Drying granular solids in a fluidized bed
Citation for published version (APA):
Hoebink, J. H. B. J. (1977). Drying granular solids in a fluidized bed. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR38366
DOI:
10.6100/IR38366
Document status and date: Published: 01/01/1977 Document Version:
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PROEFSCHRI FT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. P. VANDER LEEDEN, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN, IN HET OPENBAAR TE VERDEDIGEN OP
VRIJDAG 20 ME11977 TE 16.00 UUR.
DOOR
JOZEF HENRICUS BERNARDUS JOHANNES HOEBINK
GEBOREN TE EINDHOVEN
Prof. Dr. K.Rietema (le promotor)
Aan dit proefschrift is door velen daadwerkelijk bijge-dragen.
Experimenten zijn uitgevoerd door Joop Boonstra en
Pierre Otten, en door de afstudeerstudenten Hans Pulles, Piet Rulkens, Paul Steeghs en Jo Willems.
Sommigen zullen hun bijdragen niet direkt terugvinden in het proefschrift, maar die bijdragen zijn desalniet-temin heel belangrijk geweest,
Het bouwen en verbouwen van meetopstellingen is het werk geweest van de technische staf: Piet van Eeten, Henk de Goey, Frank Grootveld, Piet Hoskens, Chris Luyk, Jo Roozen en Toon van der Stappen. De "bijzondere werkmethoden" van Wim Koolmees hebben meerdere malen de werkzaamheden voor dit proefschrift vereenvoudigd,
Het typen van het manuscript is snel en accuraat uitgevoerd door mevrouw Ted de Meijer.
Het werk van de afstudeerstudenten Ton Bongers, Leo Hermans, Jef Jacobs, Jan Moreau, Lou Peters en Jan Roes heeft welis-waar niets met drogen te maken gehad, maar hun werk aan verschillende fluidizatie-projekten heeft zeer zeker bijge-dragen tot een beter begrip van het fluidizatie-drogen. Aan allen, ook zij die hier niet zijn genoemd, hartelijk dank.
1-2-1947
1959-1965
1965-1971
1971-1977
geboren te Eindhoven
middelbare schoolopleiding (gymnasium B) aan het Augustinianum te Eindhoven. opleiding tot scheikundig ingenieur aan de Technische Hogeschool te Eindhoven. wetenschappelijk medewerker in de vak-groep Fysische Technologie van de Tech-nische Hogeschool te Eindhoven.
1. Introduction 1
1 1.1
1.2
Basic aspects of fluidization
Fluidization applied to the drying of wet granular material
2 . Literature review 4 7 7 2.1 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.4
On fluidized bed drying
On heat and mass transport between particles and gas
The packed bed The fluidized bed
On exchange between bubbles and the
9 9 12
dense phase 17
The bubble-cloud mechanism 17 Exchange between bubbles and the dense
phase 21
Conclusions 28
3. Mass transfer aspects of fluidized bed drying 30 3.1 Mass transfer around a bubble 31 3.2 Mass transfer limitation inside the
particles 36
3.2.1 Short term response of a drying
par-ticle 37
3.2.2 Long term response of a drying
par-ticle 44
3.3 Mass transfer behaviour of the whole
bed 50
3.3.1 Mass transfer limitation by gas phase
resistance only 50
3.3.2 Mass transfer limitation inside the
particles 51
3.3.3 Batch drying of the bed 61
6.
5.1.1 The fluidized bed driers 77
5.1.2 Temperature measurement and control in
the driers 78 5 .1. 3 5 .1. 4 5.2 5.2.1 5.2.2 5.2.3
Gas humidity measurements
Experimental procedure during drying experiments
Solid material General properties
Basic fluidization data of the solid material
Data on bubble size and bubble velocity
Experimental results
6.1 Mass transfer aspects of fluidized bed drying
6.2 Heat transfer aspects of fluidized bed drying 79 83 84 84 86 88 95 95 110
7 . Discussion of experimental results on mass
transfer 117
7.1
7.2
Exchange between bubbles and the dense phase
Exchange between particles and gas
8 . General conclusions
Appendix A Transfer between a single sphere and stagnant gas; a simplified
117 121
131
approach 135
Appendix B Change of the temperature of a
rising bubble 139
Appendix C Velocities of the three phases in
a fluidized bed 141
Appendix D Diffusion of moisture in silicagel
particles 143
Appendix E Data of the drying experiments 147
References List of symbols Samenvatting 150 156 161
1. INTRODUCTION
1.1 Basic aspects of fluidization
F'luidization is the phenomenon in which c a
gravitat-ional force acting on a dense swarm of particles is coun-teracted by an upward fluid stream which causes these par-ticles to be kept more or less in a floating state [1) • The fluid is either a gas or a liquid, the particles
usual-ly are solid.
This thesis deals with gas-solid fluidization only, which means that solid particles are fluidized in a gas flow. Some basic properties of fluidization will be des-cribed here with the help of what is called "fluidization characteristics" (fig. 1.1) , which show the bed pressure drop and the bed height as a function of the superficial gas velocity.
preaaure drop
---l)loo.,... gaeveloclty
fig. 1.1 Fluidization characteristics
The pressure drop over a packed bed is given by equation (1.1) when the flow resistance is caused mainly by friction; the latter usually holds when the Reynolds number, related to the particle diameter, is small, which usually is the case in gas-solid fluidization (order of magnitude Re
=
1).!J.P (l-e::)2 3 uo H
e::
liP = bed pressure drop E bed porosity
i3 tortuosity factor uo superficial
gas-]J gas viscosity velocity
H bed height
dp mean particle diameter
The gravity force acting on the particles is compensated when the pressure drop equals the weight of the bed per unit cross-sectional area:
( 1. 2)
pp = particle density g
=
gravity acceleration In equation (1.2} buoyancy forces have been neglected be-cause of the large difference between particle density and gas density. The gas velocity, at which the bed starts fluidizing, is called minimum fluidization velocity umf' and i t can be estimated by combining equations (1.1} and(1. 2). For u
0 > umf the pressure drop remains constant, which
means (from equation (1.1)) that both the bed porosity and the bed height must increase with increasing gas velocity: the bed expands (see figure 1.1}. From a certain gas veloci-ty on expansion cannot continue without breaking some con-tacts between particles; at this so-called bubble point ve-locity (ubp) voids generally called "bubbles" arise in the bed, which move upwards at high speed. The bed is now hete-rogeneously fluidized, while the range from umf up to ubp is called homogeneous fluidization. From ubp on the bed height may continue to increase gradually, or may decrease in a certain velocity interval before further expansion occurs (dotted curve in fig. 1.1); this depends on the homo-geneous expansion that can be reached. At high velocity the particles are entrained by the gas flow, and the bed is blown out. The onset of fluidization might be delayed by friction between the particles and the bed wall. In such case the pressure drop increases still linearly above umf
until wall friction has been overcome;~at that gas vela;.. city the bed will expand shock-wise.
