A stochastic model of bubble distribution in gas-solid fluidized beds

Journal of University of Science and Technology BeijingVolume 13, Number 3, June 2006, Page 222MetallurgyA stochastic model of bubble distribution in gas-solid fluidized bedsYanping Zhang.2)and Li Wang'1)Mechanical Engineering School, University of Science and Technology Beijing, Beijing 100083, China2)Shougang Research Institute of Technology, Beijing 100043, ChinReceiveed2005-03-03)Abstract: On the basis of the Langevin equation and the Fokker- Planck equation, a stochastic model of bubble distribu-tion in a gas-solid fluidized bed was developed. A fluidized bed with a cross section of 0. 3 mx0.02 m and a height of 0.8m was used to investigate the bubble distribution with the photographic method. Two distributors were used with orificediameters of 3 and 6 mm and opening ratios of 6.4% and 6.8%, respectively. The particles were color glass beads withdiameters of0.3, 0.5 and 0.8 mm ( Geldart group b particles). The model predictions are reasonable in accordance with theexperiment data. The research results indicated that the distribution of bubble concentration was affected by the particlediameter, the fluidizing velocity, and the distributor style. The fluctuation extension of the distribution of bubble concen-tration narrowed as the particle diameter, fluidizing velocity and opening ratio of the distributor increased. For a givendistributor and given particles the distribution was relatively steady along the bed height as the fluidizing velocitychangedKey words: fluidization; bubble; stochastic force; bubble distribution1. Introductionprocess and the Langevin equation, the distribution ofbubble concentration along the axis of a fluidized bedhas typical random characteristics [1-31. In recent years, Can be described by the Fokker-Planck equationmany researchers have further studied such phenomenaof bubbles, with chaotic characteristics and dissipativeatstructures [4-12, and have analyzed the stochastic where c is the distribution of bubble concentration inforces in fluidized beds [13-14]. Research on the rules fluidized bedof movement and distribution of bubbles caused bstochastic force in a fluidized bed will help to under- cstand the mechanism of fluidization on stochastic level2. Stochastic model of the distribution of where p is the bubble concentration, Hr is the expandbubble concentration in gas-solid fluidized ed bed height; D: is the axial diffusion coefficient; andu, is the bubble rise velocity.Considering the bubble generation, growth andas a result of the effect of interior and exterior sto-bruption and the coalescence of two bubbles, Eq (1)chastic complications the bubble distribution in a flu-can be modified asidized bed is instable and inhomogeneous. The insta-bility is caused by the bubble diffusion and bubble exgo+ uhC -ucursion and it is inhomogeneous because of the bubblecoalescence and bubble abruption. The diffusion and where qo=Fro/ pdz, Fro is the bubble generationexcursion of bubbles always go with their coalescenceand abruption, and this is the cause of the fluctuation offrequency on the distributor; D is the bubble dif-bubble distributionfusion and中国煤化工ion; uhC IsSuppose the distributing of bubbles in a fluidizedbed was a Markov process. On the basis of the markov bubble attenuCNMHGbble attenuaCorrespondingauthor:YanpingZhang,E-mail:zyp@meustb.edu.cnY P Zhang et aL, A stochastic model of bubble distribution in gas-solid fluidized beds223tion is expressed by the following equation,conditions the model of the distribution of bubble con-uh=kh Frocentration is constructedwhere k, stands for the effect of bubble coalescence3. Experimentaland a break of the bubble numberThe behavior of bubbles in a two-dimensional gas-On the basis of the two-phase theory the gas superfi- solid fluidized bed was investigated by the photoial velocity Wem in the emulsion phase equals the graphic method. The fluidized bed is a rectangularminimum fluidizing velocity umf. Thereforecross-section bed, made in plexiglass, with an innercross-Section of 0. 3 mx0. 02 m and a height of 0.8 mNAVE(1-NV)Solid particles are color glass with diameters of 0.3, 0.5where N, is the number of the bubbles per unit volume; and 0. 