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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. Eindhoven: Technische HogeschoolEindhoven. https://doi.org/10.6100/IR38366
DOI:10.6100/IR38366
Document status and date:Published: 01/01/1977
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DRYING GRANULAR SOLIDS IN A FLUIDIZED BED
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
DAUK: W16RO HELMOND
Dit proefschrift is goedgekeurd door de promotoren:
Prof. Dr. K.Rietema (le promotor)
Prof. Dr. Ir. W. P. !·1. van Swaay ( 2e promotor)
Dankbetuiging
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.
Curriculum vitae
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.
Contents
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
4. Heat transfer aspects 69
5. Experimental equipment and solids properties 77
6.
5.1 Equipment 77
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 -
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 H
3 uo e::
(1.1)
- 2 -
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 it can be estimated by combining equations (1.1} and
(1. 2).
For u0
> 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
- 3 -
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
- 4 -
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; it may become
elongated as well as flattened. The rising velocity Ub of
a single bubble is related to the bubble volume:
ub = 0.71 g 1/ 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, it 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
- 5 -
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 bed 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.
- 6 -
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
- 7 -
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.
- 8 -
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
[33-35j.Plug flow of solids is approached in shallow beds wi·
- 9 -
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] .
- 10 -
Nu 2 + 0.7 Re 1 / 2 Pr1/3 (2. 1)
Sh = 2 + 0.7 Re 1 / 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. Reported 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)
- 11 -
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 its neigh
~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] Re 1/ 2 scl/3
q = (1-e:)l/3
As 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]
t
- 12 -
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.
- 13 -
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 homogeneous 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 measured by
Kettenring e.a. themselves with a bare thermocouple.
- 14 -
'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 embedding 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 is the 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
- 15 -
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 consum
ing numerical calculations because of its complexity.
- 16 -
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
it 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.
- 17 -
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 phase. 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;
- 18 -
- 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:
- 19 -
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 called 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- 1
For 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
- 20 -
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.
- 21 -
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.
- 22 -
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
- 23 -
the same volume. Proposed values for A range from o
[86-88] to 3 [14]: the minimum value reported forB
is 0.36 x s0
(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 are ve
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-
Table 1.1 heat and mass trans-~ c c "' ~e~ discrete c c c
~ .;; 0 c 0 0 -....~ three-
"' ·~ '" w 'M W '" "'. dimensional fluidized • ~ " " .w..; +'-< "' c"' +' " "' " •• •• ., ~"' "' I'd t:: n:l ::.1 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 re
measur::-. :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 18.90 100 catalyst 66 Injection of single based on
ub = I bubb~'; concentration measured via local e.a. yes volume of bubbles, containing I ~~ii~~g. in the bed; dense phase concen-[ 90] tracer; different tra !~~ected 0,71~ zero
Toei {91] lOxlO <100 glass 161,216~Injection of measured 1 measured temperature of bubble measured locally in e.a.
into "'~~ld yes via two I v~; capa- the bed with 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 Injection of yes via lvi~;:'~pa- sampling in the bed; phase concen-51 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- no o, 71\/gde at upper bed local dense concentration; plug ' ble tracer is mercury level flow of dense" phase gas assumed
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
Symbols used: d bubble diameter
- 25 -
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 phase concen-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 equivalent bubble diameter d ; the gasflow Q through the bub-e g ble is indicated as Q /u f according to the theories of
g m 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
and Harrison [14] by superimposing diffusional transfer
I ..
-I ..
-2 II
-3 ..
a;. 2 Umf'm
t (2) 161
curves 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
0
" I
-I ..
-2 .. I
-· ..
Q/umf' m2
f
(1) 127
I I I I I I I I I I I I
(3) 145
B
curves 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 I
- 27 -
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 surface .'!!. d 2
4 e • (de= equivalent bubble diameter)
When calculating K from the results in figure 2. 5 A and B
it 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.
- 28 -
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), it 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° 10 2 10 4
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 particle p
diameter. As particle sizes usually are small, the assump-
tion that gas and particles are in equilibrium in the den
se phase, 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.
- 30 -
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.