Homogeneous fluidization is observed especially when fine powder is fluidized; it does not occur with coarse material. Some authors [2,3] stated on theoretical grounds
that homogeneous fluidization cannot exist at all, which is in contradiction to many experimental observations. According to Rietema and Mutsers [4] a homogeneous bed is stable due to interparticle forces which play an important role when fluidizing fine powders. Homogeneous fluidization is mainly subject of fundamental studies with the final aim to predict stable bubble sizes in a heterogeneous bed; it is not applied in practice, as some outstanding advantages of a fluid bed disappear in the homogeneous state, and be-cause the gas flow through the bed is much too low to reach -the capacity for economic use of the process.
In a heterogeneous bed an almost particle free bubble phase and a dense phase are distinguished. For coarse ma-terial all bed expansion is due to bubbles and the dense phase porosity equals the packed bed porosity; the surplus feed gas, that exceeds the flow needed for minimum fluidi-zation passes the bed in the form of bubbles. For fine par-ticles the dense phase porosity ~d is higher than in the packed bed, and the dense phase velocity ud is in between umf and ubp' From what is called a collapse experiment ~d
and ud can be determined [S,o] •
~~ny studies have been made on bubbles. Their mean size is not predictable at this moment. Due to coalescence and
splitting of bubbles in a heterogeneous bed a large spread in bubble size arises. Usually bubbles are small (< 0.3 em)
near the distributor plate when an even distribution of gasis applied, for instance via a porous plate; higher in
the bed bubbles can become quite large (> 5 em) due to co-alescence. In high and narrow beds the bubble size may ap-proach the diameter of the vessel containin,g the bed(the bed is called to be slugging), but this will not occur when
equilibrium between bubble coalescence and bubble split-ting is reached. Some criteria for slugging have been presented [7,8] • Coalescence and splitting are being ex-tensively studied [9-11] • ~he form of a bubble ressem-bles a spherical cap with an indented base [12] • Many deviations of this form occur as the bubble may change itsshape continuously during the rising-up; i t may become
elongated as well as flattened. The rising velocity Ub of a single bubble is related to the bubble volume:
ub
=
0.71 g1/2 vb1/6 [12,13] , but in a swarm of bubbles the velocity will be much higher.~ theoretical approach [14-17] of the flow pattern of gas and solids around a bubble will be treated in Chapter 2. 'l'O some extent bubble gas bypasses the bed because of less good contact between bubble gas and solids; the ex-change between solids and gas is discussed also in Chapter 2. As a result of bubbling strong solids movement and mix-ing occurs in the bed, which is the main reason that the bed temperature is very nearly homogeneous in heat
trans-fer processes.
1.2 Fluidization applied to the drying of wet granular material
Compared with other drying techniques fluidized bed drying of granular solids offers many advantages.
High heat and mass transfer rates are possible because of a very good contact between particles and gas; Chapter 2 deals with this subject. Although bubbling may cause by-passing of gas, i t also causes intensive solids mixing with a nearly homogeneous bed temperature as a result. This makes temperature control of the bed easy, and allows operation of the bed at the highest temperature that is permissible from the viewpoint of solids thermal degrada-·tion. The fluid character of the bed facilitates solids handling especially in continuous operation. In case the drying rate is limited by diffusion inside the particles
long solids residence times are required, which can be achieved easily in a fluidized bed; the apparatus still remains relatively small when compared with other equip-ment, because of its large hold-up of solids. The
appara-tus is rather simple as there are no moving parts. The pressure drop across the bed is restricted, in spite of high gas throughputs. The solids mixing causes a consi-derable spread of the residence time of individual
parti-cles, which is a disadvantage as the product will consist of relatively dry and wet particles. This problem, when serious, is usually solved by installation of a multiple-stage apparatus.
Since most fluid bed driers operate at very high gasvelo-city entrainment of particles by the gas flow occurs. Cyclones and other dust separating equipment are often needed. Partly this problem is overcome by use of a
disen-gaging zone above the bed, with a diameter larger than the bed diameter. Due to abrasion and friction between
parti-cles fines may be produced in the bed which makes entrain-ment even more serious. Friction between particles and the bed wall may cause severe abrasion of the bed wall.
Only free-flowing powders can be fluidized. Fluidized beds should not be applied for drying of sticky material unless the solids feed can be spread evenly over the whole bed content in some way; impeller mixers are sometimes insert-ed in the binsert-ed for such purpose. Due to the lowmoisturecon-tent of the well-mixed fluidized mass an evenly spread sticky material may become dry at its surface fast enough to keep the solids free-flowing and the bed fluidizing. When the solids feed cannot be spread evenly over the bed cmtent, a less concentrated slurry feed should be
prefer-red; there are ample examples of spraying slurries and pas-tas directly on the surface of a fluid bed drier. As
in-tensive solids mixing is essential in such situations, spouted beds are often applied, which have a conical base with the gas feed in the center.
An extensive description of equipment for practical purposes is given by Vanecek e.a. [18] , Romankow [20] and Sen Gupta [21] • Apart from special arrangements for practical problems three basic designs can be distinguish-ed; as indicated schematically in figure 1.2 a-c. In a horizontal arrangernentof the stages cross-flow of gas and solidscan be applied, while the vertical arrangement is
used for countercurrent operation. 'I'he heat necessary for drying may be supplied to the bed in two ways:
- via the fluidizing gas, which is preheated in some way before it is fed to the bed;
via the vessel wall by means of a steam jacket, or via internal heat exchanging surfaces like steam coils. 'Ihe former way of heating will be adopted in shallow beds, the latter in deep beds.
1 gas Inlet 2 gas exit l solids feed 4 solids discharge s bed I distributor 7 downcomer figure 1.2c 4 figure 1.2a figure 1.2b
2. LITERATURE REVIEW
2.1 On fluidized bed drying
The literature on drying granular solids in a flui-dized bed concerns mainly global descriptions of such processes in practice. Many examples of these have been put together by Vanecek e.a. [18] and Sen Gupta e.a.
[21] ; Romankow [20] describes several types of equip-ment in practice. In most cases the data presented are far from complete. It is amazing that the major part of the investigations does not mention at all the equilibrium conditions for the drying solids under consideration. unly few fundamental studies have been reported.
