Article history:Received 22 April 2011Received in revised form 3 September 2011Accepted 23 October 2011Available online 29 October 2011
Keywords:Volume-change of mixingBinary-solid mixtureIncipient fluidizationBed void fractionPacking models
The component additive rule for the prediction of the volume of fluidized mixtures of binary-solids does nothold owing to the phenomenon of the volume-change of mixing. A significant contraction in the volume ofthe mixed bed is often observed. This phenomenon has an important bearing on the hydrodynamics of thebinary-solid fluidization. In the present, the volume-change of mixing is experimentally investigated for sev-eral liquid-fluidized binary-solid mixtures for a wide range of compositions at the incipient fluidization.Seven different binaries are composed from eight different solid species that differ in their sizes as well asdensities, and therefore exhibit a wide range of stratification pattern. The use of packing models reportedin the literature results in rather poor predictions of the observed volume-change of mixing. Incorporatingthe hydrodynamic aspects of the fluidization however helps to substantially improve predictions. A two-level integration of the hydrodynamic information is implemented here. The level one involves using the hy-drodynamic drag diameter evaluated from the expansion of individual solid species instead of the commonlyused volume-equivalent or the packing-equivalent diameter. The level two in addition involves accountingfor the expansion of either of the solid species should it occur at the incipient fluidization of the mixture.
A great degree of uncertainty in the specification of bed void fractionoften arises in the case of binary-solid mixtures. This is due to the factthat the void fraction of the mixed bed is significantly different fromeither of its constituent solid species or any linear combination thereof.Depending upon the relative composition of the binary-solid mixtureand the difference in physical properties of constituent species, themixed bed often shows a substantial degree of the volume-change ofmixing. That is, the overall height of the mixed bed is mostly differentfrom the sum of the twomono-component beds of individual solid spe-cies comprising the binary mixture [1]. The volume-change of mixingmainly arises as a result of the smaller solid species occupying the inter-stitial spaces of their larger counterpart, thereby leading to a contrac-tion of the volume of the mixed bed structure.
An accurate description of the bed void fraction is an important pre-requisite for determining various hydrodynamic aspects of the fluidizedbed including the minimum fluidization velocity of binary-solid mix-tures. There is nonetheless a lack of literature on the phenomenon ofthe volume change of mixing even though its existence at the incipientfluidization has been reported [2–6]. Using the limited data available inthe literature, Asif [7] showed that incorporating the bed contractionarising from the volume-change of mixing leads to a substantial im-provement in the prediction of the minimum fluidization velocity.
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In the present study, experimental data involving the volume-changeof mixing at the incipient fluidization is presented for binary-solid mix-tures. Eight different solid species are used to compose sevendifferent bi-naries. Their constituent species differ in the size aswell the density suchthat the larger species is lighter whereas the smaller species is heavier.Using water with its temperature carefully controlled as the fluidizingmedium, slow defluidization runs were carried out to obtain the voidfraction of the mixed bed at the incipient fluidization. As a first step to-ward the prediction, models proposed in the literature for the packingof unequal solid species are employed here without any modification.In order to improve their relevance to the present case, the hydrodynam-ic information was incorporated in these models. This was implementedin a two step approach. First, the hydrodynamic drag diameter was eval-uated from the expansion behavior of individual species. The drag diam-eter thus obtained was used instead of the volume-equivalent or thepacking-equivalent diameter commonly used with packing models.The capability of this approach was further enhanced by accounting forthe fact that the species with the lower minimum fluidization velocitycould be partially fluidized at the incipient fluidization of the binary-mixture. The overall error in each of these cases is computed to evaluatethe efficacy of the approach suggested here.