- 31 -
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 -----. ( C ) TI(R -R )K S d6 + 2TI Rc Kc Cgd- g + 3 c b og s
<c; -cg> = o (3.2)
- 32 -
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
F.xchange zone in a cloud
e Qg
cg
cgd
c* g
Kc
Rc Rb
- 33 -
= 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-basis
The 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 equals c* as stated earlier. 'l'he flow through the cloud g 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 d e (l+H v )2 Po m
- 34 -
When the equations above are inserted into equation 3.2
it follows:
Qa * dH [ R 2 ~ n(R 3-R 3)K' sine( 1+H Po Vm)de + 2n c Kc + 3 c b g
(H*-H) = o (3. 3)
if the overall mass transfer coefficient K is replaced og 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 humidity b
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 R ) 1 s c
(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 at e rr/2
Calculations 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 density pp = 1350 kgjm3 •
- 35 -
_, 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 .. \ \
\
-3 r" ..
J •• c
\
4
Figure 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:
1T (Rc 2 - ~ 2)
- 36 -
Here Q is the flow of solids through the cloud; s
according to Davidson and Harrison it holds that:
0 0 The chosen bed temperatures were 40 C and 60 C; the
equilibrium humidity H* corresponded to 100% relative
humidity at the bed temperature.
Figure 3.2 clearly shows that the gas re-entering the
bubble is very near to saturation. ~he differences in ab
solute gas humidity, as predicted by the theory of
Davidson/Harrison and the theory of Murray, are comple
tely negligible.
3.2 Mass transfer limitation inside the particles
In fluidized bed drying mass transfer between parti
cles and gas occurs both in the dense phase and in the
clouds.
As stated earlier equilibrium between particles and gas
exists everywhere in the dense phase when only gas phase
transferresistance is present; this means that particles
can get drier only when they pass a cloud, since a driv
ing force for mass transfer is present in the clouds
only.
In case that diffusion resistance inside the particles
limits the drying rate complete equilibrium between par
ticles and gas in the dense phase is not reached in gene
ral , since concentration profiles inside the particles
are present. ~he development of these profiles is a slow
ly proceeding process. Section 3.2.2 considers the long
term response of particles to changes of the gas concen
tration.
Particles which pass a cloud are exposed suddenly and for
a very short time to a low gas concentration. This pro
cesswhich is repeated very frequently, is considered as
a short term response of particles to a change of the gas
concentration; it is treated separately in the next section.
- 37 -
3.2.1 Short term response of a drying particle
When wet particles are brought into contact with dry
air for a short time, the particles get drier at their
outer surface only if diffusion resistance dominates.
The thickness of the relatively dry layer at the par
ticle's surface will be very small, not only because of
the short contact time, but also because of the usually
large value of the partition coefficient (order of mag
nitude: m = 104). This will be illustrated below for par
ticles passing a cloud; the contact-time of solids and
gas in the cloud has an order of magnitude of 0.1 s.
For unsteady-state diffusion in a sphere, 99% of the fi
nal equilibrium concentration is reached in the whole
sphereafter a contact-time tc between sphere and surroun
ding gas that corresponds with a Fourier number Fo =
ID t /d 2 = 0.4 [101]. Taking a moisture diffusivity ID in
sid~ the particle of 10-10 m2/s and a particle diameter
dp 100~(which both are conservative values) it is seen
that the maximum Fourier number to be reached in the cloud
is about 10- 3 ; this means that diffusion occurs only in an
outer shell of the particle.
The effect of the partition coefficient m is explained
with the aid of figures 3.3 and 3.4.
X
t
fig. 3.3 fig. 3.4
- 38 -
suppose a particle enters the cloud with uniform moisture
content Xd. Hhen leaving, its mean moisture content is
~e; the particle has dried in a thin outer shell o, in
which a linear moisture profile may be assumed when o is
small. X. is the moisture content at the particle surface ~
when leaving the cloud. A moisture balance over the cloud
gives:,
where Ps and p0
are the densities of dry solids and dry
gas. and Qa are the flows of solids and gas through the
cloud.