Angelino e .a. [22] measured ad- and desorption. of mois-ture in air by silica-alumina catalyst under non-lsother-wal conditions. 'ihey found in a 18.5 em diameter bed that the relation between outlet gas humidity(measured) and mean solid moisture content (calculated from a gra-phically integrated mass balance) is always the same when the bed height is more than 5 em; only small varia-tions of gas flow were applied. It is suggested that the relation mentioned is the equilibrium curve, and that complete equilibrium between gas and solids is reached at the upper bed level.
Several authors [23-25] applied fluidized bed drying to the measurement of gas-particle heat transfer coefficients; these results will be considered in section 2.2.
Vanecek e.a. [26] studied the influence of particle size on the drying of fertilizers in a fluidized bed. They
showed experimentally that the mean solids moisture con-tent in dimensionless terms (related to the initial and
2 equilibrium moisture content) is a function of t/R only
(t = time, R
=
particle radius). This result indicates diffusion limitation inside the particles to occur.Much work concerned the translation of batch drying re-sults into predictions for continuous driers [18,20,27]. ;rhe residence time distribution of particles in the
con-tinuous drier therefore has to be taken into account. Reported results in this respect cover the whole range from plug flow to ideal mixing of solids, depending mainlyon gasflow and drier geometry and construction.
A general review concerning experimental results on so-lids mixing in fluidized beds was given by Verloop e.a.
[28] (see also [29] ) • On theoretical grounds solids mixing, which is a result of bubbling of the bed, has been ascribed to three mechanisms:
- Solids are moving upwards in the wake of a bubble, and during the rising-up there is a continuous exchange be-tween solids in the wake and solids in the dense phase [30] • The wake volume amounts to about 25% of the bub-ble volume [12] • The upward flow of solids is compen-sated by a downward flow in the dense phase.
- \'ihen a bubble rises up, the solids in its neighbourhood are drifted upwards. 'l'heir position after the bubble has passed is higher in the bed than it was before the bubble arrived [31] • Solids far away from the bubble will move downwards a little.
- '.there is some tendency for bubbles to move towards the bed center. As a result the bed density is lower in the center than it is near the walls; this causes overall circulation in the bed, and an increase of the bubble movement to the bed center.
under practical conditions for fluid bed driers the solids are quite near to ideal mixing, especially in beds with height over diameter ratio of about unity, operat-ed at high gasvelocity. This has been found for single stage apparatus [18,27,321 and multiple stage designs
crossflow of gas and solids [36-37]. In high beds plug flowwith axial mixing or overall circulation will occur.
Some design methods have been proposed for continuous driers by Vanecek e.a. [18,38] , Rornankow [20] and Sen Gupta [21] • In case that the drying rate is limited by gas phase resistance only, these methods are based on total heat- and mass balances only, and ideal mixing of solids is mostly assumed. When diffusion inside the par-ticles limits the drying rate, it is proposed that an equation is developped from batch drying experiments, which expresses drying kinetics: e.g. moisture content of the particles as function of time and in dependence of gas flow, oed height etc. Such relation is combined with solids residence time distribution and external balances to meet the design specifications.
The simple combination of batch drying results with so-lids residence time distribution may lead to improper de-sign of the continuous dryer, as the gas concentration is not included in the calculations. when a still relatively wet particle leaves a drier with ideally mixed solids
af-ter a fixed time,it will be drier than according to a re-lation based on batch drying results, since the
has been exposed to a larger driving force in the continu-ous drier.
Exchange of heat and mass between particles and gas in packed beds was extensively studied; some reviews in this field were presented [39-41] .
Bxperimental results show a general agreement when the Reynolds number Re is larger than 10 [39] . These results are conveniently correlated by equations 2.1 and 2.2, the general form of which was originally presented for single spheres by Frossling
[421
and applied to chemical engi-neering by Ranz and Marshall [43] .Nu 2 + 0.7 Re1/2 Pr1/3 (2. 1)
Sh = 2 + 0.7 Re1/2 (2. 2)
The constants in the above relations refer to the work of Rowe e.a.[41]. The Reynolds number exponent may be as low as 0.4 for Re near to 10, and as high as 0.6 for Re is
4
about 10 • The dependence of the Nd- and Sh-number on the Prandtl- and Schmidt-number (Pr and Sc respectively) is based more on theoretical grounds than on experimental evidence. Small deviations from the correlations 2.1 and 2.2 may be expected due to:
- the influence of the bed porosity on the transfer rate;
- the influence of any regularity in the packing; for ran-domly packed spheres and spheres in ordered arrays dif-ferent results were reported;
- the influence of the particle shaoe.
'l'hose factors have not been quantified, and deviations are within the accuracy of relations 2.1 and 2.2.
In the range of low Reynolds numbers (Re < 10) experimen-tal Nu- ~nd Sh- numbers differ over more than three deca-des. They often fall far below the value 2 [44-46] ,which is the minimum value for one single particle in an infini-te stagnant fluidum. Reporinfini-ted Reynolds number exponents scatter up to values of 1.3. Apparently low Nu- and Sh-numbers have been ascribed to gas mixing in the bed, to chanelling, and as far as heat transfer is concerned, to heat conduction via the packing [40,47] , but correction of data for such effects was often not sufficient to ex-plain low Nu- and Sh-values [40] •
Nelson and Galloway [40] suggested that correlations like 2.1 and 2.2 are valid only for single spheres {in an infi-nite fluidum or embedded in an array of inert particles)
and for beds packed with coarse material. In a dense swarm of fine particles, which all take part in the trans-fer process, the mean interparticle spacing becomes very small; this might mean that the concentration (or tempe-rature) gradient in radial direction around a particle be-comes zero at a very short distance, instead of becoming zero at infinite distance as is assumed in deriving Sh
(or Nu)
=
2. Working out this idea the authors showed that in beds of fine particles the transport from an indi-vidual particle is hindered by transport from itsneigh-~ours, nindering becoming stronger with decreasing bed po-rosity e:. They derived equation 2.3, vlhich is shown in figure 2.1 as taken from their paper.
[ 2 - 2 ] tanh 2P + 2P g p Sh ( 1-g:) 2 ( 2. 3)
[
]
- tanh P p 0.3[~-
1] Re1/2 scl/3 q = (1-e:)l/3As seen from figure 2.1 (Nu- and)Sh-numbers may be much smaller than 2. For large Reynolds number or bed voidage e: approaching to unity relations 2.1 and 2.2 are found.
Figure 2.1 plot of
relation (2.3) as taken from [40]
The work of Nelson and Galloway signalizes an effect that may be very important for transfer processes in packed and fluidized beds. Nevertheless their theory gives no com-plete satisfaction since other factors may be involved. Recently Schllinder [48] has shown that low Nu- or Sh-numbers may be observed because of irregularities in the packing. When in a bundle of parallel pores a small spread
of pore diameter exists the contact efficiency in such a bun-dle will be much lower than to be expected on basis of the mean pore diameter.