8 mm(Geldart group B particles). Two distribu-and Vh is the volume of a bubble. The whole volume of tors were used with orifice diameters of 6 mm(for typebubbles in a fluidized bed can be obtained by,A distributor)and 3 mm(for type b distributor) andopening ratios 6.8%(for type A distributor) and 6.4%Hr-Ho=NaDh(5) (for type B distributor), respectivelwhere Ho is the initial bed heightFig. 1 is the schematic of the experimental equipment. The digital video camera was set in front of theFrom Eqs. (4 )and (5), we can getfluidized bed, and the spotlight was set at the back ofthe bed. The video camera recorded the bubbles' forkhpo=d,3=umfmation,motion, merging, and abruptionIn addition, supplementary equations area processing software for photographic data wasdeveloped. Its function was dispersing the consecutiveimages, determining bubbles' coordinates. and statisti-Po=l(u-lmfcally comparing experimental datal b=u-umf+k(gDb)6C2There Dho and Dh are the initial bubble diameter andbubble diameter, respectively; uo is the initial bubblevelocity; and K=0.71182/5Schematic of the experimental equipment.10)1-compressor; 2-transducer; 3-pressure measuring vessel; 4-spotlight and lens hood; 5-vidicon; 6-computer;A)477077-anemoscope;8 bellows; 9 fluidized bed; 10 grid.1/7b=1.28h4. Comparison of model predictions with ex-perimental datawhere a is the area of the distributor and no is the numFigs. 2-4 compare the model predictions with thee distributorexperimental data. The model can basically describThe boundary and initial conditions arethe trend of the bubble concentration distributionig. 2 shows the distribution of bubble concentrationu=u=00≤x≤H(12)along bed height for diversified experimental condtions witha distributor. It can be seen from thego u>lmf(13) figures that the particle diameter, the fluidizing veloc-ity and the distributor style affect the distribution ofwhere Zo is the initial bubble generating position from bubble concent V凵中国煤化工 tribute inhCNMHGthe particleBy Eqs. (2), and(6)-(9), the boundary, and initial diameter increasing ar'ine peak or ine dubble concen-224J Univ Sci. Technol. Beijing, VoL 13, No. 3, Jun 2006tration the distribution lowers because the entire num- ce number of the type b distributor is larger than that ofber and volume of bubbles decrease for the same flu- the type A distributor, the average bubble concentra-idizing velocity. Therefore the probability of the coa- tion in the bed increases, so the distribution peak of thelescence and abruptions reducebubble concentration with the type b distributor is lower than that of the type A42Model predictionExperiment data with type a distribModel prediction10Experiment data with type A distributo410180.751.25Model predictionExperiment data with lype A distributorModel predictionExperiment data with type A distributor00.100.18力/m0.751.752.256Model predictionExperiment data with typc A distributorModel prediction15Experiment data with type A distributor0.050.100.150.20u/(mS-)Fig. 2. Trend of the bubble concentration along bed height(a)=17ms,dp=03mm;(b)u=175m',d=0.5mm;(c)u=1.7 m/s, d=0.8 mmExpcriment data with type A distributorFig 3 shows the distribution of bubble concentrationalong with fluidizing velocity at some certain heiglfrom above the type A distributor. The distribution ofbubble concentration is basically steady with thechanges of fluidizing velocity. With the fluidizing ve-locity increase the numbers of bubbles in different parts0.751.25of the bed and the number of bubbles in the entire bedincrease so the change of the distribution of bubbleconcentration is smooth. On the other hand. as the bub- Fig. 3. Trend of the bubble concentration along with fluble numbers increase the coalescence enhances so the idizing velocity:(a)h=60 mm, d, =0.03 mm;(b)h=80 mmincrease of the distribution of bubble concentration is dp=0.03 mm;(e)h=60 mm,dp =0.05 mm;(d)h=100 mm,restrainedd=0.05mmFig. 4 compares the distributions of bubble concen5. Conclusiontrations between two kinds of distributors The bubble中国煤化工concentration peak with the type B distributor is lower(1)A stochasCNMH Gon of bubblethan that with the type a distributor. Because the orificoncentration in a gas-Solid lluluizeu Deu was proposedY P Zhang et aL, A stochastic model of bubble distribution in gas-solid fluidized beds225The model predictions showed good agreement with [2] N. 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