This is n open ccess rticle published under n ACS AuthorChoice License, which permits copying nd redistribution of the rticle or ny dpttions for non-commercil purposes. Cite This: Ind. Eng. Chem. Res. 218, 57, 2658 2669 pubs.cs.org/iecr Models for Predicting Averge Bubble imeter nd Volumetric Bubble Flux in eep Fluidized Beds Cornelius E. Agu,*, Christoph Pfeifer, Mrinne Eikelnd, Lrs-Andre Tokheim, nd Britt M. E. Moldestd eprtment of Process, Energy, nd Environmentl Technology, niversity College of Southest Norwy, 3918 Porsgrunn, Norwy eprtment of Mteril Sciences nd Process Engineering, niversity of Nturl Resources nd Life Sciences, 119 Vienn, Austri ownloded vi 148.251.232.83 on November 11, 218 t 9:28:1 (TC). See https://pubs.cs.org/shringguidelines for options on how to legitimtely shre published rticles. ABSTRACT: The verge bubble dimeter nd volumetric bubble flux give indictions bout the overll bed expnsion in fluidized bed. As these properties depend on the prticle properties nd fluidized bed regime, their ccurte predictions hve been chllenge. A new set of models for predicting the verge bubble properties within the bubbling nd slugging regimes in deep fluidized bed is proposed, where bubble flux is modeled by G c.66, bubble dimeter is modeled by d =.848G.34 nd trnsition = ( ).27.35 t bs t velocity is modeled by = 1 + 2.33 ( φ c 1). The models re developed using the informtion obtined from n experimentl setup equipped with dulplne electricl cpcitnce tomogrphy nd porous distributor plte. Although they re empiricl, the proposed models re bsed on the two-phse theory used in describing the bubble flow in fluidized bed. These models hve been vlidted, nd the results show tht they cn be used to predict the behvior in different regimes t different gs velocities. 1. INTROCTION ue to numerous dvntges, fluidized bed technologies hve wide industril pplictions. To ensure sufficient residence time for the recting gses, fluidized bed rector cn be operted in bubbling or nonbubbling regime. Nonbubbling fluidiztion is lso regrded s prticulte fluidiztion, nd it is often desired when high gs residence time is required. In the bubbling fluidized bed, there is higher trnsfer of het nd mss due to higher degree of solid movement, but this is t the expense of gs residence time. Prticle size is mong the fctors tht influence the fluidized bed regimes. For Geldrt A prticles, 1 fluidized bed psses through the prticulte regime before it begins to bubble when the gs velocity is further incresed, wheres for Geldrt B prticles, which cn be fluidized esily, bubbles pper in the bed s soon s the minimum fluidiztion velocity is exceeded. Mndl et l. 2 show tht bed of Geldrt B prticles cn exhibit nonbubbling fluidized bed behvior t higher gs velocity when it is formed within the interstitil void spce of lrge nd sttionry prticles. Similr to internls such s verticl tubes nd bffles, the lrge prticles serve s bubble brekers, preventing rise nd flow of bubbles in the binry beds. In this study, the focus is on the bubbling fluidized beds often pplied in smll-scle rectors. esigning bubbling fluidized bed rector, especilly in the preliminry stge, my require knowledge bout the verge bed properties. For given gs velocity, the verge bubble dimeter nd volumetric bubble flux re importnt prmeters tht give n indiction of bed expnsion. Severl correltions 3 6 found in the literture provide the bubble dimeter t ny position long the xis of the bed. For b h ( ).588 given superficil gs velocity bove the minimum fluidiztion velocity, these correltions give the sme bubble dimeter independent of the prticle chrcteristics. This my probbly be becuse most of these models re developed bsed on the two-phse theory proposed by Toomey nd Johnstone. 7 According to the two-phse theory, the gs velocity in excess of the minimum fluidiztion velocity constitutes the bubble flow in the bed. On the bsis of this ssumption, different beds of prticles hve the sme volumetric bubble flux t the sme excess gs velocity independent of the prticle properties. However, studies of Hilligrdt nd Werther 8 nd Grce nd Cliff 9 showed tht the ctul volumetric bubble flux is lower thn tht given by the two-phse theory. These findings indicte tht mny of the existing bubble dimeter models my not be pproprite for ll systems. Moreover, the bubble dimeter nd volumetric bubble flux t the sme excess gs velocity hve been observed to vry between different types of prticles. Severl fctors, which include prticle shpe nd size distribution, cn be responsible for this devition. Accurte prediction of bubble dimeter in deep fluidized beds hs lso been chllenge for most of the vilble models becuse they re developed for freely bubbling beds. For deep bed where there is possibility of slug flow, none of these models hs been found to predict the behvior in the slugging regime. Even though they re developed for freely bubbling Received: October 22, 217 Revised: Jnury 3, 218 Accepted: Februry 1, 218 Published: Februry 1, 218 218 Americn Chemicl Society 2658 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