2.2.2 The fluidized bed
Many correlations were proposed to relate the Nusselt-or Sherwood number to the particle's Reynolds number, as
has been done also for packed beds. For fluidized beds the agreement is very poor, even in qualitative respect, as can be concluded from reviews in this field [49-51] . compared with the packed bed aheterogeneously fluidized bed has two new aspects that influence exchange between particles and gas: the presence of bubbles and intensive solids mixing. These effects are related to each other as explainedin section 2.1.
In most experimental work the fluidized bed was treated as homogeneous and bubbles were not considered. Bubbles seemed often unimportant, as several authors [23-25,52-57] concluded from their measurements that equilibrium between gasand solids was reached after the gas had penetrated a
few centimeters at mostinto the bed. ~specially in gas-to-particle heat transfer an apparent disappearing of the driving force was reported in the bulk of the bed, even if
rather shallow beds were applied. In some experimental work however equilibrium was not reached in even deep beds
[58-60]. This discrepancy deals partly with wrong inter-pretation of measurements.
It is generally accepted now that the bed temperature, which is measured by inserting a bare thermocouple, in a fluidized bed, is in between the gas temperature and the solids temperature. The solids temperature is meas-ured with a bare thermocouple after closing the gas flow [53,58] ; due to the large difference in heat ca-pacity between solids and gas the thermocouple indica-tes almost immediately the solids temperature. Gas con-centration and temperature should be measured via suction of gas [48,52,53] ; this method provides no way to dis-stinguish properly between bubble gas and dense phase gas. Therefore,aconclusion of equilibrium being reached in the
bed cannot be based on an observation of either homogene-ous bed temperature or homogenehomogene-ous gas temperature ( or concentration) as was done in the majority of published results. v;arnsley and Johanson [58] clearly showed that such conclusion is wrong. They studied heating of the bed via the fluidizing gas. 'J.'hey found a uniform bed temperat-ure (indicated by a bare thermocouple) and an equal,uni-forrn gas temperature (measured via suction of gas) from 1 ern above the distributor on. 'l'he solids temperature was measured after closing the gas flow for a short while, and was found to be lower than the bed temperature. The latter indicatesthat bypassing of gas occurred to some extent.
Wamsley and Johanson were able to show that bypassing gets less when coarser particles are fluidized because of a de-creasing fractional bubble flow.
"l'he same authors remarked that bypassing also must have taken place during the experiments of Kettenring e.a [23], who studied heat- and mass transfer in a bed of particles that dried at constant rate; from total heat- and mass balances it can be clearly shown that the solids temperat-ure must have been appreciably lower than meastemperat-ured by
'J.'he foregoing demonstrates that bubbles are involved in the transfer process. Further evidence is found in the work of Petrovic and Thodos [ 61]. 'l'hey studied mass transfer between gas and rather coarse porous particles, which contained an evaporating liquid. 'Ihe bed was weighed at regular times, and e.xit gas concentrations were calcu-lated via a mass balance; the equilibrium gas concentra-tion was determined via the temperature of the evaporat-ing liquid, which was measured by embeddevaporat-ing a thermo-couple in one of the particles. Assuming plug flow of gas and ideal solids mixing the authors determined a mass transfer coefficient which was presented as the Colburn factor :
Sh Re
Figure 2.2, adopted from their paper, shows the results.
im
t ::::
0,01 0,04 U..---L----..1--~.-LJ 100 200 400 100 --...:)loa- Re parameter isthe particle size in )l
Figure 2.2 Results of Petrovic and Thodos [61]
For all particle sizes investigated one single straight line was found when the bed was in the packed state. For the fluidized state each particle size corresponded with a different line. 'I'he intersection of each line for the fluidized bed with the packed bed line occurred at a par-ticle Reynolds number which exceeds the value at minimum fluidization with about 20%. Petrovic and Thodos refer to
the intersections as being bubble points. They introduced somekind of effectiveness factor that compares the perfor-mance of a fluidized bed with a packed bed under the same conditions; such factor should account for bubble bypassing. Accordingto Petrovic and Thodos [61] fluid bed performance may be as low as 20% compared with a packed bed.
Similar results as expressed in figure 2. 2 were reported more often for coarse particles [62,63,64] • In more recent work [65-67] the Archimedes number is used in correlating Nus-selt- or Sherwood-number with Reynolds. This fact may also
point to an effect of bubbles on the transfer between gas and particles, as the same Archimedes number is involved in the transition from homogeneous to heterogeneous fluidi-zation [ 4] •
Data on bubbles never were reported in relation with trans-fer between particles and gas. Nevertheless some authors [51,68-70] made re-interpretations of data in this field to incorporate the effect of bubbling. Unknown bubble para-meters were adjusted such as to fit the classical data.
Kunii and Levenspiel [68] applied their "Lubbling bed model". '.1.'he model assumes some effective bubble diameter as para-meter, which is constant in the whole bed. Eor the
re-interpretation Kunii and Levenspiel had to assume real smallbubble sizes (0.3 - 1 em) to fit classical data. As
the effective bubble size may include many effects (see 2.3.2) it is not related to actual bubble size in a simple way. A better approach was made by Kato e.a. [69,70] who used their "bubble-assemblage model". The bed is divided into compartments of different height, each having a mean bubble diameter. Diameters in subsequent compartments are related by a coalescence model. To fit classical data it was assumed that bubbles were not present in the compartment nearestto the distribution ; sometimes the height of
that compartment exceeded the total bed height. Apart from many doubtful assumptions the model requires time
In both models some statements had to be made on the ex-change between bubbles and the dense phase; this subject
will be considered in general in section 2.3.2.
When bubbles in the bed are very small, exchange between particles and gas will be very effective. Such small
bub-bles m'lSt have been present during the experiments of Angelino e.a. [22] , Heertjes e.a.[24]and Richardson e.a.
[6], who all found a high degree of equilibrium between particles and exit gas.
As in packed beds low values of the Nusselt and Sherwood number (below the value of 2} were also reported for fluidized beds by authors who treated the bed as being homogeneous [47,49,57]. Richardson and Szekely [57] show that axial mixing of gas may account for this effect; Kato e.a. [69,70] and Kunii and Levenspiel [68] ascribe i t to bubbling. Another suggestion, which holds for heat transfer only, is that transient heating of particles oc-curs in the bottom region of the bed, as happens in heat transfer from the vessel wall to the bed. In such situation the residence time of individual particles or particle packetsnear the distribution plate may be controlling the
transfer rate.
