Fractal Geometry of Particle Aggregates Formed in Calcium Sulfite Slurry

第7卷第2期过程工程学报Vol.7 No.22007年4月The Chinese Jourmal of Process EngineeringApr.2007Fractal Geometry of Particle Aggregates Formed in Calcium Sulfite SlurryNI Wei-min(倪伟敏)，WU Zhong-biao(吴忠标)，GUAN Bao-hong(官宝红),ZHAO Wei-tong(赵伟荣), ZHENG Ping(郑平)(Department of Environmental Engineering, Zhejiang University, Hangzhou, Zhejiang 310027, China)Abstract: The solid-liquid separation is an important operation for the regenerated slurry of dual alkali FGD system, and calcium sulftecould predominate in particle aggregates of the slurry. The setting velocity of calcium sulfite particles is a key parameter for thesolid-liquid separation design. However, the sttling velocity predicted by Stokes' Law could be sutable only for a spherical aggregate,but not for the iregular one. In this work, fractal geometry was introduced in order to cbaracterize highly iregular gcometric shapes. Thesizes of calcium sulfte particle aggregates were analyzed using a meallographic pbase microscope and image analysis. The resultsshowed that particle aggregates had fractal features. The fractal dimensions could reveal the charaterstics of the aggregates' geometryand aggregation process. An exponential relation between the fractal dimension D2 and the particle size 1 was determined as ActP2.According to fractal theory, a parameter can be used to modify Stokes stling velocity close to actual setting velocity. The resuls couldbe valuable for the design of solid-liquid separation processes.Key words: sedimentation; factal geometry; Stokes' Law; dualalkali FGDCLC No.: TQ028.5，Document Code: AArticle D: 1009- -606X(2007)02- 0360- 061 NTRODUCTIONproperties similar or identical to those of impermeablespheres. So， when the particle aggregates in theFlue gas desulpburization (FGD) has become oneregenerated slurry are irregular, such assumptions makeof the hot research topics of environmental science andit difficult to reconcile measured and predicted settlingengineering in China recentlyI1,2. The advantages ofvelocities5].dual-alkali FGD as compared with those of wetMany studies show that the irregular aggregatelimestone one are lower corrosion potential and lessshapes could be described in terms of fractal geometry.scaling or plugging. The dual-alkali FGD system occursPrevious research in this area included fractalin arrangement similar to limestone forced oxidationcharacterization of particles generated from wastewaterprocess, but involves the use of sodium sulfite as thetreatment', bacterial and yeast aggregates fromabsorbing solution. The absorbing solution islaboratory batch experiments' ，and phytoplanktonrecirculated through the absorbing tower to remove SO2aggregates in a simulated oceanic systemls. It wasfrom flue gasb. The spent solution is then mixed withbelieved that most particle aggregates in nature andlime, simultaneously forming calcium sulfite sludge andengineering systems were fractal in their morphologicalregenerating the spent sodium sulfte solution. So thestructurel9. So far, few studics bave been concernedsolid-liquid separation is an important operation towith the physical aspects of particle aggregates formedmake the regenerating slurry recirculated in dual-alkaliin the regenerated slurry, and ltle is known regardingFGD system, and calcium sulfite could predominate inthe morphological characteristics of aggregates and theparticle aggregates of the slurry. The determination ofrelation between their structure and settling properties.an appropriate settling velocity model can provide anIn this work, an experiment by microscopic andimportant tool for designing a solid-liquid separation.settling test is presented to gain knowledge on theBesides size, particle shape affects the behavior ofphysical and morphological aspects of the aggregatesaggregates, particularly their settling velocities. Thand relationship between the structure of aggregates andequations such as Stokes' Law4I to model particletheir settling velocity. Fractal dimensions are used tosettling in sedimentation tanks, however, are usuallydescribe geometrical characteristics of iregular particlebased on assumptions that the aggregates have stting aggregates that are_ not well defined by Euclidean中国煤化工Received date: 2005- 07- 28; Accepted date: 2006 _06-12CNMHG,2030-1; New Ceatury ExclanlFoundation item: Supported by the National Hi-tech Research and Developmeat ProgramScholar Program of Ministry of Education of China No.NCET-04-0549); The Key Research Project of Zhejiang Province (N0.010007037)Blography: NI Wei-min(1978- -), malc, native of Hangzhou City, Zhejiang Province, Ph.D, candidate and specializing in environmental eogineering;WU Zhong-biao, coresponding author, E-mail: zbwu@zj.cdu.cn.