Figure 1. () Physicl view of cold fluidized bed with dul-plne ECT sensors for mesurement of the solids frction distribution. (b) Cross section of the bed divided into 812 pixels. 2659 beds, the predictbilities of these different models lso differ from one system to nother. Krimipour nd Pugsley 1 reported tht bubble dimeters in the beds of Geldrt B prticles cn be best predicted using the models developed by Choi et l. 3 nd Mori nd Wen, 4 while for Geldrt A nd prticles, the correltion of Ci et l. 11 is best suited. As most of these correltions re either fully empiricl or semiempiricl, the mesurement techniques used to cquire the experimentl dt for their developments my lso influence their ccurcies. Although the use of photogrphic techniques (video imging) in two-dimensionl (2) beds provides dequte informtion bout the bubble growth, bubble shpe nd bubble sptil distribution, 12 this informtion my not be pplicble for threedimensionl (3) beds. Most of the techniques used for 3 systems do not mesure the bubble dimeter directly. epending on the technique, the informtion cquired during the bubble pssge is nlyzed to obtin the relevnt bubble properties such s bubble holdup, bubble size, nd bubble rise velocity. Since these properties re inter-relted, mesurement of ny one of them is often used to determine the other properties. 12 X- nd γ-ry bsorption techniques re widely pplied to obtin properties of single rising bubble. 5,13,14 For beds with lrge dimeter, these methods fil to provide ccurte bubble properties due to difficulties to identify prticulr bubble in the presence of lrge number of different bubbles. ifferent types of probes re lso employed to mesure the bubble properties. 3,5,15 18 The needle cpcitnce probes re extensively used 5,15 despite their low signl-to-noise rtio. In generl, the techniques bsed on the use of conductivity, inductnce, nd cpcitnce probes provide informtion bout the locl bubble size, but to obtin the verge bubble dimeter t ny cross-section of the bed requires considerble effort. 12 Being intrusive devices, probes lso hve the cpcities to chnge the hydrodynmics of their surroundings, thus cquiring flse results. Viswnthn nd Ro 12 obtined the bubble holdup from pressure mesurements, nd provided n itertive procedure for determining the bubble dimeter by bck-clcultions using the relevnt correltions relting the bubble rise velocity, bubble holdup, nd bubble dimeter. The im of this study is to develop set of models for obtining the verge bubble volumetric flux nd bubble dimeter in deep fluidized beds. Frshi et l. 19 highlighted four different wys to compute the verge bubble dimeter, which is lso clled the effective dimeter. Ech of these methods depends on the totl bed height, nd the simplest of them is by finding the bubble dimeter t the middle of the bed. ue to vriety of concepts involved, these different methods my give different results. This pper presents models tht re independent of the totl bed height for obtining the verge bubble dimeter nd verge volumetric bubble flux. The models re bsed on the nlysis of informtion obtined from n experimentl setup equipped with dul-plne electricl cpcitnce tomogrphy (ECT). Being noninvsive technique, number of reserchers 2 22 hve used ECT in their studies to chrcterize bubbling fluidized beds. ECT sensors provide dequte informtion bout the solids frction distribution, which cn be nlyzed to obtin different bubble properties t given plne in fluidized bed. In this study, sets of ECT dt re cquired nd nlyzed with different MATLAB codes. From the dt nlysis, relevnt bubble properties re found, which re then used to develop the models for determining the gs velocity t trnsition between bubbling nd slugging regime, the verge bubble volumetric flux nd the verge bubble dimeter t different gs velocity. The experimentl method used for the dt cquisition is presented in the following section, while the detils for the proposed model development re given in the subsequent sections. 2. ATA ACQISITION 2.1. Experimentl Setup. In this pper, the experimentl setup used to cquire the necessry dt consists of cylindricl OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
Tble 1. Properties of ifferent Prticles Investigted in This Work mterils men prticle dimeter [μm] density [kg/m 3 ] Geldrt group column of 14 mm internl dimeter equipped with dulplne ECT sensor s shown in Figure 1. The sensors re locted t two different positions: 15.7 nd 28.7 cm bove the gs distributor. Ech sensor consists of 12 electrodes, uniformly distributed round the mesurement plne. The cross-section of ech sensor is divided into 32 32 squre pixels, of which 812 pixels lie within the bed s shown in Figure 1b. Ech pixel holds normlized reltive permittivity between nd 1. The normlized reltive permittivity ϵ r is mesure of volume frction of solids in the bed. The volume frction of prticles ε s t ny point in the plne is obtined from ε s = ε s ϵ r, where ε s is the fixed bed solids frction. More detil bout this setup cn be found in Agu et l. 23 The experiments were conducted using seven different types of prticles. The properties nd Geldrt clssifiction of these prticles re given in Tble 1. The minimum fluidiztion velocity of these different prticles were obtined from this setup. The prticle densities for the different powders were mesured with