The results of Heertjes and coworkers [24,25,53] show that the latter may take place. In studying heat transfer toa bed of particles which dried at constant rate Heertjes e.a. [53] observed that the distribution plate transfers heat to the particles. 'I'he temperature of the feed gas
dif-fered considerably from the temperature of the gas that left the distributor plate and entered the bed. The tem-perature drop over the distributor plate depended on the gas flow through the bed, and on the distributor design
(both plate material and construction} • 'I'he temperature difference across the distribution plate may be as high as 50% of the temperature difference between feed gas and bed. This same effect may explain why [24] observed an analogy between heat- and mass transfer in deep beds only.
In shallow beds heat transfer between particles and dis-tribution plate may dominate, and as there is no equiva-lent in mass transfer the analogy will not hold anymore, heat transport becoming a more rapid process. The influ-ence of the distributor on heat transfer was reported nowhere else.
In mass transfer between particles and gas diffusion li-mitation inside the particles may lower the transfer rate.
~his effect was reported by Richardson and Szekely [57] and by Hsu and Molstad [71] who both studied adsorption of carbontetrachloride in air onto active carbon. Trans-fer coefficients (based on plug flow of gas and ideal so-lidsmixing) were found to decrease with increasing time.
Diffusion limitation was also observed by Vanecek e.a. [26] in fluidized bed drying (see section 2.1). In the experiments of Richardson and Bakhtiar [56]and Angelino e.a. [22] diffusion limitation did not occur. Unsufficient data are available to compare these experiments; the on-ly obvious fact is that authors who observed diffusion li-mitation used very shallow beds, while the others used rather deep beds.
2.3 On exchange between bubbles and the dense phase 2.3.1 The bubble-cloud mechanism
'i.'he well-known bubble cloud concept has been introduc-ed by Davidson and Harrison [14] on theoretical grounds;
experimental evidence for it was presented by Rowe e.a. [72] • Although often criticized as will be discussed later, the basic idea is still of great importance for the understanding of the exchange between bubbles and the den-se phaden-se.
Davidson and Harrison analyzed the flow of gas and solids around a single rising bubble in a fluidized bed under the next assumptions:
- the bubble has a spherical shape;
- the dense phase porosity is uniform, the gasphase is incompressible;
- the pressure gradient in the dense phase is related to the slip velocity between particles and gas via a
Darcy type of equation;
- solids movement can be treated as potential flow around a sphere.
'rhe continuity equations for both gas and solids were solv-ed in combination with the gas phase momentum balance and as a result the stream pattern of the gas phase is found. 'l'wo types of flow have to be distinguished; they are shown in figures 2. 3 and 2. 4, ·where, as usually done, a statio-nary bubble is presented in a downflow of solids that mov-es with the bubble velocity downwards.
GAS SOLIDS GAS
Fig. 2.3 a, > 1 Fig. 2. 4 a, < 1
Flow pattern of gas and solids around a spherical void schematically
When the velocity Ub of the rising bubble is lower than the linear gas velocity ud in the dense phase the sphe-rical void acts as a bypass for the dense phase flow; at the equator the flow through the void is three times the flow that would pass the same area if no void was present:
where Q is the gas flow through the void, £ the dense
g
phase porosity, db the void diameter.
v~hen the bubble velocity is larger than the dense phase velocity, the same throughflow through the bubble exists, Lut the gas is recirculated from the top of the bubble to its bot tom via the dense phase • 'I' he dense phase re-gion in which the bubble gas can penetrate is restrict-ed and is callrestrict-ed the cloud. According to Davidson and Harrison [14] the boundary between cloud and dense phase is a sphere, concentric around the bubble, and its dia-meter d0 is related to the bubble diameter db:
=
\3~
V
a - 1For single bubbles the velocity ub depends on the dia-meter db : Ub "' vdb [13] • l>- rough estimate of the dense phase velocity Ud is the minimum fluidization velocity, which depends on the mean particle diameter d :Ud"'d 2 •
p p
From this it follows that the cloud diameter is much lar-ger than the bubble diameter when the bubbles are small or the particles coarse (a. + 1) .'vvnen a is much larger than unity (large bubbles in beds of small particles) the cloud diameter approaches to the bubble diameter. Several modifications of the basic idea were proposed
(Jackson [15] 1 Murray [16,17]) because of the
contra-aictory assumptions that potential flow of solids is ap-plicable and that the pressure inside the bubble is con-stant. Murray [16, 17] studied non-spherical bubbles, in which the pressure is not taken constant anymore, while
Jackson [15] allows the dense phase porosity around the bubble to vary; both authors maintain the assumption of solids potential flow. Rietema 1 in a recent paper [ 731 ,
criticizes the applicability of potential flow theory, especially for fine particles, on both theoretical and
experimental grounds, and he presents a qualitative but general proof of thecloud'sexistance without any
assump-tion about the type of solids flow.
At this moment the Davidson/Harrison approach must still beconsidered as the best one available for making
quan-titative estimates of the flow pattern around a bubble. Its obvious imperfections have not been overcome yet by theproposed modifications, which only made the description more complicated without basic improvements and without doubtless experimental support. The present uncertainty about the real flow pattern around a bubble justifies an even more simplified description than proposed by Davidson and Harrison, particularly when their theory is applied to a special topic.
2.3.2 Exchange between bubbles and the dense phase Hass transfer and to a minor extent heat transfer be-tween the dense and the bubble phase has been discussed by many authors [76-101]. In describing the phenomena in-side the bed two completely different approaches can be distinguished in the literature, each with its own merits and shortcomings [ 74,75] •
The first approach concerns the so-called two-phase mo-dels, which were originally proposed for fluidized bed re-actors [76-78] • The bed is divided schematically into a particle-free bubble phase and a dense phase. The bubble phase is usually assumed to be in plug flow, while diffe-rent mixing patterns are proposed for the dense phase. The most general approach (van Deemter [78] ) describes the beds performance with an overall mass transfer coefficient between bubbles and the dense phase, and with an axial mi-xing coefficient for the dense phase. ':l'hese parameters in-clude all kinds of fluid bed phenomena like bubble split-ting and coalescence, bubble formation, cloud shedding, overall solids recirculation), so they will depend on many variables (bed height and diameter, gas velocity and dis-cributor design, particle size and size distribution). Helative simple tests are available to measure the parame-ters [ 78-80] •
'.l·he second approach splits up any fluid bed process in many sub-processes, for each of which the behaviour of in-Yividual bubbles is studied separately. As such sub-proces-ses can be considered bubble formation in connection with distributor design, bubble splitting and coalescence and exchange between a bubble and the dense phase. The dense phase mixing is mostly treated as in two-phase models, and transfer between particles and gas in the dense phase is assumed to occur at very high rates [81,82]. Integration of these sub-processes over all bubbles in the bed yields a fluidized bed model (often called bubble model). Because of the complexity simplifications are often made by ne-glecting or combining sub-processes.