2. Predictive models
The volume-change of mixing, ΔV, can be defined as follows:
362 M. Asif / Powder Technology 217 (2012) 361–368
Here, V1 and V2 are specific-volumes of pure species 1 and species2, respectively while VM is the specific-volume of the mixture. Notethat the specific-volume is the volume of the bed per unit volume ofthe solid species. The term in the parentheses represents the simplearithmetic averaging of the specific-volume of the two speciesweighed according to their fluid-free volume fraction. Thus, a nega-tive volume-change of mixing represents a contraction of the volumeof the mixed bed. It is customary in the fluidization literature to usethe bed void fraction instead of the specific-volume. The two arehowever related as:
V ¼ 11−ε
: ð2Þ
The prediction of the specific-volume or the bed void fractionusing the packing models has already been discussed in connectionwith the expansion behavior of binary-solid fluidized beds [8–10]. Itis nonetheless worthwhile to briefly present the same here in orderto make the present discussion self-contained. Needless to mention,a good deal of literature concerning the packing behavior of binary-solid mixtures has been reported [11–15]. These models account forthe observed volume-change of mixing arising from the mixing oftwo unequal solid species. For example, the simple model based onthe Westman equation is given as [16]:
VM−V1X1
V2
� �2þ 2G
VM−V1X1
V2
� �VM−X1−V2X2
V1−1
� �
þ VM−X1−V2X2
V1−1
� �2¼ 1:
ð3Þ
The parameter G depends upon the size ratio of the two compo-nents of the packing. It is easy to see that setting G=1 in the aboveequation yields VM=X1V1+(1−X1)V2, thereby meaning that thevolume-change of mixing is zero. Using a large base of data, Yu etal. [17] have proposed the following relationship for G:
1G
¼ 1:355 r1:566 r≤0:824ð Þ1 r > 0:824ð Þ�
ð4Þ
where r is the size ratio (smaller to larger) of the two solid species. Onthe other hand, Finkers and Hoffmann [18] have suggested anotherexpression for the parameter G in the Westman equation. Their ap-proach makes use of the structural ratio rather than the diameterratio, and is equally applicable for both spherical and non-sphericalparticles. This is given by,
G ¼ rkstr þ 1−ε−k1
� �; rstr ¼
1=ε1ð Þ−1ð Þr31−ε2
" #ð5Þ
where a value of exponent k=−0.63 has been recommended by theauthors. Another model, suggested by Yu et al. [14], is applicable forpredicting the porosity of non-spherical particle mixture with twoor more components. This model, known as the modified linear-packing model, is given by,
VTi ¼ Vi þ
Xi−1
j¼1
Vj− Vj−1� �
g rð Þ−Vi
h iXj þ
Xnj¼iþ1
Vj−Vj f rð Þ−Vi
h iXj ð6Þ
V ¼ max VT1;V
T2;::::::;V
Tn
h ið7Þ
where V is the overall specific-volume of the multi-solid packing sys-tem of different size, and Vi is the mono-component specific-volume
of the ith component. The functions f and g in the above equationare given by
where r is the size ratio. Zou and Yu [19] proposed the following rela-tionship for non-spherical particles,
dvidpi
¼ ψ2:785i exp 2:946 1−ψið Þ½ � ð9Þ
where dvi and dpi are, respectively, the volume-equivalent diameterand the packing-equivalent diameters of the ith component. It canbe seen here that the overall specific-volume of the mixed bed de-pends mainly upon three factors; the specific-volumes of pure spe-cies, the composition and the size ratio of the constituent species.
It is worthwhile to mention at this stage that the volume-changeof mixing was initially believed to be negligible in the fluidization lit-erature [20]. Its occurrence was however clearly established forbinary-solid mixtures with large particle size difference, and its influ-ence on the bulk density of the mixed bed was highlighted [8]. The ca-pability of the packing models in predicting this phenomenon wasalso pointed out. It is important to point out here that even the aver-age particle diameter could qualitatively predict the volume-changeof mixing. It is therefore not surprising to find that approachesbased on using the average particle diameter are capable of describ-ing the layer-inversion phenomenon. Nevertheless, this approach ofpredicting the volume-change of mixing at the incipient fluidizationis not pursued here in the light of the well-established superiorityof packing models [9, 21]. Employing the Westman's packing modelfor predicting the void fraction of binary-solid fluidized beds at theonset of the layer-inversion, Escudié and Epstein [22] proposed thefollowing relationship for the parameter G:
G ¼ 7:05r0:1γ−3:83Ar0:121 ð10Þ
Here, r is the size ratio, γ ¼ ρs2−ρf
ρs1−ρf
!is the density ratio and Ar1
is the Archimedes number based on the physical properties of solidspecie 1. The parameter G in the above equation is deduced fromthe fluidized bed data of binary-solid species. It is interesting tonote that Eq. (10) shows a strong dependence on the density ratioof the two species while a relatively weak dependence on the sizeratio is observed.