For the layer 6 the following relation is applicable:
X) e
It follows from combination of the above equations:
== 2 Po 0a 3 Ps 0s
If gas phase transfer resistance is absent, H. can be ap~n
preached by Ps Xi/m p0
• The largest value of o is found
when Hb is taken zero. It follows that:
< 2 3
The flows of gas and solids through the cloud are of the
same order of magnitude, so Q /Q z 1. The partition coef-g s 4
ficient has an order of magnitude 10 . If Xi == 0.99 Xd
is assumed, which means that a high degree of equilibrium
is reached in the cloud, it is found that:
which indeed is very small.
- 39 -
In the following a mass transfer coefficient will be de
rived to describe the transport between particle and gas
during a very short contact time.
Suppose a spherical particle (radius R 1 uniform concen-o p
tration cs ) is surrounded by a closed volume of gas.
At time t=o the gas concentration Cg is zero~ at t > o
the gas concentration is uniform and in equilibrium with
the surface concentration of the particle. For constant
diffusivity ~ transport of moisture inside the particle
is described by the following differential equation with
boundary and initial conditions:
a c ~ a 2 2 ar (r ----af-) r
t 0
t > 0
0 < r
r > R p
r = o
r == R p
$. R c c 0 p s s
c g 0
The solution of these equations is known from the litera
ture [101,102], and allows calculation of a partial mass
transfer coefficient Ks for the particle, which is defined
by the next equation:
K <c* - c ) s s sR
c* is the concentration inside the particle when complete s equilibriumbetween particle and gas has been reached.
From a mass balance it follows:
c* s ==
c 0 s
v 1 + 4/31T
c 0 s m
100
- 40 -
where E denotes some kind of an extraction factor.
In dimensionless terms Ks
number Sh = K R /~ and is s p
1 3 E
2 'f -qi Fo 1 e
Sh 'f e - qi2 Fo 1
is expressed as the Sherwood
found to be:
q. 2 J.
E2 2 + 9 (E+1) q; ( 3. 5) 1
E2 qi 2 + 9(E+l)
in which Fo = ~ t/Rp2 , the Fourier time, and qi are all
positive roots of the equation:
(3 + E
In figure 3.5 E Sh is
parameter.
F..Sh
• I
2 q. ) tan q. = 3 q. J. J. J.
plotted versus Fo/E2 , with E as a
E = 0. 82
10-l
E = 0.082
Figure 3. 5
E .. Sh versus Fo/E2
according to equation 3~ 5
- 41 -
As seen from this figure distinction should be made be
tween large and small values of E. In the former case
E Sh initially is proportional to 1/ .JFo/E2 , as is well
known from the penetration theory [103], and tends to a
constant value for large Fo/E2 • When E is small, the pene
tration period is followed by a large interval in which
E Sh is proportional to 1/(Fo/E2), before E Sh becomes 2 constant at large Fo/E • Small values of E are very like-
ly in fluidized bed drying; since the bed porosity is
about 0.5, the volumetric ratio of gas and solids is near
to unity, which means E ~ !. The proportionality E Sh ~ 2 m
1/(Fo/E ) means that the mass transfer coefficient Ks is
independent of the diffusivity of moisture inside the
particles. This seems surprising but is caused by the
fact that the gas phase concentration has become quite
near to its equilibrium value before the diffusion pro
cess inside the particle has properly begun. Appendix A
explains this phenomenon via a simplified approach, which
also takes into account transfer limitation by gas phase
resistance, as is likely to occur for small values of
Fo/E2 •
The foregoing will be applied now to the transfer be
tween particles and gas in clouds.
Since particles which pass a cloud, get drier only in a
very thin layer at their outer surface, the existance of
a concentration profile in the particles is not important
any more from the moment on that these particles enter a
cloud. Inside the cloud they may be considered as having
a uniform profile with a concentration equal to the sur
face concentration on the moment of arrival in the cloud.
This means that the equilibrium gas humidity H* of equa
tion 3.3 can be approximated by the humidity that is in equilibrium with the surface moisture content of the par
ticles.