Some bubble models were mentioned briefly in section 2.2. A complete review on fluid bed modelling {especially for chemical reactions) is given by Yates [83] • A summary will be presented here on exchange between single bubbles and the dense phase; the literature in this field mainly deals with mass transfer. Typical for theoretical work are the many assumptions that can and have been made, which makes comparison of different approaches quite
dif-ficult. Davidson and Harrison [14] suggest that transfer occurs, as a superposition, by the gas flow through the bubble and by diffusional transfer across the bubble boun-dary. In contradiction with their own cloud theory the cloud is not considered as a closed envelope around the bubble. Hovmand [84]and Walker [85] present modifications of the model.
Authors who assumed the cloud to be a closed envelope, introduced several resistances for mass transfer, e.g. in the cloud (Chibah and Kobayashi [86] ), in the dense phase
(Rowe and Partridge [ 87] ,'l'oei and Matsuno [88]), or com-binations of these resistances (Kunii and Levenspiel[30]). They started from the bubble-cloud model of either
Davidson and Harrison [14] or Murray [16,17]; moreover transfer coefficients were derived from boundary layer theory as well as from penetration theory.
When the cloud diameter is small compared to the bubble diameter and when the solids are not porous or adsorbing, most theoretical results can be conveniently expressed as:
as can be concluded from the work of Drinkenburg [6) • Here ud is the superficial dense phase velocity, ID the gas phase diffusivity (which should be the effective dif-fusivity (Drinkenburg [6] ) instead of the molecular one), g the gravity acceleration, d the equivalent bubble
dia-e
meter and K the overall mass transfer coefficient for the bubble, which is based on the surface of a sphere with
the same volume. Proposed values for A range from o [86-88] to 3 [14]: the minimum value reported forB is 0.36 x s
0 (s0 is the dense phase porosity) and the
maximum is 0.975 ( [88] and [14] respectively).
when the cloud is large compared to the bubble A and B will depend on cr
=
Ub/Ud' and on cloud and bubble dia-meter: several functions were proposed (see [6] ) . In case of porous or adsorbing solids multiplication fac-tors for the mass transfer coefficients were derived [6,87,89] . Drinkenburg [89,90]in a numerical approach, aoes not assume on forehand that the transfer resistance is concentrated somewhere. His work includes different cloud theories, and the possibility that tracer trans-fer from the bubble occurs via porous or adsorbing par-ticles. It is found that the concentration in the cloud changes severely in tangential direction, and that no specific transfer resisting areas can be indicated. Whenparticles are adsorbing mass transfer rates areve-ry much increased due to larger concentration gradients in the dense phase (thin clouds) or transfer between particles and gas inside the cloud (thick clouds) which effects become dominant. Toei and Matsuno [91] also in-dicate the importance of adsorption in both heat and masstransfer; they also take into account that particles
may rain through the bubble, as was considered by Wakabayashi and Kunii too [95]. It is obvious that ad-sorption of the transferable component by the particles is an important factor in fluidized bed drying.
Table 2. 1 summarizes schematically methods and condi-tions of experimental work on exchange between bubbles and the dense phase in three-dimensional beds. Concentration and temperature measuring techniques are not included, as they are too diverse;among them there are spectrophoto-meters [ 87,88] , chromatographic equipment [6,93,94,95] , dewpoint meter [92], thermocouples [82,91] • Local bubble concentrations were measured via sampling [6,94] or via probes inserted in the bed [83,88,91,95] • Dense phase
con-• " " "' +' "
"'
"
••
••
.,
~"' "' I'd beds w "' ... 0 00 C-< C.-<"'
C.Q H ".Q u c . +'9
S' ..;ru~f~
'0"' ·-<"' '" .Q :::: ~~" ·M ::.1 C ... u fJl <C m o.- ..j..l·,-,j p.. g"' • e "' il"' t g"'.n a
a~ lle"'
..
p...;" ~.53 "' " u "'"
'OS'"" c ~ kO'- w.o .':: !:6 ~ ... 0' \lie u ·.-1 c.
Q u
.,.
0 ro ro +' .-;" ..j..IQ())
~~~m
Author '0 '0 oro .,..,
.
~ ~~ '•l r') O'k" • . ... 0
..
~ • 00" 111""'"'"'
'00>"'
0 >"' "'
"'
"" 'OC,_.+J"'
""'
+'Barile e.a~ [Bi] 9. 5 8.5 glass 127' 365 Chain of cold bubbles yes \lb ~1. 38 _ (uo -umfl !:lubble- and dense phase tempera tt' re were from one orifice into
(9b.f 0.6 \lb measured with bare thermocouples locally
warm homogeneous bed in the bed; plug flow of both phases
y >sum,•d
Walker e.a. [85] 10.' 62 sand 53,97' Ciiain of ozone contai vb = m~a:sureu measured exit concentration bubble and 150 ning bubbles from one na via capa- dense phase ...,..,;
orifice into homogene ? ~A city city prob assuming plug phases
I
~~~bed, catalytic remeasur::-. :neasure~- 1 :mbble measured lo:~.l~~. in the Chibah e .a~ {86] 10 60 glass 140, 210 Injection of single no via light via light bed; dense
bubbles, containing I ozona tracer
probe probe assuming both
Drinkenburg e.a. 18.90 100 catalyst 66 Injection of single yes based on ub = I bubb~'; concentration measured via local
volume of
bubbles, containing
I
~~ii~~g. in the bed; dense phaseconcen-[ 90] tracer; different tra !~~ected 0,71~ zero
Toei e.a. {91] lOxlO <100 glass 161,216~Injection of measured 1 measured temperature of bubble measured locally in into "'~~ld
yes via two I v~;
capa-the bed with capa-thermocouple; r.lense phase 270 hot bubbles thermo-
I
city probe temperature as reference
bed counl~s
Wakabayashi e~a. 20 <80 catalyst I ~~~in of moisture con vb measured counted at exit concentration measured; ci.ense phase (92] from via bed concentration zero
a into ~A city pi:-abe h.omoqeneous bed
Kato e.a. (931 10 <16 glass 192,324 Chain of bubbles into yes
vb 2 1. 38 ub ~ exit concentration measured; dense phase homogeneous I;mi:oor 0. G
0.71~ gas saturated
fraction of (~) !Ja.rticles evaporati!'!9'
o• •ly
in dense phase measured measured bubble concentration via local Hoebink e.a. [94] 45 90 66 51 Injection of yes via lvi~;:'~pa- sampling in the bed; phase
concen-ethene ,i~g city probe . orob .:ration zero bubbles no
Pereira e~a* (SS) 15.4 70 cokes 92 Injection of single no measured vi meas~~~ucl measured bubble concentration and dense phase con-five
t.l-t1~bbles containing he vity prcbes ti~i~y conduc- centrat1on measured i.n the bed ium tracer ci;;itv probes probes
Stephens e.aJ96l 5,15 30 glass ~~(/~~0 Chain of bubbles fn;=.a tJb observed , __ exit measured, as well as 368,' 590'homogeneous bed; bub-ble tracer is mercury no o, 71\/gde at upper bed local dense level flow of dense" phase gas assumed concentration; plug
Szekely [97] lO <21 catalys 60 c~~i~r of bubbl into yes vb = tJb ~ observed at exit ~t_ltration measured; dense phase homogeneous tra- ~A
0.71~ upper bed I conce zero
cer is '"'" level f
'"
Davies e.a. [lOll catalyst~60 I In of single ub =
P.v.c. 16,142 1
co;
containing bubbles ?0. 71 v-g-;r diakon 128
centrations were mostly zero, while some authors [85,86] assumed plug flow of dense phase gas for calculating its concentration.