gs pycnometer nd the prticle sizes were obtined from the sieve nlysis. The verge sphericity of the prticles is difficult to mesure, lthough the pproximte vlue cn be obtined by fitting experimentl pressure drop dt to pressure drop model 24 such s Crmn Kozeny 25 nd Ergun 26 equtions. However, the vlue of sphericity obtined from this method my differ from one pressure drop eqution to nother. For the purpose of model development, the sphericity vlues given in Tble 1 correspond to the verge of those listed in the literture for the sme mterils. The initil bed height in ech of the experiments lied between 4 nd 65 cm. For the Geldrt B glss nd limestone prticles, the experiments were performed with three different initil bed heights, 52, 58, nd 64 cm, to obtin the influence of bed height on the fluidized bed behvior. In the experiments, dry compressed ir ws used. The ir velocity bove the minimum fluidiztion velocity for the different types of prticles ws vried in the rnge of.5.4 m/s. For ech ir velocity, the imges of the solids distribution t the mesurement plnes were cptured. The imge dt were recorded for 6 s t frequency of 1 Hz. Figure 2 is n exmple of the solids frction distribution obtined during the experiments. The higher vlues on the figure color br indicte higher solid concentrtions. The flow of bubbles cn be observed in the regions where the solid concentrtion pproches zero. Considering tht bubbles contin some mount of solids, 24 ny region bounded by the solids frction between nd.2 is regrded s bubble in this work. On the bsis of this bubble solid threshold, different bubbles re identified. The bubble properties re clculted using the imge processing toolbox in MATLAB. The number of pixels occupied by bubble t ny given time is obtined nd mpped into the ctul bubble projected re using A = A N b b N, pix sphericity [-] fixed bed solids frction [-] minimum fluidiztion velocity [cm/s] glss beds 188 25 B 1..63 4. glss beds 261 25 B 1..62 8.15 limestone 293 2837 B.65.51 14. snd 483 265 B.72.55 17.5 glss 624 25 B/ 1..62 23.3 limestone 697 2837.65.49 39.24 moleculr sieve 217 13 1..6 76.85 266 Figure 2. Fluidized bed behvior obtined t the ECT lower plne for the 261 μm glss prticles. () istribution of solids where numbers in the color br give the solids volume frction. (b) Region occupied by the ctul bubble (white) nd region defined by pproximtely sphericl bubble (bounded by red circle). where A is the cross-sectionl re of the bed, N b is the number of pixels occupied by the bubble, nd N pix = 812 is the totl number of pixels within the plne. The chnges in the vlues of A b with time re used to obtin other properties such s bubble frequency nd bubble volumetric flow rte s described in the following section. 2.2. Mesurement of Bubble Properties. Anlysis of the experimentl dt shows tht the pssge of bubbles through given plne is in regulr periodic mnner. Figure 3 is the Figure 3. Evolution of bubble-projected re, showing the ctive nd idle periods in deep bed. Symbols: T b verge ctive bubble period, T i verge idle period, T b totl bubble period, nd A b verge bubble cross-sectionl re. vrition of bubble-projected re with time, which is typicl for ll the beds studied in this work. The projected re increses from zero to pek vlue nd then decreses to zero s the bubble psses through given plne. This vrition indictes tht bubble shpe is either sphericl or ovl. When bubble first rrives plne, its projected re is zero. The bubble projected re decreses to zero from pek vlue immeditely the bubble leves the plne. The pek of the projected re represents the bubble cross-sectionl re through its center. OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
Figure 4. Effect of bed height on the bubble dimeters mesured t 28.7 cm bove the distributor: () 188 μm glss prticles, (b) 261 μm glss prticles, nd (c) 293 μm limestone prticles. The time intervl between when the bubble rrives nd when it completely leves the plne is described s the ctive bubble period. The bubble dimeter cn be best determined from the bubble equivlent volume. 27 In this study, the 2 ECT sensors employed only provide informtion bout the bubble crosssectionl re nd none for the bubble height, mking it difficult to mesure the bubble volume directly. Assuming sphericl bubble, n pproximte bubble size cn be obtined from the pek of the projected res. The time-verge bubble dimeter cn therefore be described by 1 4A b,i db = n π (1) where n is the number of peks of the projected res recorded over the mesurement period nd A b,i is the pek of the projected res during the individul bubble pssge. The ctive bubble frequency f b is obtined s the reciprocl of the ctive bubble period: 1 f = b Tb (2) T b = 1 n T bi Here T b is the time-verge of the individul ctive bubble periods, T bi. It should be noted tht the concept of ctive bubble period nd frequency re introduced in this work, nd (3) tht the true bubble period T b is represented by the sum of the ctive nd idle periods s shown in Figure 3. The true bubble frequency is lower thn the ctive bubble frequency since T b > T b. The volumetric bubble flux G is mesured by considering the volume of bubble tht psses through n observer plne of unit re in unit