第2期NI Wei-min, et al: Fractal Geometry ofParticle Aggregates Formed in Calcium Sulfite Shury361geometry. A model modified from Stokes' Law couldmF=P5lb',(2)predict settling velocities of aggregates according to thewhere ρo is the density of primary particle (kg/m').value of the aggregate fractal dimensions. The resultsSo the mass of an aggregate, m, is obtained ascould be vahuable for the design of solid-liquidseparation processes.m=Nm-=pwP56-DpP.(3)2 THEORETICAL ANALYSES2.2.2 Aggregate volumeIt could be defined in two ways: as an encased2.1 Fractal AggregatesFractals could be defined as disordered systemsvolume, Ve, or as an occupied or solid volume, V.with a non-integtal dimension. A fractal dimensionl0l, According to the frst deinition,n V。is caculated asDn, is used to describe the structure of the particleV=5P,(4)aggregate in n-dimension. Its value varies from 1 to 3.The higher the value of Dn, the more densely theV is calculated as the total volume occupied by allaggregates pack. With fractal dimension of 3, theprimary particles in the aggregate. The differenceaggregates are close to a solid spberical structure.between V。and V is that the former includes both theFractal aggregates have an important property,volume of particles and the volume of the pores. Theself- similarity. If part of an aggregate is cut out, thensolid volume for a fractal aggregate is related to its massthis part is magnified, the resulting object seems to beby m=Vp, or by using Eq.(3).the original one. It could also be seen that as the size ofV=yP2513-D2pP3.the aggregate increases, the size of the pores betweenthe primary particles also increases. Consequently, theAs indicated in Eq.(5)， V does not scale to andensity of the aggregate decreases as it gets bigger. Thiinteger power of 3. These changes make thintroduces the second important property of fractalrelationships between aggregate characteristics such asaggregates, power law behavior.density, porosity, and settling velocity different from2.2 Characteristics of Fractal AggregatesEuclidean characteristics.In order to use fractal geometry to describe the2.2.3 Setling velocitycharacterization of the aggregates' geometry， theThe relationship for describing the settling velocityproperties of the fractal aggregates must be cast in termsof impermeable spheres is based on a force balance of aof fractal dimensions2. Those properties, such assettling aggregate asmass, volume, density and porosity, can influenceV&(P- P:w)g=1/2ApwCoU",(6)settling velocity of aggregates. For a fractal aggregate,Logan et al.chose packing and shape factors thatwhere Pa is the bulk aggregate density, which includeswould be reduced to their Euclidean counterparts, andhe mass of both primary particles and liquid in theobtained the number of particles, N, in an aggregate asencased aggregate volume (kg/m), Pw is the fluidfollows:density (kg/m*), g is the gravitational constant (m/s), Avr=1/011)P,(1)is the projected surface area of the aggregate (m), Cp is .a drag cofficient (dimensionless), U is the settlingwhere v=s与/品 is a packing and shape factorvelocity of aggregate (m/s).(dimensionless), 5 is a packing factor (dimensioness), ξTo specify a similar relationship for the setlingand 品are shape factors of aggregate and primaryvelocity of fractal aggregates, three assurmptions areparticle (dimensionless), I is the characteristic length ofnecessary. Firstly, it is assumed that the advection flowa fractal aggregate, here defined as the longestthrough the highlyporous aggregate does notaggregate length (m), lo is the length of primary particlesignificantly affect settling velocity. Secondly, thin the aggregate (m).projected surface areallal is assumed to be a function ofThe following equations are derived from fractalan additional fractal dimension defined asaggregates based on relationship used for Euclideanobjects.