As to be expected results we~e interpreted in many diffe-rent ways. For comparison all results were recalculated in terms of an overall exchange flow Q (after [14,84,85] defined by the next equation:
with ub bubble vb bubble cb bubble
=
Q (C-c )
b d velocity volume concentration h = height coordinate cd dense phaseconcen-tration
Whenever possible recalculations were made from direct ex-perimental results [92-95] ; otherwise the presented calcu-lated data had to be used[81,85,86,90,91,96] .Assumptions made by the authors were always adopted. Reinterpreted data are presented in figures 2.5 A and B as Q/umf versus equi-valent bubble diameter d ; the gasflow Q through the
bub-e g
ble is indicated as Q /u g m f according to the theories of Davidson and Harrison [14] and of Murray [16,17] . Not in-cluded are results of Szekely [97] who states most transfer to occur during bubble formation. Most results in figure 2.5 include transfer during bubble formation (except [91] and [94] ). In mass transfer this effect was reported for t~odimensional beds [98b for small bubbles (diameter < 2 em) the driving force reduces as much as 50% during bubble for-mation, while the effect practically disappears for large bubbles (de > 8 em) .
Figures 2.5 A and B suggest that heat transfer occurs at higher rates than mass transfer at approximately same par-ticle size. Transfer rates are higher than according to the circulatinggasflow Q alone, when the particles are small
g
(approximately d < 100~). This is explained by Davidson
p
-I
..
-2 II -3..
a;.
2 Umf'mt
(2) 161curves refer to heat
trans-(l) Javies e.a. t 101);
(2) Toei e.a. ($1] t 216,
"
\3) Pereira e.c..(JS]; cokes 93 u
(4} Drinkenburg e.a.(90];: (5) (6) Ch1bah e.a.(o6J' 10 IZ 1401 (2x)
1-..
66~_; 11 -I..
-2..
I-·
..
Q/umf' m2f
(1) 127 I I I I I I I I I I I I (3) 145 Bcurves refer to heat
trans-(l)Barile e.a.[8l); glass 127, 365 J..1
(2)Walker e.a.[85]; sand 53,':)7~150 lJ
including chemical reaction (3)Wakabayashi e.a.[92); catalyst
145 " (4}Kato e.a. [93]; glass 192, 324 1J {S)Stephens e.a.[96]; glass 130, 250,
290,368,590 "
11 14 11
upon the transfer via the gasflow. Walker [85] , whose re-sults fall below the general trend, suggests that Murray's prediction of Qg [16,17] is a better approach, although, some of his results are not in accordance with Murray's theory. The assumption of combined diffusional and convec-tive transfer can be correct only, if the gasflow through the bubble is much smaller than expected at this moment. The exchange gasflow Q might be considered as the product of an overall transfer coefficient K and the bubble's sur-face .'!!. 4 d 2 e • (de= equivalent bubble diameter)
When calculating K from the results in figure 2. 5 A and B i t is found that in most cases K increases with increasing bubble diameter, or otherwise K is constant~ this is in contradiction with theoretical results, whichmostly predict a decrease of the transfer coefficient with increasing bub-.ule diameter.
Hoebink and Rietema [94] suggest high transfer rates to oc-cur as a result of unstable bubble motion, due to shape changes and zig zag movement of the bubbles during the
ri-sing-up. Such effects occur more likely with large bubbles and in beds of small particles, causing higher transfer ra-tes in such situations. 'l'his suggestion is probably related to cloud shedding that has been observed by Rowe e.a.
[99,100] in twodimensional beds.
For bubbles with large clouds (as happens with small bubbles in beds of coarse particles) a description of the transfer processwhich is based on a bubble cloud model seems
reaso-nable. When transfer between particles and gas in the cloud occurs, a severe change of the gas concentration in the cloud might be expected in tangential direction [6]. Such a situation is likely to occur in fluidized bed driers.
2.4
Heat- or mass transfer in a fluidized bed between par-ticles and gas is influenced by bubbling phenomena.
~ear the distribution high exchange rates may be present; these are possibly due to transfer during the formation of bubbles, or due to the fact that bubbles may be very small near the distribution plate.
Once that bubbles have become large in the higher regions of the bed exchange between particles and gas becomes less effective, as the transfer between bubbles and the dense
phase becomes limiting.
When transfer between and gas in the dense phase
is described with packed bed relations (assuming only gas phase resistance present), i t is seen that the height of a
transfer unit is very small. Irrespective of the correla-tion that is used the height of a transfer unit HOT equals some particle diameters at most, especially in the range of low Reynolds numbers that mostly pertain in fluidiza-tion (Re ~ 1); differences between different correlations are not important for practical purposes. 'i:·he tabel below is based on the work of Nelson and Galloway (40] (equation 2. 3) and is ment as an illustration.
Re 10- 2 10° 102 104
Sh 3.10- 4 6.10- 2 4.82 62.0
HOT ,tip 10.9 5.56 6.92 53.7
The calculation is based on a dense phase porosity 0.5 and the Schmidtnumber was taken Sc=l. HOT is defined as HOT
=
umf/Kg S, Kg the gas phase transfer coefficient and S the particle's specific surface; d is the mean particlep
diameter. As particle sizes usually are small, the assump-tion that gas and particles are in equilibrium in the den-se phaden-se, is reasonable.