3. Experimental
The test section used in the present experimental investigation con-sisted of a 1.5 m tall transparent Perspex column of 60 mm internal di-ameter preceded by a distributor. A flow-through circulation coolerwasused to maintain the temperature of the tap water at 20±0.2 °C in therecirculationwater tank. This was important in view of the fact that anychange in thewater temperature affects the viscosity, and consequentlythe pressure drop and the bed height. The flow rate of thewaterwas ad-justed using one of three calibrated flow-meters of a suitable range. Bedheights were read visually with the help of a ruler along the length ofthe column. The overall pressure drop along the bed was measuredusing two different manometers of significantly different ranges. Forsmall pressure drop measurements, an inverted air-water manometerwas used. A mercury manometer, on the hand, was employed for mea-suring high pressure drops. Observations included measuring the flowrate, the bed height and the pressure drop across the bed.
A critical component of the present experimental set up is theproper design of the distributor. This is important for the even distri-bution of the liquid flow and the elimination of dead zones and the
Table 1Physical properties and fluidization characteristics of solid samples used.
Solids Material Shape Size range Mean diameter Density Umf εmf
fluid channeling. Toward this end, a 90 mm thick perforated platewith a high density of 1.5 mm circular perforations drilled on a squarepitch was used. The open area of the distributor plate was kept at 4%by adjusting the number of perforations accordingly. Such a configu-ration ensures enough pressure-drop even at low fluid flow to ruleout the existence of dead zones in the distributor region [23–24].Both faces of the distributor were covered with a nylon mesh of neg-ligible pressure drop to prevent raining down or clogging of the dis-tributor with solid particles. The distributor was preceded by a0.5 m long calming section, which was packed with 3 mm glassbeads to eliminate any entry effects. In order to further verify theelimination of distributor effects in the present data, an indirectmethod based on the measurement of the pressure drop was alsoused here. This consisted of comparing the (−Δp/L) for the lower re-gion of the bed, located immediately above the distributor, with theoverall pressure gradient of the whole bed for the case of a mono-component packed bed of the sand for different liquid flow rates. Aclose agreement between two readings was noted, thus indicating auniformity of the liquid velocity in the bed.
Eight different types of solid samples were used in the presentstudy. Their physical properties, fluidization velocities and bed voidfractions at the incipient fluidization are tabulated in Table 1. Glassand sand samples were mostly the fraction retained between twoconsecutive standard sieves. The reported mean particle diameter ofthe sample was the arithmetic mean of the two sieve openings. Thesample A on the other hand consisted of 33 vol.% of the sand sampleof 212–250 μm range and 67 vol.% of sand sample of 250–300 μmrange. The volume-mean diameter of the sample A was computedto be 257 μm. Measured with the help of a vernier, both the meanequivalent volume diameter and the shape factor of the two largersolid species are also reported in Table 1. Although similar in thesize as well as the shape, their densities were significantly different,thereby leading to an almost 150% difference in their minimum fluid-ization velocities as seen in the table.
A total of seven different binary systems were composed using theeight different solid samples here. All these binaries were carefullychosen so that the larger constituent was lighter than its smallercounterpart. Such binary systems often exhibit a greatly varied strat-ification pattern as a function of the liquid superficial velocity andtheir composition. Moreover, the mono-component bulk-density
Table 2Binary systems studied.
Binary Components Size ratio,r
1 2
1 P A 0.092 P B 0.103 P C 0.174 P E 0.245 Q E 0.226 Q D 0.197 Q F 0.26
versus velocity curves of the two constituent species of each binaryhere also intersect, and are therefore capable of showing the layer-inversion phenomenon. In all cases, a wide range of binary makeupwas studied as shown in Table 2. The size ratio (r) and the reduceddensity ratio (γ) are also reported in the table.
A typical experimental run for a binary consisted of first adding aknown mass of the smaller species and carrying out both fluidizationas well as defluidization experiments. A known proportion of thelarger solid species was then added to the column and experimentswere again carried out by gradually changing the flow rate and re-cording the pressure drop. Usually, two or more runs were carriedout. Since the defluidization runs provided reproducible results inmost cases, the minimum fluidization velocities and the bed voidfraction at the incipient fluidization condition were evaluated usingthe pressure drop data during the defluidization. All data reportedhere were obtained by carrying out a gradual defluidization process.