中国煤化IP(7)2.2.1 Aggregate masswhelH.C N M H Gwo-dimensional space,The mass of a primary pril1,12, mo, is definedD2 is ne Iractal aumension unat relates aggregate size toasprojected area and k=52h2~ D3. The area defined by the362过程工程学报第7卷two-dimensional fractal dimension D2 is not equal to anthe lime slurry is about 100 g/L and the sizes of the limeencased area Ae. By the same reason, the fractal volumeparticles are less than a mesh of 100. The purity of SO2is not equal to the encased volume. Thirdly, theis more than 99%. When the pH value of the slurryapproximate expression for the drag coefficient for areaches between 5 and 6, the sludge in the slurry can befractal aggregate isseparated by settlingfor tests and the solidCb= =aRe~",(8)concentration of sludge for tests is 20%~ 30%.3.2 Settling ExperimentsRe=U/v,(9)Settling experiments are performed in 8top-loading column"o, as shown in Fig.l, containing awhere Re is Reynolds number, assumed valid for fractalsettling column with 10 cm in diameter and 100 cm inaggregates, a and b are determined for different rangesheight, an upper reservoir at the top to transferof Reynolds number as listed in Table 1, v is the fluidaggregates into the settling column and a cone at thekinematic viscosity (m/s).bottom to withdraw the aggregates. The volume of theTable 1 Parameters a and b in different ranges ofreservoir is about 250 mL. A plate is placed at theReynolds numberl51bottom of the reservoir and can be split with the_ ParameterRes0.10.12),(11)noted before, it failed when the aggregates were so bigthat they settled too fast for the measurement. Stirring towhere H is the height of the settling column (m), Oh; iskeep aggregates suspended might result in restructuring,the descent of water level due to withdrawal of water inwhich could change their fractal dimensions. Usingsample i (m). .sedimentation instead of light scattering to determine Dnwill allow large dense aggregates to be characterized.Plate-Upper reservolrThe two-dimensional fractal dimension is calculated byregression analysis of the logarithm of the projected下号一Stting column口1.area versus the logarithm of the characteristic length, asr Supportdescribed by Eq.(7).,Sample cone3.1 MaterialsCalcium sulfte could predominate in particle中国煤化工aggregates of the regenerated slury ftom dual-alkaliFGD process5!. So the regenerated slurry sample canYHCNMHGbe obtained by dissolving SO2 in the lime slurry at roomFig.1 A schematic diagram of the toploading columntemperature of about 20C. The lime concentration of第2期NI Wei-min, et al: Fractal Geometry of Particle Aggregates Formed in Calcium Sulfite Slurry3633.3 Measurement of Fractal Dimension D2evident. Fig.4 presents the result of the log -log plot ofAfter settling test, each sample is picked uparea vs. longest size for all samples. Twenty particles ofindividually and randomly with a wide mouth pipetteeach sample are analyzed. The equivalent diametersand put in a Petri dish with distilled water. Care is taken[d=(4A/t)"7] and the fractal dimension calculated forto maintain its initial morphology. Image acquisition iseach sample by image analysis method are displayed instarted after the complete immobilization of liquidTable 2. The equivalent diameter ranges from 3.20 tooccurs. In this work, the projected area is captured by34.87 μm in the experiment.metallographic phase microscope and analyzed byAccording to Eq.(7), D2 is calculated by the slopeSigmaScan Pro5. A schematic flow sheet is shown howof a plot between lgA and lgl. The fractal dimension D2to do image analysis by SigmaScan and fractalis very close to each other with a 98% of confidencedimension measurements in Fig.2. To interpret imagelevel. So the particle aggregates of calcium sulfite havesizes accurately, a lattice with known size isfractal features. The result of least squares fitting is thenphotographed to determine the number of pixelsA=0.0054/62. The value of D2 is 1.62 and the value ofcorresponding to a given standard length and a givenk\ is equal to 0.0054.standard area for each set of experiments.Image analysis systemImage capture and acquisition (JPG format)- one?aggregate per imagea)(bc)Manual threshold selectionFig.3 Digitized images of particles (a), aggregates (b) andstandard for dimensions of pixel (C) (The sizedimensions of a lttice are 1/16 mmx 1/25 mm)Grey to black and white image1000Measurements (area, longest size andequivalent diameter)| Fractal dimension calculation based on area-longest100size power lawFig.2 Procedures of image acquisition and fractal measurement4 RESULTS AND DISCUSSION104.1 Fractal Dimension D2Longest size (um)Two kinds of representative digitized particles areFig.4 Log-log plot of area- size power law for total aggregatespresented in Fig3, and the different morphologies areTable2 Equivalent diameters and fractal dimensions for aggregatesSampleSetting