time. Considering tht the ctive bubble period is the time for complete bubble pssge s shown in Figure 3, the volumetric bubble flux cn be expressed s vb G = ATb (4) where v b is the volume of bubble tht psses through given plne within the time period T b. For sphericl bubbles, v = π b d 6 b3, nd eq 4 cn be rewritten s π G = 6A f d b b 3 (5) 3. MOEL EVELOPMENT, RESLTS, AN ISCSSION As stted erlier, the proposed models for predicting the verge bubble volumetric flux nd bubble dimeter re independent of initil bed height within the bubbling nd slugging regime. This sttement is first discussed here. Figure 4 shows the bubble dimeters mesured t 28.7 cm bove the distributor in different beds of prticles: 188 μm glss prticles, 261 μm glss prticles, nd 293 μm limestone 2661 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
prticles. For the beds of 261 μm glss prticles, the bubble dimeter is independent of the initil bed height t the sme gs velocity. For the 293 μm limestone nd 188 μm glss prticles beds, there re lso no differences in the corresponding bubble dimeters when the initil bed height is incresed from 58 to 64 cm. However, for the height of 52 cm, the bed of 188 μm glss prticles shows significnt increse in bubble dimeters while tht of 293 μm limestone prticles shows decrese in bubble dimeters compred with the vlues recorded t the bed height of 58 cm, lthough this effect seems to decrese with incresing gs velocity. These results show tht the bubble dimeter is independent of bed height when the bed height is reltively high. Therefore, the models developed in this section re to be pplied in deep beds with lrge spect rtios (bed height to bed dimeter rtio). 3.1. Model for Averge Volumetric Bubble Flux. According to Grce nd Clift, 9 the volumetric bubble flux cn be expressed s G = k (6) Eqution 6 is form of modified two-phse theory describing the bubble flow rte in fluidized bed, where the prmeter k ccounts for devition of the theoreticl bubble flow rte from the ctul bubble volumetric flow rte. The vlue of k my vry depending on the superficil gs velocity, bed properties nd verticl position in the bed. 28 In freely bubbling beds, Choi et l. 29 obtined correltion between the vlue of k nd the gs velocity rtio s given in eq 7, where nd c re constnt with vlues of.62 nd 1., respectively. = k c (7) In the present work, vlues of k re obtined nd nlyzed. ( G) From the mesured volumetric bubble flux, k = cn be determined. Figure 5 shows the verge vlues of k ginst the gs velocity rtios for four different beds: 483 μm snd, 293 μm limestone, 261 μm glss, nd 188 μm glss. In ech bed, the trend of vrition in k chnges s the bed trnsits from bubbling to slugging regime with incresing gs velocity. The dt in the two different regimes cn be fitted with seprte stright lines s shown in the figure. The extension of the fitting lines beyond the dt points is rbitrry. For exmple, t the verticl intercept where = 1, slugs re never observed. The vlue of = 1 gives the minimum possible velocity for bubble to flow, nd depending on the prticle size, bubble my or my not exist t this velocity. However, since the lines re used to describe the behvior in the different regimes, the verticl intercepts s well s the line slopes re essentil. Also, s shown in Figure 5, the reltive stndrd error for ech of the fitting lines is smll, indicting tht the dt points cn be well described by the liner functions. The slope of ech line increses s the flow regime chnges from bubbling to slugging. In the bubbling regime, the intercept on the verticl xis is closer to zero for the lrger prticles, but increses s the prticle size decreses. This vrition is s expected since smller prticles require significntly higher vlues of for the bubble to rise in the beds. In ddition, the line slopes in this regime differ between the different beds. The slope decreses between 188 μm glss nd limestone nd increses therefter towrd the snd prticles. This behvior cn be ttributed to the vrition in size nd shpe between these prticles. However, in the slugging regime, snd nd glss prticles hve lmost the sme line slopes, which differ significntly from tht of limestone prticles. On the bsis of these liner reltionships shown in Figure 5, the expression k = c ( ) cn be used to describe the behvior in both bubbling nd slugging regimes, where nd c re the line slope nd the line intercept, respectively. As described bove, the vlues of nd c depend on the prticle size, shpe, nd fluidiztion regime. Further nlysis of these behvior (Figure 5) provides different expressions for vlues of nd c s given in Tble 2. For the slugging regime, the Tble 2. Correltions for the Proposed Model Prmeters nd c prmeters expressions vlidity Bubbling Regime φ 1.5 (4.168 1.389 log(ar)) log(ar) < 3.5 φ 1.5 (.329 1.156 1 3 Ar.9 ) log(ar) 3.5 c (1.321 + 8.161 1 4 Ar 1.4 ).83 log(ar) > Slugging Regime.725 +.23 log(ar) log(ar) < 3.9 1.184 + 8.962 1 4 Ar 1.35 log(ar) 3.9 c.42 +.18 log(ar) log(ar) < 4. (.978 1.964 1 2 Ar.8 ) 4.88 log(ar) 4. Figure 5. Vrition of k =( G)/ with gs velocity rtio /. Solid lines, bubbling regime; dshed lines, slugging regime. correltions re bsed on the beds of 188 μm glss, 293 μm limestone, snd, 697 μm limestone, nd the 3 moleculr sieves prticles. The correltions for the bubbling regime re bsed on the glss prticles (188 nd 261 μm), the snd prticles nd two of the Geldrt prticles (697 μm limestone nd 2.17 mm moleculr sieve prticles). Tble 2 shows tht the expressions for nd c vry between the bubbling nd slugging regimes nd tht these prmeters depend on the prticle Archimedes number, Ar = d 3 p ρ g (ρ p ρ g ) g/μ 2 g, where d p is the prticle dimeter, φ is the prticle sphericity, ρ p is the prticle density, nd g is the ccelertion 2662 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
due to grvity. ρ g nd μ g re the gs density nd dynmic viscosity, respectively. 3.1.1. Vlidtion of the Proposed Model for Volumetric Bubble Flux. sing the expressions for the prmeters nd c s given in Tble 2, the verge volumetric bubble flux in deep fluidized beds cn be obtined from = G c Figure 6 compres the verge volumetric bubble flux obtined from eq 8 with the experimentl dt. As shown in the figure, Figure 6. Computed verge volumetric bubble flux bsed on G = c ( ) (lines) compred with the experimentl dt (strs) used in the model development. these results re for the prticles used in developing the models in Tble 2. Quntittively, it cn be seen tht the model results re in good greement with the experimentl dt. The trnsition from bubbling to slugging regimes re well-cptured, nd the trends of the bubble flux in both regimes correspond with those of the experiments. Figure 7 shows the computed verge volumetric bubble flux ginst the gs excess velocity (8) for other sets of prticles lso studied in this work. It cn be seen tht the model prediction is lso in good greement with the experimentl dt within the given rnge of gs velocities. 3.2. Model for Gs Velocity t Bubble to Slug Trnsition. To pply the models in Tble 2 successfully t ny given gs velocity, model t the boundry between the bubbling nd slugging regimes is required. ifferent bubble slug trnsition models re vilble in the literture. 3 33 The trnsition models provide the velocity t the onset of slugging. Among the vilble models, the Beyens nd Geldrt 3 model is commonly used. = +.16(1.3 h ) +.7( g) ms.175 2.5. The Beyens nd Geldrt 3 correltion (eq 9) shows tht the minimum gs velocity required for slug to flow in fluidized bed depends on the prticle minimum fluidiztion velocity, the bed height, nd the bed dimeter, but the excess velocity ms is independent of the fluid nd prticle properties except where h chnges with these properties. In this section, model where ms is fully dependent on fluid nd prticle properties is developed. As shown in Figure 5, the trnsition from bubbling to slugging regime occurs t the point of intersection between the two different regime lines. At the intersection, the vlues of k from the two regimes re the sme: b s ms ms cb = cs (1) Here, b nd s re the corresponding vlues of in the bubbling nd slugging regimes, c b nd c s re the respective vlues of c, nd ms is the superficil gs velocity t the trnsition. With the vlues of nd c known in the respective regime, eq 1 cn be simplified: ms = c (9) t t (11) where c t = c b /c s nd t = 1/( s b ). Figure 8 compres the trnsition velocity rtios computed from eq 11 with those obtined in the experiment for the different powders. For the sphericl prticles, the results show tht the computed dt gree very well with the experimentl dt. The results differ significntly when the prticles re nonsphericl. This indictes tht t the onset of slugging Figure 7. Computed results (lines) bsed on the proposed model = ( ) G c from different beds. compred with the experimentl dt (strs) Figure 8. Computed vlues of gs velocity rtio t the trnsition from bubbling to slugging regime for different prticles using eq 11. 2663 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
regime, prticle shpe plys significnt role. In imtti et l., 34 the minimum slugging velocity is lso reported to depend on the prticle sphericity. Therefore, eq 11 cn be modified to ccount for the influence of prticle sphericity t the trnsition. By introducing fctor of φ.35 in eq 11, the errors ssocited with the computed vlues of ms for the nonsphericl prticles re minimized. ms = φ.35 c t t (12) Contrry to the Beyens nd Geldrt model, the results from eq 12 re independent of the bed height nd bed dimeter. This shows tht the trnsition velocity described by this model cn be ccurte when the bed is reltively deep, tht is, where h 4. To be ble to utilize eq 12 in beds with smller spect rtios, some modifictions re needed. Agu et l. 23 show tht the onset of slugging depends on the bed height especilly in the bed of smller prticles, nd s given by eq 9, this in generl should depend on both h nd. ms / 1 Figure 9 shows the rtio,, computed ginst the.35 ( φ c t t 1) Figure 9. Vrition of normlized gs velocity t slugging with bed height. t points: experiment; lines: Beyens nd Geldrt model, eq 9. bed spect rtio h for the different beds: 188 μm glss, 261 μm glss, nd 293 μm limestone prticles. For the bed of 188 μm glss prticles, the experimentl dt show continuous ms / 1 decrese in the vlue of with n increse in h,.35 ( φ c t t 1) but for the two lrger prticle beds, some degrees of sctter cn be observed. However, the results from the Beyens nd Geldrt model suggests tht the vrition of this normlized slug velocity rtio with h is liner with constnt slope when h < 6.5 for ll the beds. On the bsis of this linerity, the following reltionship cn be derived: ms/ 1 t ( φ c 1) h = β.35 t α (13) Here, α is the slope of the line, tken to be constnt for ll the beds, nd β is the intercept on the verticl xis, which decreses with incresing prticle size s cn be seen in Figure 9. Figure 1 shows tht the three sets of the experimentl dt cn be fitted with different stright lines of the sme slope. It Figure 1. Normlized minimum gs velocity for slugging fitted with constnt slope lines. cn be seen clerly tht s the prticle size increses, the degree of dt sctter increses. imtti et l. 34 lso reported similr sctter vrition. The sctter vrition indictes tht the dependency of the minimum slugging velocity on the bed height my be insignificnt when the prticle size is lrge. The slope of ech line in Figure 1 is α =.588. The vlue of the intercept β is found to depend on the prticle minimum fluidiztion velocity by the expression β = γ θ, where γ = 2.33 nd θ = 1.27. From these results, the onset of slugging velocity cn be obtined s function of bed spect rtio s described by eq 14..588 ms.27.35 h h = 1 + 2.33 ( φ c t t 1) ; 1.5 < < 7.2 (14) The coefficient.27 in eq 14 ccounts for the bed expnsion bove the height t fixed stte during the trnsition. The bed height expnsion is lso ccounted for in the Beyens nd Geldrt model by replcing h with h. Note tht in both eqs 9 nd 14, is mesured in m/s. Equtions 14 nd 9 gree very well within the spect rtio rnge of 1.5 7.2, nd this is tken s the rnge of vlidity of this model until further verifiction is obtined. 3.2.1. Vlidtion of Proposed Model for Onset of Slugging Regime. Eqution 14 shows tht both ms nd ms depend on the fluid nd prticle properties. This mkes the model more robust to predict the onset of slugging velocity in different systems with vrying operting conditions, including temperture nd pressure. However, relibility of this model lso depends on its performnce when compred with results from other setups or correltions. Figure 11 compres the minimum slugging velocity computed from eq 14 with those obtined in the experiments reported by Singh nd Roy. 35 The vlues bsed on the Beyens nd Geldrt model re lso shown for comprison. The bed height t minimum fluidiztion condition, = ε sh h, used in 1 ε eq 9 is bsed on the verge bed height h = 55 cm chrcterizing the present work. The vlues of solids frction ε s in fixed stte re given in Singh nd Roy 36 for the sme set of prticles. The void frctions t minimum fluidiztion ε re 2664 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
Figure 11. Computed minimum gs velocity for slug flow t different bed heights nd bed dimeters. Figure 12. Response of the model R = γ obtined ccording to Wen nd Yu 37 bsed on sphericity of.7 for ll the powders. As shown in Figure 11, the proposed model, eq 14 grees very well with the Beyens nd Geldrt model, nd the results from both models re in good greement with the experimentl dt. With different bed height, h = 25 cm, nd the bed dimeter = 1.16 cm reported in Singh nd Roy, 35 eqs 9 nd 14 lso gree very well with ech other. Compred with eq 9, the proposed model responds very well to the vritions in the bed height to bed dimeter rtio. The results from both models lso show tht the effect of bed height is insignificnt when incresing the prticle size. 3.2.2. Sensitivity of the Model, Eqution 14, to the Fitting Prmeters α, γ, θ, nd. Although the results presented in Figure 11 show tht the proposed bubble-slug trnsition model cn predict the onset of slugging regime with good ccurcy, the model vlidity depends on the prticle size due to h θ h ( ) α for chnges in the model prmeters t different bed heights. () 188 μm glss prticles (b) 293 μm limestone prticles. Colored lines: = 5. (solid), = 5.6 (dshed), nd = 6.2 (dotted). h h uncertinty in mesurement. In Figure 1, the slopes of the ctul lines tht cn fit seprtely the dt from the three different sets of prticles differ from the verge vlue,.588 used in the proposed model. The ctul intercept of ech line 1.27 lso differs from tht given by the correltion β = 2.33 following the devition in the corresponding line slope. ue to these devitions, the mximum error ssocited with the righthnd side of eq 13, R = γ( ), lies between 15 nd θ h α +1% for ll the bed heights. It should be noted tht chnges in the model prmeters α, γ, θ, nd from their bse vlues my cuse significnt chnge in the model mximum error. On the bsis of this, it will be interesting to check the model sensitivity to these prmeters within possible rnge of chnges. Figure 12 shows how the model responds to smll chnges in ny of the four prmeters. These results show tht the model 2665 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
sensitivity is not ffected by chnges in the bed spect rtio for chnges in ny of the prmeters within ±1%. For the chnges in the prmeters α, γ, nd, the sensitivity is independent of the bed prticles within the ±1% chnges. Any slight increse in the prmeter θ from the nominl vlue results in rpid increse in the model output, lthough this effect seems to decrese with n increse in the prticle size. Hence, due to this high sensitivity, the nominl vlue θ = 1.27, should be mintined in the model. As cn be seen, chnge in α within ±15% hs the sme effect on the model output s the sme chnge in. Within ±1%, chnge in γ hs the sme mgnitude, but the opposite effect s n equl chnge in α or. This mens tht ny chnge pplied to α should be pplied to γ to minimize the model error. Since is lso vrible in the model, it follows tht the vlue of γ cn be vried from the bse vlue ccording to the uncertinty in mesurement or estimtion of. 3.3. Model for Averge Bubble imeter. As shown in Figure 3, the ctive bubble frequency depends on the bubble size. As the bubble size increses, the time tken by the bubble to pss through given plne increses. Figure 13 shows how the bubble frequency chnges with the bubble dimeter. The plotted dt re obtined from nine different beds of three different types of prticles, 188 μm glss, 261 μm glss, nd 293 μm limestone, with three different bed heights, 52, 58, nd 64 cm. The plot includes ll the dt obtined from both plnes (15.7 nd 28.7 cm bove the distributor) for ech bed. The result indictes tht the reltionship between the bubble frequency nd the bubble dimeter is independent of bed height nd cn be described by curve with the following function: f b = 1.927 d b 1.48 (15) Eqution 15 shows tht when the bubble dimeter is s lrge s the bed dimeter, the ctive bubble frequency is reduced to 1.93 s 1. With eq 15, the volumetric bubble flux described by eq 5 cn be written s π G =.321 A d b 1.48 db 3 (16) Substituting π 4 2 for A, eq 16 cn be simplified to = G m d b 1.52 (17) Figure 13. Reltionship between the ctive bubbling frequency nd bubble dimeter. where m is constnt with vlue of 1.285 s 1. Keeping the units of d b nd the sme, the unit of G is thus m/s, cm/s, or mm/s depending on wht unit ssigned to the bed dimeter,. The results from eq 17 re compred with the experimentl dt s shown in Figure 14. The verge bubble dimeters used in these results re those obtined from the experiments with the different types of prticles. As cn be seen in Figure 14, the model predicts the behvior in the different beds with resonble ccurcy. For the beds of prticles shown in Figure 14b, the model ccurcies re s good s those obtined from the three beds used in the model development, prticulrly in the bubbling regime. Moreover, the results show tht the model Figure 14. Computed verge volumetric bubble flux bsed on = ( db ) Comprison with dt from other different beds. 1.52 G m () Comprison with dt used in the model development. (b) 2666 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
predicts well the similr behvior observed in the different beds of the sme mteril. Since the results from both models, eqs 8 nd 17, gree very well with the experimentl dt, combintion of these models cn be used to obtin the verge bubble dimeter in deep fluidized beds t different gs velocities. Assuming tht ll the bubbles pssing over given bed t given gs velocity is represented by single bubble with verge dimeter d b, combintion of eqs 8 nd 17 gives db c = 1.285 1.52 (18) Recsting eq 18, the model for verge bubble dimeter is given by db =.848 c.66.34 (19) 3.3.1. Vlidtion of the Proposed Model for Averge Bubble imeter. For given gs velocity, the verge bubble dimeter in deep fluidized bed cn be predicted using eq 19. The unit of gs velocity in this empiricl model must be in per second nd must correspond to ny unit ssigned to the bed dimeter. To vlidte this model, the computed verge bubble dimeters for different beds of prticles re compred with the experimentl dt s shown in Figure 15. These results re ( )[ db db 1.132 h] +.474 g ( d db ) = 1.63 d = [ Ac ( )].4 b.2 g (2).4 db =.652[ A( )] (.652[ A( )] db) h exp.3 2 db =.376( ) (21) Here, h [cm] is verticl position in the fluidized bed, A c [cm 2 ] is the ctchment re described in rton et l., 5 g is in [cm/ s 2 ], nd nd re in [cm/s]. The bed verge bubble dimeters bsed on these models, eqs 2 nd 21, re obtined by integrtion tken between the two mesurement plnes, 15.7 nd 28.7 cm bove the gs distributor. The results from these three models, the present work, the Choi et l. 3 model, nd the Mori nd Wen 4 models, re shown in Figure 16 for three different beds of prticles. The figure.5 b 1.5 1.5.4 Figure 16. Predictbility of the proposed model d b =.848 c.66 ( ).34 compred with those of existing models eqs 2 nd 21. Figure 15. Computed verge bubble dimeter bsed on the proposed.66 model =.34 d c b.848 ( ) compred with the experimentl dt used in the model development. bsed on the set of prticles used in formulting the model. The results show tht strong greement exists between the model nd the experimentl dt within the rnge of gs velocities shown. Further vlidtion of this model for verge bubble dimeter is obtined by compring its results with those from the existing models. In this cse, the models proposed by Choi et l. 3 nd Mori nd Wen 4 re considered since both models re widely pplied in predicting the bubble dimeters. The Choi et l. nd Mori nd Wen models re s described in eqs 2 nd 21, respectively. shows tht the bubble dimeters computed with the present model gree very well with the experimentl dt in ll the beds. Ech of the Choi et l. nd Mori nd Wen models predicts the sme bubble dimeter in the different beds t the sme excess gs velocity,. Within the rnge of the excess gs velocities shown, the results from the Choi et l. model re closer to the experimentl dt if verged compred with those from the Mori nd Wen model. While neither Choi et l. nor Mori nd Wen model predicts the behvior in the slugging regime, the present model resonbly predicts this behvior. This bility to predict the bubble dimeters in different regimes of the deep fluidized beds mkes the present model superior to these two other models previously described in the literture. 