4. Results and discussions
In the following, experimental data are first compared with pre-dictions of the conventional packing models. The hydrodynamic as-pects of the fluidization are bound to affect the volume-change ofmixing otherwise not noted with packed structures of unequal solidspecies. Incorporating local hydrodynamic details will entail a de-tailed description of the local bed behavior. An attempt is howevermade here by incorporating overall hydrodynamic features in orderto improve the model predictions.
4.1. Comparison of experimental data with packing models
Experimental data and corresponding predictions of differentmodels are shown in Figs. 1–7. Here, the ordinate is the percentvolume-change of mixing (χ) defined as:
χ ¼ ΔVVM
� �� 100: ð11Þ
On the other hand, the abscissa is the fraction of the larger species(X1) of the binary as shown in Table 2. Thus, X1=0 implies a bed con-sisting only of smaller species, and likewise X1=1 represents a
Fig. 2. Comparison of experimental data and predictions of models for Binary 2.
Fig. 4. Comparison of experimental data and predictions of models for Binary 4.Fig. 1. Comparison of experimental data and predictions of models for Binary 1.
364 M. Asif / Powder Technology 217 (2012) 361–368
mono-component bed of the larger species. The V1 and V2 at the incip-ient fluidization are evaluated as:
V1 ¼ 1−εmf1
� �−1;V2 ¼ 1−εmf2
� �−1 ð12Þ
where εmf1 and εmf2 are experimentally obtained values at Umf1 andUmf2, respectively as presented in Table 1. Note that a negative valueof the volume-change of mixing will indicate a contraction of thevolume indicating that the height of the mixed structure will beshorter than the corresponding height of the two segregated mono-component layers at the incipient fluidization.
Experimental data as well as predictions of different models areshown in Fig. 1 for the case of Binary 1, which is characterized bythe most significant difference of the size of constituent speciesamong all binaries considered here. A substantial volume-change ofmixing is clearly seen in the experimental data. This trend is qualita-tively described by packing models presented earlier. Note thatthe value of the exponent k=−0.345 is used in Eq. (5) instead ofk=−0.63, which was originally suggested by Finkers and Hoffman[18]. This results in a superior predictive capability of Eq. (5) as
Fig. 3. Comparison of experimental data and predictions of models for Binary 3.
compared to others. This value of the k is therefore used for allother binaries as well.
The case of Binary 2 is shown in Fig. 2. The trend of the experimen-tal data is similar to Binary 1. This is expected as the physical proper-ties of these binaries are similar except that the size spread for thesmaller constituent of Binary 2 is narrower as compared to Binary 1.
Experimental data and model predictions for Binary 3 are pre-sented in Fig. 3. Note that the smaller constituent of Binary 3 is largeras compared to ones of Binaries 1–2. This means that the size differ-ence of the constituent solid species is now less pronounced whilethe density ratio is comparable. This leads to the development of seg-regation tendencies in the bed especially at the higher concentrationof the larger species. This phenomenon is reflected in the volumechange of mixing. The degree of the volume-change of mixing is ini-tially seen to be significant as a small quantity of larger species getsdispersed in the lower layer consisting of 463 μm glass resulting inthe contraction of the volume. But, as the fraction of the larger speciesis increased, the degree of the dispersion remains unaffected, and anupper layer almost exclusively consisting of the larger species pro-gressively develops. The upper layer keeps increasing in the sizewith the increase in the fraction of the larger species. Thus, a strongerstratification pattern develops in the bed at higher X1 thereby causinga reduced degree of mixing. A reduced degree of volume contractionis therefore seen here. As far as predictions of models are concerned, agood agreement is seen for the small X1 while the comparison is poorfor large values of X1 due to the development of the stratificationpatterns.