29
-Heat transfer may occur at a higher rate than mass transfer. Heat transport between particles and the distribution plate accounts for this effect, as it has no equivalent in mass transfer.
3. Mass transfer aspects of fluidized bed drying
This chapter deals with a theoretical approach of the mass transfer in fluidized bed driers. Heat transfer as-pects will be discussed separately in chapter 4, and an-ticipating on that discussion, a uniform bed temperature will be assumed in the following analysis on mass transfer. Mass transport will be considered both for systems, in which the transfer rate is limited by gas phase resistan-ce, and for systems in which diffusion limitation inside the particles occurs.
The analysis deals mainly with processes in which solids are drying batch-wise. It is assumed that the drying of particles in a fluidized bed is a quasi-stationary process, when considered from the gas phase. Under practical con-ditions such assumption is allowed, since the change of the' mean solids moisture content will be negligeable during a time comparable with the residence time of the dense phase gas in the bed.
For the sorption isotherm of the drying solids a linear re-lationship is assumed:
c
s (3 .1)
being the moisture concentration inside the solids, Cg the moisture concentration of the gas and m the
parti-tion coefficient. For non-linear isotherms a linear
ap-proach is usually allowed over intervals that are sufficientl small. Concentrations mentioned in this thesis refer to weight-concentrations (kg/m3).
In view of the long residence time needed to dry solid particles, the solids in a free-bubbling bed may be consi-dered as ideally mixed: chapter 2 has already dealt with this subject, and experimental evidence on this point will be given in chapter 6.
3.1 Mass transfer around a bubble
Figure 3.l.A shows a bubble and its cloud as predicted by the Davidson-Harrison theory. For fluidized bed drying moisture exchange between dense phase, clouds and bubbles is assumed to take place according to figures 3.l.B and 3.l.C. Gas leaves each bubble at the top, passes the cloud cocurrently with the solids coming from the dense phase, and re-enters the bubble at its bottom. In the cloud ex-change between particles and gas takes place; moreover dif-fusional transfer occurs across the boundary between dense phase and cloud. The bubble's humidity is considered ideal-ly mixed. Diffusive transfer across the boundary bubble-cloud is neglected; humidity changes inside the bubble are due only to the convective flow of bubble gas through the cloud. For the present purpose of drying in a fluidized bed the bubble-cloud model will be simplified by the fol-lowing assumptions:
- The flow of both gas and solids through the cloud is con-stant and equal to the flow at the bubble's equator. - Gas and solids pass the cloud in plug flow.
- The zone of the cloud, where exchange between gas and solids takes place, is restricted to the hatched area of figure 3.l.B for which zone TI/4 <
e
< 3TI/4.The rather simplifying assumptions find their justification partly :i.n the present uncertainty about the real flow pat-tern around a bubble. On the other hand the results of the following analysis show that a more detailed description of the transfer process is somewhat superfluous when applied to fluidized bed drying.
Changes in moisture concentration of the gas in the cloud are described by equation 3.2:
~
sin
e
d
ccr
2 2 3 3---. d6 + 2TI Rc Kc Cgd- g ( C ) +
3
TI(R c -R b )K og s S<c; -cg>
=
o (3.2)I I
dense phase cloud bubble
I
-+'--....L
solids gas gas -:}solid
~
Figure 3.1B Mass transfer around a bubble schematically
GAS SOLIDS
Figure 3 .lA bubble and cloud
Figure 3 .lC
e
Qg cg cgdc*
g Kc Rc Rb=
tangential coordinate gas flow through the cloud gas concentration in the cloud dense phase gas concentration equilibrium gas concentration= transfer coefficient cloud/dense phase cloud radius
bubble radius
= specific surface of particles per unit of bed volume
ss
Kog
=
gas-to-particle transfer coefficient on overall gas-basisThe first term represents the moisture pick-up by the cir-culating gas flow Qg, "tihile the second and third term deal with exchange between cloud and dense phase, and with drying of solids in the cloud respectively.
Equation 3.2 will be worked out firstly for the situation that mass transfer resistance is completely in the gas phase~ the situation of mass transfer limitation inside the parti-cles will be treated in section 3.2.1.
For gas-phase resistance only the dense phase concentration
c
d equalsc*
as stated earlier. 'l'he flow through the cloudg g
Qg consists of a constant contribution Qa of dry air and a contribution of water vapour:
where V denotes the volume of a unit mass of moisture. The m
moisture concentration of gas and the gas humidity H are re-lated by:
where Po is the density of dry air. From this it may be derived: d
c
Po dH ___9. = d6 de
(l+Hv
)2 Po mWhen the equations above are inserted into equation 3.2 it follows:
Qa sine( 1+H Po Vm)de + * dH [ 2n c Kc + 3 R 2
~
n(R 3-R 3)K' c b g (H*-H)=
o (3. 3) if the overall mass transfer coefficient K is replacedog
by a gas-side transfer coefficient K'g• and if the dense phase humidity is taken equal to the saturation humidity H* (corresponding to
c; ).
Integration of equation 3.3. with boundary condition 6
=
n/4, H=
H (the bubble's humidity) gives the humidityb
Hin of the gas re-entering the bubble at 6
=
3n/4, so that:exp [-R 3_[-R 3 c b R 3 c K' g S s R ) c
1
(3. 4)In figure 3.2 (H. - H*)/(Hb-H*) is plotted versus the bub-J.n ble radius Rb as calculated from equation 3.4. Both the theory of Davidson and Harrison (drawn curves) and of Murray (dotted curves) were used in the calculations. According to the former [14] :
1/3 Rc = [ a+2] ~ a-1 According to Murray [ 16,17]: =
[~]1/3
a-1 ate
rr/2Calculations were made for three different particle diame-ters. Dense phase velocities ud were taken equal to the minimum fluidization velocity, determined from Ergun's equation with the porosity E = 0.45 and the particle den-sity pp
= 1350 kgjm3 •
_,
II
H:•Hro llip:ZII~ dp:JH!l dp:UO!L
H•Hb
t
11°C drav.:n curves: avidson/Harris n dotted curves Murray -r..
\ \ \ -3r"
..
J
•• c \ 4Figure 3~2 Gas saturation in a cloud
Bubble velocity was calculated from Ub = 0.71Jg 2 ~· Kc was taken from the relation of Chibah and Kobayashi
[86] , and Kg from the work of Nelson and Galloway [40], assuming a Schmidt-number Sc = 1 and a slip velocity be-tween particles and gas in the cloud which equals:
Q8/(l-e:)- Qg/e:
