increased with the super ficial gas velocity.With no jet effect atZ=10 cm,the dispersion coefficientsDxandDyshow a linear trend.Because of the jet effect atZ=5 cm,when the super ficial gas velocity is greater than 3.5Umf,the dispersion coefficientDxis larger than the value obtained atZ=10 cm.

3.3.Effect ofwalls on the gas dispersion coef ficients(Dx and Dy)atdifferent tracer positions

Fig.11(a)shows the 3Dsurface plotofthe concentrations ofthe tracer gas when the tracer's injection position is in the center ofthe bed.The operating conditions areU=5Umf,Hs=20 cm,andZ=10 cm.Subsequently,Fig.11(b)shows the nonlinear least squares surface fitting on the 3D surface plot of the experimental data in Fig.11(a)from whichDxandDyare estimated.The estimated dispersion coefficientsDxandDyare listed in Table 2.

Compared with Fig.11(a),which has similar operating conditions(U=5Umf;Hs=20 cm;Z=10 cm),Fig.12(a)-(c)shows the 3D surface plot of the tracer gas concentrations when the location of the tracer gas injection point is near the wall.Subsequently,Fig.13(a)-(c)shows the nonlinear least squares surface fitting on the 3D surface plot of the experimental data from Fig.12(a)-(c).The estimated dispersion coefficientsDxandDyare listed in Table 1.

Under ideal conditions,at the same horizontal plane with similar working conditions and operating parameters,the dispersion coefficientsDxandDyshould notdiffer.Theoretically,by modeling the analytic solution and assuming thatDx=Dy=46.0 cm2·s-1,which is given in Fig.11(b),the simulation of the tracer's concentration profile is presented in Fig.14.By comparing Figs.11(a)and 14,we can see that the tracer gas concentration profiles differ experimentally and theoretically.

Fig.10.Gas dispersion coefficient with various fluidization numbers at different sampling heights(H s=20 cm).(a)Dx vs.U/U mf;(b)Dy vs.U/U mf.

Table 2Summary of the experimental conditions and results with different location of the tracer

However,in Fig.13(a)-(c),as the tracergas injection pointshifted to near the wall from the center,the dispersion coefficients differed.From Table 2,we can see that given similar operating conditions,the value ofDxdecreased,whereasDyremained almostunchanged.The ratio ofDxtoDywas 1.4 in the case when the tracer injection point was at the center.As the tracer injection pointshifted to near the wall,the ratio ofDxtoDywas 1.1.This change could be attributed to the fact that the concentration of the tracer gas(Ca/Co=28.9)was greater near the walls of the chamber.Therefore,the effect of the side walls on both dispersion coefficients in theXandYdirections is similar.Conversely,the tracer that was released at the center can diffuse much farther in theXdirection(Dx=46 cm2·s-1)because the walls are distant compared with theYdirection(Dx=32 cm2·s-1).In other words,if the length to width ratio of the rectangular bed is not appropriate,the mixing in the center area of the bed will not be excellent.

Painet al.[26]discussed that the particles may slip at the wall or bounce off the wall,which creates complicated boundary conditions.These authors found that the time-averaged distribution of particles demonstrated that bubbles move in the central region of the bed and particles fall down along the bed wall.Liet al.[27]discussed that the solid-phase wall boundary condition needs to be specified with great care when gas mixing is modeled,because free-slip,partial-slip and no-slip wall boundary conditions result in substantial differences in the extent to which gas is transported downwards at the wall.Liet al.[28],in another study,found that both 2D and 3D simulations capture the general flow behavior in bubbling fluidized beds,but the wall effect in an experimental“two-dimensional”column was importantand must be considered when quantitatively comparing numerical simulations and experimental data.

3.4.The effectof the gas injection rates on the dispersion coef ficients Dx and Dy in a horizontal plane

To observe the effect of the side walls in the case of different gas injection rates,we changed the tracer gas injection rate from 0.20%to 0.10%of the fluidizing air flow rate.Fig.15(a)shows the 3D surface plot of the tracer gas concentrations when positioned at the center of the bed.The operating conditions were similar tofig.11(a),asU=5Umf,Hs=20 cm,andZ=10 cm,but the gas injection rate was 0.10%.Subsequently,Fig.15(b)shows the nonlinear least squares surface fitting on the 3D surface plot of the experimental data from Fig.15(a).The estimated dispersion coefficientsDxandDyare listed in Table 2.Evidently,we can see thatDychanged slightly from 32 to 29.8 cm2·s-1,butDxchanged from 46 to 26.5 cm2·s-1.At a tracer gas injection rate of 0.10%of the fluidizing air flow rate,the value ofDxwas approximately equal toDy.

This result may be because the difference between the ambient and referenced carbon dioxide concentration was 1950×10-6(i.e.,2350×10-6-400×10-6)when the gas injection rate was 0.20%of the fluidizing air;the difference was 950×10-6(i.e.,1350×10-6-400×10-6)when the gas injection rate was 0.10%of the fluidizing air.This difference between the ambient and the referenced carbon dioxide concentration in the bed may be significant.The reduction in the ambient carbon dioxide from the background in the case when the gas injection rate was 0.20%ofthe fluidizing air improves the detection sensitivity.Therefore,the value ofDxcan be detected more accurately and is greater thanDy.In contrast,when the gas injection rate was 0.10%ofthe fluidizing air,Dxwas approximately equal toDydue to the lower detection sensitivity.

3.5.Effect of the static bed heights on the gas dispersion coef ficients(Dx and Dy)

Figs.16(a)and 11(a)show the 3D surface plot of the tracer gas concentration with a super ficial gas velocity of 5Umfat static bed heights of 15 cm and 20 cm,respectively.Subsequently,the nonlinear least squares surface fitting on the 3D surface plot of the experimental data in Figs.16(a)and 11(a)are shown in Figs.16(b)and 11(b).The estimated dispersion coefficientsDxandDyfor both bed heights(Hs=15 cm and 20 cm)are listed in Table 2.Fig.17(a)and(b)show the 2D representation of the tracer concentration for different bed heights(Hs=15 cm and 20 cm)in theXandYdirections,respectively.Clearly,the effect of static bed height onDxappears to be insignificant.This result is in agreement with our previous study where the effect of different bed heights onDrwas insignificant[29,30].However,the value ofDyincreased from 32 cm2·s-1to 37.3 cm2·s-1as the bed heightdecreased from 20 cm to 15 cm.Overall,the estimation ofthe optimal length to width ratio,bed height,and gas injection rate for good mixing in a rectangular bed is helpful for designing and controlling the fluidized bed reactor.

3.6.Effect of the super ficial gas velocities on the gas dispersion coef ficients(Dx and Dy)with the multi-horizontal nozzle distributor

Fig.18(a)and(b)show the 3D surface plot of the tracer gas concentration at super ficial gas velocities of 3.5Umfand 5Umfand static bed heights of 25 cm,respectively.It can be seen clearly that in the case of a multi-horizontal nozzle distributor,the maximum concentration of tracer gas is not at the center of the bed as was observed in the case of perforated plate distributor.The maximum concentration is shifted to the direction of nozzle orientation rather than stable at the center.We would like to validate our dispersion model in the case of a multihorizontal nozzle distributor.For this purpose,we rearranged the scale in theXdirection while inYdirection,the scale remains same.Fig.19(a)and(b)show the nonlinear least squares surface fitting on the 3D surface plot of the experimental data from Fig.18(a)and(b)and the results are listed in Table 2.It can be seen thatDxis considerably larger thanDyfor the multi-horizontal nozzle distributor compared to the perforated plate distributor in Table 2.The ratio ofDxandDyis 1.8.This shows that dispersion in theXdirection is higher for the multi-horizontal distributor than for the perforated plate distributor.

Fig.12.The experimental 3D surface plot of the tracer gas concentration at 0.20%of the fluidizing air flow rate.The tracer gas injection was near the wall(H s=20 cm;Z=10 cm;U=5U mf).(a)Corner view;(b)side view;(c)front view.

Fig.13.Nonlinear least squares surface fitting matrix on the experimental 3D surface plot of the tracer gas concentration at 0.20%of the fluidizing air flow rate.The tracer gas injection was near the wall(H s=20 cm;Z=10 cm;U=5U mf).(a)Corner view;(b)side view;(c)front view.

4.Comparison With the Conventional Method

Fig.14.The simulated 3D surface plot of the tracer concentrations based on Dx=Dy=46.0 cm2·s-1 where H s=20 cm,Z=10 cm,and U=5U mf.

Fig.15.(a).The experimental 3D surface plot of the tracer gas concentrations at 0.10%of the fluidizing air flow rate(H s=20 cm;Z=10 cm;U=5U mf).(b).Nonlinear least squares surface fitting matrix on the experimental 3D surface plot of the tracer gas concentration at 0.10%of the fluidizing air flow rate(H s=20 cm;Z=10 cm;U=5U mf).

Fig.16.(a).The experimental3Dsurface plotofthe tracergas concentration at0.20%ofthe fluidizing air flow rate(H s=15 cm;Z=10 cm;U=5U mf).(b).Nonlinear least squares surface fitting matrix on the experimental 3D surface plot of the tracer gas concentration at 0.20%of the fluidizing air flow rate(H s=15 cm;Z=10 cm;U=5U mf).

Fig.17.Experimental 2D concentration profile of the tracer gas obtained using various static bed heights at 0.20%of the fluidizing air flow rate(Z=10 cm U=5U mf).(a)C/C o vs.location along the X-axis;(b)C/C o vs.location along the Y-axis.

Fig.18.(a).The experimental3Dsurface plotofthe tracergas concentration at0.20%ofthe fluidizing air flow rate for a multi-horizontal nozzle distributor(H s=25 cm;Z=10 cm;U=3.5U mf).(b).The experimental3Dsurface plotofthe tracer gas concentration at0.20%of the fluidizing air flow rate for a multi-horizontal nozzle distributor(H s=25 cm;Z=10 cm;U=5U mf).

Fig.19.(a).Nonlinear least squares surface fitting matrix on the experimental 3D surface plot of the tracer gas concentration at 0.20%of the fluidizing air flow rate for a multihorizontal nozzle distributor(H s=25 cm;Z=10 cm;U=3.5U mf).(b).Nonlinear least squares surface fitting matrix on the experimental 3D surface plot of the tracer gas concentration at 0.20%of the fluidizing air flow rate for a multi-horizontal nozzle distributor(H s=25 cm;Z=10 cm;U=5U mf).

This approach is compared with the well-known conventional method[13]for estimating dispersion coefficients.In Table 1,by comparingDrwithDxandDy,the values and their order of magnitude generally remain in the same domain.These results show good support for this model.Previously,due to the assumption of uniform tracer concentration in theXandYdirections and the limitation of the solution,i.e.,Eq.(9),itwas only possible to calculateDrby neglecting the effectofthe side walls.With this new solution and using the 3Dsurface fitting method,which further decomposesDrintoDxandDy,it is possible to analyze the nature ofthe dispersion in both directionsas wellas the effect ofthe side walls in the rectangular chamber.It can be considered that there is always disparity between the theoretical and experimental dispersion behaviors due to the design of the chamber.

5.Conclusions

In this study,a mathematicalmodel for tracer dispersion in a rectangular fluidized bed is formulated.An analyticalsolution using the Pitheorem of dimensional analysis was obtained for 3D surface fitting of the experimental data.The dispersion coefficients,DxandDy,are estimated and analyzed.The experimental results obtained in this study indicate that the effect of super ficial gas velocity on the extent of gas dispersion is significant.The surface fitting shows that the estimatedDxandDyincrease with the super ficial gas velocity.The results also discussed the domination of the jet effect at a lower sampling height with higher super ficial gas velocity.The jet effect becomes insignificant as the sampling height increases.The estimatedDxandDyare not equal due to the effect of the side walls.The experimental results obtained in this study indicate that the effectof the side walls is significant.With similar working conditions,as the tracer gas injection point shifts near the wall from the center,DxandDychanged.The width to length ratio in a small rectangular fluidized bed affects the mixing in the centerofthe bed.Ata lowertracergas ratio ofthe injected gas to the totalgas flow rate,DxandDyare approximately equal.The results also show that the effect of bed height onDxis insignificant,whereas bed height may have an effect onDy.This model is also able to estimate the dispersion coefficients in the case of a multi-horizontal nozzle distributor.The results from this model are compared with the conventional method,which shows reasonably good agreement.

Acknowledgements

The financial support from the Ministry of Science and Technology under Grant MOST 105-3113-E-033-001 is greatly acknowledged.

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