velocity, UProjected area,A. Longest size,1Re=Ul/v .Average equivalent diamneter,Fractal dimension,(103 m/s)_(um2)(um)d (u10)__D6.90~8.114.877.79x10-*3.201.750.2119.89-115.0214.012.94x10~30.3395.32-227.2720.017.011.600.55189.08~489.4247.172.59x10-229.631.550.71595.70-954.2864.38 .4.57x10-234.87Total1.62+0.13Note: The water kinematic viscosity at room temperature, 1. is 1x10-* m'/s.4.2 Setting Velocity of Aggregates Based on FractalTable中国煤化工equal to 24 and 1. SoTheory:TYHCN M H Gows:From Table 2, the values of Re range from(=[(12):)J(0- P)YP16+0-0D15P+1-5 (12)0.000779 to 0.0457 and all are below 0.1. According to364过程工程学报第7卷Therefore, the reltion'51 between U and I is obtained asStokes' Law. Here, the Stokes' velocity predicted by anU=krl",(13)equivalent impermeable sphere, Us, is calculated byUa=aA-pAw)g&f/(18au).(15)where a=D3+1-D2 and k=[g5/(12pw)](ar -pw)yPs310+D2-D3v".Using Eqs.(12) and (15), a dimensionless ratio ofUIf using least squares fiting between lgU and lgl,to Ust, r; is calculated bythe value of a is obtained as 0.92 with a 99% oconfidence level and the value of D3 is 1.53, less thanr=U/U=L[g(120v)](ao Pw4V~51b1+Dr DpPt+1-D2/ .the value of D2. According to the conclusion drawn byMeakin!"7] and Logan et al.!2, D3 is not less than D2[585)(cu0+1-2D2.(16)and when the value of D3 is less than 2, D3 is equal to ;D2. So, for fractal dimension about calcium sulfiteHere, because D3 is equal to D2,「can be given byaggregates, the value of D3 is equal to the value of D2,H3m5/(85)y+"+(1)-D3.(17)1.62. Therefore the value of a is equal to 1 and thevalue of kz is equal to 25.66.Let[=σ1!-),(18)Substituting Eq.(1) into Eq.(12), the settlingwhere =(5/5)4P23160-, a modified shape factor, andvelocity of fractal aggregates is given asσ is equal to 3m8 as a constant.U=(g5(2pw)[(o- paw)/ yJ(l)-D2.(14)So r is related to the aggregate size and irregularaggregate shape. Using r as a modified shape factor, aBased on the fractal theory, the way for promotingmodel modifed from Stokes' Law can be used to predictstling velocity can be proposed. As particles start tosettling velocity of aggregates. These differences inaggregate during flocculation, average aggregate sizesettling velocities are likely a consequence of thegrows and fractal dimensions decreae8. In this way,heterogeneous distribution of primary particles in anan increase in I can result in an increase in U, but U isaggregate and described as厂. On the other hand thenot proportional to P. It appears that flocculant dosagevalue of r can be calculated byand particle concentration affect D2. Otherwise, as moreand more particles are collected, aggregate growthr=U1Uz=,1448(Po- Pa)]=_9k,P.V一 1-B. (19)results from restructuring effects and the value of ψ will18p.vT2(Po-P.)gkincrease and result in increase in k2. So increase in I orψ can result in the increase in U, but it occurs irSubstituting Eq.(18) into Eq.(19)， 0=9kzAwWdifferent floculating conditions.[2d(a-Pw)gki], so the value of θ is equal to 0.00328.4.3 Setting Velocity of Aggregates Based on Stokes'According to fractal theory, the actual stting velocityLawcan be modified with I ^from Stokes' Law.The actual settling velocities are consistently5 CONCLUSIONS .higher than those predicted from Stokes' Law shown inFig.5. The settling velocity calculated by Eq.(14) is inThe structure of calcium sulfite aggregates can beagreement with the actual one and more precise thandescribed with fractal dimensions. Fractal dimension ofcalcium sulfite aggregates may include a description ofthe aggregates not only in geometric forms but also inActual setting velocty with!some other physical parameters such as aggregate, Setting velocity based on fractal theo1E-3density. And the value of D3 is equal to the value of D2,being 1.62. Based on fractal theory, U is proportional tol. Thus a settling velocity model is modified fromStokes' Law, which is valuable for the design of1E-4solid- -liquid separation process. The results of themo中国煤化工. with the actual one. So1E-5theYHto predict the setingveloCNMHGodel,r,theratioofUdor1(m)and Us, is defined as a modified shape factor. TheFig.5 Setting velocity of agegates vs. prediction by Stokes' lawdimensionless variable r related to aggregate size and第2期NI Wei-min, et al: Factar Geometry of Particle Aggregates Formed in Calcium Sulfite Slurry365irregular aggregate shapes shows the form of primaryWater Res, 2002, 36(4): 1056 -1066.particles in an aggregate. 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