Binary 4 (P–E) has a bigger size ratio (r=0.24) as compared toBinary 3 (r=0.17) while their density ratios are same. Stronger seg-regation tendencies are observed in this case irrespective of the com-position of the binary. Note that a small size ratio can mitigate theeffect of a large density ratio by effectively decreasing the bulk-density difference of the two mono-component layers, thereby pro-moting the mixing of constituent species. Thus, Binaries 1–2 posses-sing the least size ratios (r≈0.1) exhibit a strong mixing tendency
Fig. 5. Comparison of experimental data and predictions of models for Binary 5.
Fig. 6. Comparison of experimental data and predictions of models for Binary 6.
365M. Asif / Powder Technology 217 (2012) 361–368
whereas Binary 3 with a bigger size ratio of 0.17 shows segregationfor the high X1. Due to the strong segregation in Binary 4, thevolume-change of mixing is rather insignificant as seen in Fig. 4.The two layers of individual species are almost segregated duringthe defluidization process that is controlled by the hydrodynamics.Only a small mixing was observed at the interface of the two layersdue to the dispersion phenomenon. As a result, packing models failto predict this behavior.
In the case of Binary 5 (Q–E), the smaller species E is the same butthe larger species is different. The larger species Q has a higher densi-ty as compared to species P. As a result, the difference between thebulk densities of the two layers is not as significant as for Binary 4at the incipient fluidization. This causes a small degree of mixing tooccur. This fact is clearly reflected in Fig. 5 at low X1. But, the behavioris almost same as before at higher fraction of the larger species due tothe formation of an almost segregated layer species Q. It appears thatthe behavior is more influenced by the size rather than the density ofthe constituent species. As far as model predictions are concerned,trend is almost same as seen before. Eq. (5) with k=−0.345 betterdescribes the volume-change of mixing as compared to other models.
The data of Binary 6 (Q–D) is presented in Fig. 6. Binary 6, is sim-ilar to Binary 5, except for the size of the smaller constituent. It is seenhere that there is mainly a negative deviation at small X1, and positivedeviation at large X1. However less pronounced, a similar behaviorwas sometimes (at X1=0.86) observed before as seen in Binary 1and Binary 2. In this connection, it is worthwhile to point out thatas the fraction of the larger component (X1) increases so does theminimum fluidization velocity of the mixture. Not all the smaller spe-cies could be accommodated in the interstitial spaces of the matrix ofthe larger particles in some situations, and are thus left to form amono-component segregated layer above the lower mixed layer.Keeping in mind that the minimum fluidization velocity of smallerspecies is substantially lower than its larger counterpart and thoseof mixed beds also, its mono-component layer, when present and flu-idized at liquid velocities greater than its own Umf, possesses a muchhigher void fraction than its own mono-component εmf. This causes
Fig. 7. Comparison of experimental data and predictions of models for Binary 7.
the bed of mixture to show a positive deviation. It is therefore notsurprising that the conventional packing models here fail to describethis behavior. The notable exception is Eq. (10), which is able to de-scribe this behavior by predicting the absence of the volume-changeof mixing.
The behavior of Binary 7 is shown in Fig. 7. The smaller constituentof the binary in this case is of the largest size, and therefore the sizedifference of constituents of the binary is smallest of all binaries con-sidered here. This leads to a smaller difference in the bulk-densityprofiles of the individual species. Therefore, a good mixing is ob-served in this case, causing a contraction of the bed volume. Mostpacking models are able to describe this behavior. Eq. (10) howeverdoes not predict any volume-change of mixing in this case eitherthough the size ratio is larger than Binary 6. This is due to the factthat the Eq. (10) is relatively insensitive to the change in the sizeratio, which is evident from the power of r that happens to be 0.1 inthe equation.
In the foregoing, the packing-equivalent diameter was used forcylindrical particles (solid specie P and Q) in spite of the fact thatthe difference between the volume-equivalent and the packing-equivalent diameter was rather insignificant. Using Eq. (9), the differ-ence between the two diameter definitions was approximately foundto be 1% for ψ=0.85 and 0.5% for ψ=0.87. The comparison withmodel predictions is shown in Table 3 by computing the error foreach case shown in Figs. 1–7. The error definition used here is basedon the difference between the predicted and experimental values ofthe percent volume-change of mixing, and is given as:
E ¼ 1N−2
XN−1
i¼2
χe−χp
��� ��� ð13Þ
Here, subscripts e and p represent experimental and predictedvalues, respectively. The summation excludes the experimentally de-termined boundary points. Eq. (13) can also be rewritten as:
E ¼ 1N−2
XN−1
i¼2
εe−εp1−ε ΔVM¼0ð Þ
����������� 100: ð14Þ
Here, ε(ΔVM=0) is bed void fraction in the absence of mixing effectsand is obtained using G=1 in Eq. (3). It is evident from Table 3 thatpredictions of Eq. (5) are better than other conventional packingmodels. Its predictions are even superior to ones of Eq. (10) formost cases except for Binary 5 and Binary 6, which show segregationtendencies.