4. SMMARY OF THE PROPOSE MOELS The models developed in this pper for predicting the verge volumetric bubble flux, the verge bubble dimeter, nd gs velocity of trnsition from the bubbling to the slugging re summrized in Tble 3. The min ssumption of these models is tht within the bubbling or the slugging regime the verge 2667 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
Tble 3. Proposed Models for Averge Bubble Flux, Bubble imeter, nd Bubble to Slug Trnsition Velocity volumetric bubble flux nd bubble dimeter re independent of the initil bed height, h. However, the expressions for the trnsition velocity indictes tht the bed height is n importnt prmeter for determining the regime of opertion. The dependency of the model prmeters nd c on the prticle nd fluid properties mkes it possible for the model to predict unique bubble dimeter in fluidized beds of different prticles with the sme excess gs velocity,. It should be noted tht the expressions for G nd d b re discontinuous bs over the entire rnge of gs velocity 1 < <. The discontinuity over this velocity rnge is due to the expressions for nd c tht re different in the two different regimes. bs However, within ech of the regimes, 1 < < nd fluidized bed prmeter bs >, the expressions for G nd d b re continuous nd differentible. 5. CONCLSIONS A fluidized bed cn be operted in bubbling or nonbubbling regime depending on the Geldrt clss of the bed prticles. For bubbling fluidized bed, the bubble properties lso depend on the prticle properties nd fluidized bed regime (freely bubbling or slugging), mking their ccurte predictions chllenge. This pper presents set of new models for predicting the verge volumetric bubble flux, verge bubble dimeter nd gs velocity t the trnsition between bubbling nd slugging regimes in deep fluidized beds: Bubble flux: = G c Bubble dimeter: d =.848G b Trnsition velocity: bs.66.34 model volumetric bubble flux G = c ( ) bubble dimeter =.848 ( ) gs velocity t bubble to slug trnsition.66.34 d c b ms.27.35 h t = 1 + 2.33 ( φ ct 1).588 The model prmeters, c, t, nd c t depend on the fluid nd prticle properties, nd their correltions with these properties re lso presented in this pper. In the slugging regime where bs >, the sme models re pplied but with different correltions for the prmeters nd c. These models re developed bsed on the nlysis of dt obtined from cylindricl setup equipped with dul-plne electricl cpcitnce tomogrphy. Although the models re empiricl, they re lso bsed on the two-phse theory used in describing the bubble flow in fluidized beds..27 = 1 + 2.33 ( φ c 1) h ( ).35 t t.588 These models hve been tested with different types of prticles hving men dimeters in the rnge of 13 22 μm, nd their results re consistent with different experimentl dt. The models cpture the behvior in different regimes of deep fluidized beds t incresing gs velocity. The dependency of the model for verge bubble dimeter on the bed dimeter increses its pplicbility for design purposes. However, these models require further vlidtion with experimentl dt bsed on different mesurement techniques s well s bed height to dimeter rtio less thn 4. ATHOR INFORMATION Corresponding Author *E-mil: cornelius.e.gu@usn.no. ORCI Cornelius E. Agu: -2-5339-9794 Notes The uthors declre no competing finncil interest. NOMENCLATRE A bed cross-sectionl re, m 2 Ar dimensionless prticle Archimedes number dimensionless fitting index c dimensionless fitting coefficient bed dimeter, m d dimeter, m d verge dimeter, m f frequency, s 1 G volumetric bubble flux, m/s g ccelertion due to grvity, m/s 2 gs h verticl position in the bed, m h initil bed height, m i index k dimensionless two-phse bubble flow devition coefficient log logrithm function to bse 1 m dimensionless model coefficient N, n numbers R right-hnd-side of model T period, s superficil gs velocity, m/s v volume, m 3 Greek Symbols α dimensionless fitting index β fitting coefficient, s/m ε dimensionless void frction ε s dimensionless solids frction θ dimensionless fitting index ρ density, kg/m 3 φ dimensionless prticle sphericity μ dynmic viscosity, P s γ fitting coefficient, (m/s) θ 1 Subscripts b bubble b ctive bubble bs bubble to slug trnsition g ccelertion due to grvity, m/s 2 gs p prticle s solid t trnsition 2668 OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669
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A Generlized Method for Predicting the Minimum Fluidiztion Velocity. AIChE J. 1966, 12, 61. OI: 1.121/cs.iecr.7b437 Ind. Eng. Chem. Res. 218, 57, 2658 2669