4.2. Using the hydrodynamic drag diameter
In this section, the hydrodynamic information is incorporated inpacking models. Probably, the simplest approach toward this end isto use the hydrodynamic drag diameter for representing the size ofthe constituent species of the binaries. To evaluate the drag diameter,
Table 3Error in predictions using packing diameter for species 1 and volume-equivalent diam-eter for species 2.
366 M. Asif / Powder Technology 217 (2012) 361–368
the expansion behavior of individual solid species was first investi-gated. The data thus obtained was fitted with the Richardson andZaki equation:
Uo ¼ Utεn ð15Þ
where Uo is the liquid superficial velocity, ε is the bed void fraction,and Ut and n are correlation parameters. The parameter Ut can beinterpreted as the bounded terminal settling velocity owing to thepresence of wall-effects. In order to obtain the unbounded terminalsettling velocity, the following correction proposed by Khan and Rich-ardson [25] can be used:
Ut
Ut∞
� �¼ 1−1:15
dVDc
� �0:6ð16Þ
which yields the unbounded terminal velocity, Ut∞. Here, Dc is the col-umn diameter. Although there are several relationships proposed inthe literature that correlates the particle properties with the terminalvelocity, the following correlation of Khan and Richardson [25] isused here. It is given as:
Ret∞ ¼ 2:33Ga0:018−1:53Ga−0:016� �13:3 ð17Þ
where the Galileo number and the terminal Reynolds number are de-fined as:
Ga ¼ ρs−ρð ÞρD3dg
μ2
!;Ret∞ ¼ Ut∞ρDd
μ
� �ð18Þ
where Dd is the particle drag diameter. Knowing the unbounded ter-minal velocity of a solid species, it is possible to evaluate its drag di-ameter using the above equations. Data obtained from theexpansion behavior of each of the eight species are presented inTable 4. It is seen here that in all cases the drag diameter is smallerthan the corresponding equivalent volume diameter reported inTable 1. The difference between the two diameters appears to be sig-nificant as the particle size increases.
For particle species used in our experimental investigations, thecomparison is presented in Table 5 when the drag diameter is usedto characterize the size of solid species here. The percentage errorfor each of the binary species is computed and reported in the tablealong with overall mean error. Clear improvement is observed forall binaries. Using drag diameter helps improving predictions of allpacking models. It can be seen here the overall mean error decreasesalmost 20% for all the different models when the drag diameter isused. Eq. (5) still retains its superior predictive capability with itsmean error of 5.4 while the maximum error being 10 for the Binary6. On the other hand, the error with Eq. (10) remains largelyunaffected.
Table 4Bounded and unbounded terminal velocities and hydrodynamic drag diameter for solidspecies used.
4.3. Using hydrodynamic drag diameter and expansion
It is clear from the comparison in the foregoing that predictions ofpacking models are poor for the case of Binary 5 and Binary 6. Notethat the upper layer in these binaries mainly consists of smaller solids,which expands above its εmf as the mixture Umf increases above Umf2,reflecting ultimately in the positive deviation of the volume-changeof mixing. This issue needs to be accounted for by considering thatthe mono-component void fraction of the smaller constituent (spe-cies 2) will rather be equal to that of the fluidized bed of component2 corresponding to the minimum fluidization velocity of the mixture.Therefore, V2 needs to be modified as follows:
V1 ¼ 1−εmf1
� �−1; V2 ¼ 1−εmf2
� �−1; εmf2 ¼ Umf
Ut2
� � 1n2 ð19Þ
where εmf1 is as usual experimentally obtained value at Umf1, whereasεmf2 is now obtained from the mono-component expansion of thespecies 2 using Richardson and Zaki correlation as described in theabove equation (Eq. (19)). Note that correlation parameter, namelyUt and n, are already reported in Table 4. The comparison is presentedin Table 6 by incorporating the drag diameter for both componentsand the expansion of the layer of smaller species. A significant im-provement is now seen. There is an error reduction of over 50% forthe case of Eq. (4) when compared with the previous case of thelevel 1 modification. Similar improvement is also seen for othermodels except Eq. (10). The mean error is found to be least usingEq. (5). To a large extent, this improvement is due to the improvedpredictions for Binaries 5 and 6.
The comparison for Binary 6 is presented in Fig. 8 when thehydrodynamic information, as mentioned before, is incorporated.Predictions of only Eq. (5) are presented in the figure as the represen-tative case. The improvement using the drag diameter (Eq. 17) isevident here. The predicted volume-change of mixing is seen to bea little less. This is due to the fact that the size difference whileusing the drag diameter is less pronounced as compared to the caseof the volume-equivalent diameter, thereby leading to a smallervolume-change of mixing. A notable difference is observed whenthe expansion of the smaller solid species is also incorporatedin the packing models. The modified model is now capable of
Table 6Error in predictions using drag diameter and bed expansion.
Fig. 8. Comparison of predictions of conventional packing model (Eq. (5)), the modifiedmodel incorporating drag diameter (Eqs. (5) and (17)) andmodifiedmodel incorporatingdrag diameter as well as expansion (Eqs. (5), (17) and (19)) with experimental data ofBinary 6. Model prediction errors are 12%, 10% and 2%, respectively.
367M. Asif / Powder Technology 217 (2012) 361–368
predicting both positive as well as negative values for the volume-change of mixing unlike conventional packing models capable of de-scribing only a contraction of the volume. This aspect of the modifiedmodel substantially helps improving their predictive capability.Though only Binary 6 is discussed in the foregoing, other binariesalso show a similar trend.
5. Conclusions
It is evident from the foregoing that the overall bed void fraction atthe incipient fluidization is substantially affected by the stratificationpattern which is controlled by the hydrodynamic aspects of fluidiza-tion. As a result, the case of every binary is different. It is neverthelessclear that in the event of strong segregation, Eq. (1) best describes thebehavior. On the other hand, model capable of accounting for themixing effects is more effective for the case of mixed beds. Incorpo-rating hydrodynamic aspects of fluidization nonetheless leads to sub-stantially improved predictions irrespective of the degree of mixingand segregation prevailing in the bed as both positive as well as neg-ative deviations in the volume change of mixing can be accounted for.
Symbols used
Ar [−] Archimedes number=d2i ρf ρs−ρ
f
� �g
μ
24
35
Dd [mm] drag diameterDc [mm] column diameterdi [mm] diameter of ith particle speciesdvi [mm] equivalent volume diameterdpi [mm] equivalent packing diameterE [%] error defined by Eqs. (13) and (14)f [−] size ratio function defined by Eq. (8),g [−] size ratio function defined by Eq. (8)G [−] parameter G of Westman equation (Eq. (3))Ga [−] Galileo number defined by Eq. (18)n [−] Richardson–Zaki correlation indexr [−] size ratio (smaller to larger)rstr [−] particle structural ratio defined by Eq. (5)Retoo [−] Reynolds number defined by Eq. (18)Umf [mm s−1] minimum fluidization velocityUo [mm s−1] liquid (superficial) velocityUt [mm s−1] Richardson–Zaki correlation parameter (Uo=Ut for ε=1)Utoo [mm s−1] Unbounded particle terminal velocity
V [−] overall specific volume ¼ total bed volumetotal solids volume
� �Vi [−] mono-component specific volume of ith speciesΔV [−] volume-change of mixing defined by Eq. (1)X1 [−] fluid-free volume fraction of particle species 1
Greek symbolsε [−] bed void fractionεI [−] void fraction of mono-component bed of ith particle species,
γ [−] reduced density ratio¼ ρs2−ρf
ρs1−ρf
!
ρb [kg m−3] bed bulk density,ρf [kg m−3] fluid densityρs [kg m−3] solid densityψI [−] shape factor of ith particle species
This work was supported by the Research Centre, Deanship ofScientific Research, College of Engineering, King Saud University.
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