In section 3.1 the cloud has been considered as a co
current plug flow mass exchanger. When the linear veloci
ties of gas and solids in that mass exchanger are about
- 42 -
the same, the mass transfer process is comparable with
the batchwise contact between a particle and stagnant
gas. This means that equation (3.5) is applicable for des
cribing transfer rates in the cloud. As said before the
maximum Fourier-number to be reached by a particle in the -3 2 5
cloud has an order of magnitude 10 . So Fo/E equals 10
when E ~ 1/m is taken as 10-4 .
From figure 3.5 it follows that the relation:
E Sh = 0.133 (3. 6)
holds for the major part of a particle's residence time
in the cloud. Only for small values of Fo/E2
this equation
is not correct, since in that case E Sh ~ E/vfFo; however,
as shown in appendix A, transfer resistance in that period
is mainly in the gas phase.
For description of the mass transfer process in the cloud
when diffusion limitation is present, the mass transfer
coefficient K of equation 3.3 is expressed as: og
1 1 0.133 R t
Kog --+ K' m s 0.133 R K'
g p g
Tle time t is related to the tangential coordinate e in the
cloud:
t 2 3 7T
R 3 _ 3
c ~ (1/2 /2- case) Q
g
from which the local transfer coefficient K is derived: og
K og
K' I [1+57T -R~c-3~-R=b-3 g Qg
K' j ~ (1/2 Vz- case) (3.7)
With equation 3.7 the concentration profile in the cloud
is found by integrating 3.3:
H* -l1+57T
R 3 - R 3 K' r 0.4 (1-E)
H ':J b ~(1/2 Vz-cose} H*- Hb Qa(l+H*po v )
m 2n 2 l exp [- (~v2-cose} (3.8} Q (l+H* p V )
a om
- 43 -
H* here is the gas humidity in equilibrium with the sur
face moisture content Xi of the particles:
* H
with CsR the surface concentration of the particles.
vJhen a = 31T/4 is inserted in relation 3.8, the humidity
H. of the gas that leaves the cloud and re-enters the ~n
bubble is found:
* H -H. ~n
exp
- 0.4(1-£)
:· g] p .
l- 21T/2 R 2
K ] c c (3. 9)
o.sr---~--~---r---,
0.4
0,3
0
t
I
dp = 200 fl
_.;,.- Rb,cm
dp= 300 p
~ Rb.cm
dp=400iJ 0.3
-~)lo>jllllll'- Rb.om
2 4 0 2 4 0 2
Gas saturation in a cloud with diffusion limitation inside the particles (drawn curves:Davidson/Harrison, dotted curves: durray)
4
- 44 -
In figure 3.6 equation 3.9 is plotted as function of the
bubble radius. Calculations were made for the same condi
tions as mentioned for figure 3.2, where only gas phase
resistance was present.
3.2.2 Long term response of a drying particle
Suppose one single particle is exposed to a gas flow Qp
with initial moisture concentration Cg0• The particle's
radius is Rp, and its initial moisture concentration is Cs0•
Thegas around the particle is ideally mixed~ gas phase mass
transfer resistance is present in a thin film at the par
ticle's surface. Transport inside the particle is described
by the following differential equation with initial and
boundary conditions:
t 0
t > 0
t > 0
Ill 2 r
0 <
r =
r =
r~
0
R p
R p
ac
ac s
ar
s ar
- ID
= c 0
s
0
ac c s K ( sR - Cg) ar g m
When the gas volume around the particle is small, accumu
lation of moisture in that volume is negligible • All mois
ture released by the particle must be transported by the
gas flow:
The combined equations above can be solved via Laplace
transformation [1041.
The concentration profile inside the particle is given by:
c - m c 0 00 sin sin r/R 2 l.li-l.li cos \.11 lli sr g 2 E p e-lli Fo .
r/R cs
0 - cg 0
1 \.1.-sin lli cos l.li l.li m ~ p
(3 .10)
with Fo = ~ t/R 2 • p
- 45 -
~i are the non-zero positive roots of the equation:
(1-B) tan ~i = ~i
where B = K Q R /(Q +4nR 2 K )mIll = Bi /(1+N ) 9PP p p 9 9 g
The Biot number Bi and the number of mass transfer units g Ng are defined here as:
Q p
Ng should be expressed for conditions inside the fluidized
bed. If the gas flow Qp is considered as the flow per par
ticle present in the fluidized bed it follows:
R2 1 (1-E)L 3 K (1-£) L Kg ss L Ng 4'IT Kg 4/311 R 3 = p uo u R u p 0 p 0
where E: = bed porosity u 0
superficial gas velocity
L total bed height ss specific surface of particles
In figures 3.7 A until D concentration profiles for the
particle are shown (drawn curves) as well as the gas concen
tration (dotted line) ; each of the figures corresponds to a
certain value of Big and Ng' the parameter in each figure
being the Fourier number.
In figures 3.8 A and B respectively the gas concentration
and.the mean solids moisture concentration are plotted ver
sus the Fourier time for different values of Ng and Big.
Some remarkable facts are to be concluded from figures
3.8 A and B; drying proceeds much quicker when the number
of mass transfer units Ng is decreased at constant Big' or
when the Biot number Big is increased at constant Ng. Both
these facts indicate that mass transfer in a fluidized bed
drier is limited by the capacity of the gas flow to remove
the moisture released by the particles. h decrease of the
- 46 -
t Big 10
Ng 10
.. r/R
0 0 0.2 0.4 OJi 0.1
3.7A
I I I 0.41 T
1-1.64
o.l 0
io- C 8 -mCg --..;
c~-mCg0
0,6 - t -- Fo= 6.55 I
I --0.4 -
I Big"" 10 I'"
Ng= 250 I 6.2- -
I - .... r/R
0 I I I I I I I I I I
0 0.2 0.4 0,6 0,1
Figure 3.7B
0,6
0.2
0
Big 250
Ng=10
0,2
Figure 3.7C
0
c.-mcg0
c~-mcgo
t Big= 250
Ng 250
Fo= 1.14
1.2
Fi~ure 3.7D
""' 47 ""'
0.10
0.4 0.1 0.1
)1. r/R
0.4 0.1 o.a
- 48 -
Figure 3.6A Dimensionless gas concentration versus Fourier
250.,1()
-~-Fa
Figure 3.8B Dimensionless solids concentration versus Fourier
number of transfer units may mean an increase of the gas
flow; an increase of the Biot number may mean an increase
of the bed temperature, which will diminish the partition
coefficient m.
From equation 3.12 a partial mass transfer coefficient Ks
can be calculated, which is defined as follows:
d c - ID( ·. s
ar r=R p
= K (C - c ) s s sR
- 49 -
It can be derived that: 2 Fo ,, - j..ti
l: B K R ]l,2 + s2-s e
Sh _§_____,£ 1 1 ~ (3.11) = 3 s ID 2 2 "' B - J.li - j..ti Fa l: 2 + s 2-B
e 1 J.li
with Shs the Sherwood number for the particle.
Figure 3.9 shows Shs as function of Fa, with Ng and Big
as parameters. Shs becomes more or less constant from
Fo = 0.2 on. This does not mean that the drying process
can always be described by approximating Sh3
by its limiting value, since (see figure 3.8) the process might be over al
ready before that limiting value is reached.
10,250
10,10 250,250
250,10
~Fo
00 0~
Figure 3.9 Plot of equation 3-11
- 50 -
3.3 Mass transfer behaviour of the whole bed
3.3.1 Mass transfer limitation by gas phase resistance
~
Changes of the moisture content of a rising bubble are
described by equation 3.12:
(3.12)
Vb is the volume of dry air in the bubble at the gas pres
sure near the bottom. Although the volume will change a
little during the rising-up as a result of an increase of
the bubble's moisture content Hb, this will hardly change
the bubble velocity ub. Both vb and Ub are considered as
constant over the bed height L.
After substitution of relation 3.4 equation 3.12 can be
integrated with boundary condition l
is found that:
o, Hb = H0
, and it
2 R 3 - R 3 with Bm (K + .!. c b
Kg ss R ) Q (l+H* v ) c 3 R 3 c
a Po m c
The term vb ub HTU may be considered as the
Q (1-e -Bm) m a height of a mass transfer unit for the bubble.
Figure 3,10 shows HTUm as function of the bubble radius ~,
for the same conditions as mentioned at figure 3.2.
Dotted curves refer to the theory of Murray, drawn curves
to the theory of Davidson and Harrison. Since e-Bm is very
near to zero, the bed temperature has no influence on the
results, although Sm itself depends on temperature as
shown in section 3.1 (see equation 3.4 and figure 3.2).
~he exit gas humidity He is found from:
He = s Hb + (1-s) a* l=L
- 51 -
u s is the fractional bubble flow : s = b/u
0 with ub the
superficial bubble velocity. The latter equation may be
applied to both batch and continuous drying, and an over
all mass balance will show the decrease of the mean moisture content of the solids.
3.3.2 Mass transfer limitation inside the particles
Some attention will be given firstly to the special
case of diffusion limitation inside the particles, com
bined with ideal mixing of both solids and gas. Ideal
gas mixing here means' that the composition of dense phase
gas and bubble gas is the same and uniform throughout the
bed. In this special situation all particles show the same
drying behaviour. Therefore such drying process is fully
described by the long term response of a single particle,
discussed in section 3.2.2. Equation 3.12 can be used to
calculate the course of the drying process.
In practice ideal gas mixing is not likely to occur.
'l'he bubble gas is usually considered to be in plug flow;
~lug flow also is a good approximation for the dense phase gas,
when large particles are fluidized in rather low beds. In
such case the residence time of the dense phase gas in the
bed is small and the spread of residence time due to gas
mixing may be negligible. As will be shown later from the
results of the present analysis the state of mixing of the
dense phase gas is not very important for fluidized bed
drying. As a consequence particles in the bottom region of
the bed are exposed to larger driving forces than particles
in the upper part of the bed. In principle this may cause
that particles near the bottom are relatively dry when com
pared with particles in the top region; due to particle
mixing wet and dry particles are present next to each other
in each volume element of the bed. In this respect distinc
tion should be made between three kinds of mean solids
moisture concentration:
100~-r--.-~-.--.--.--.-~, /
HTU
1
10
(em) //
I f
I I
I I
/ /
/
/ /
/
/
/ /
-Rb
I I I
I I I I
~ 52 -
" / /
/ /
/ /
/ /
/ /
I I
I I
I I
I I
I I
30:1 fl
-Rb
·~~_L~L_~~~~~--~~. 0~-L--L-~~~J-~~~~ 4 0
100
drawn curves: Davidson and Earrison
dotted curves: Murray
/ /
/ /
/ /
/ /
/
I ,10 I
l I I
I I
I
4 I
I I
d = 400 f1 ?
-Rb
' 4
Figure 3.10 Be1ght of a mass transfer unit for the bubble as function of the bubble radius
Cs the mean concentration of an individual particle;
C the solids concentration, averaged over the
particles in a small volume element of the bed;
C the solids concentration, averaged over the whole
bed content.
Differences between these three mean concentrations may
arise, and will be due either to solids mixing or to dry
ing of solids in a bubble cloud, in which particles are
exposed to a larger driving force, than in the dense phase.
When both solids mixing and bubbles are absent (as occurs
- 53 -
in a packed bed) C equals , and both depend strongly on s =
the height in the bed, so deviate from Cs' When bubbles
are present but solids mixing is absent (a situation not
likely to occur in practice) differences between C and s Cs will arise, since some may pass a cloud fre-
quently and get exposed to a larger driving force, while
others will meet a cloud only seldom. Due to particle mi-
xing differences between and will become ~qualized
to some extent, but differences between Cs and Cs still
may exist due to the action of bubble clouds.
In the following differences between and will not be
considered anymore, since they cannot be calculated nor
measured; C is the moisture concentration of particles s that is measured by sampling the solids content of the bed.
A question a~ to whether solids are ideally mixed (which
means: Cs = Cs) can be answered by a comparison of the
time Tm' necessary to mix-up the bed's content completely,
and the relaxation time 'p of a particle, which is the time
that a particle needs for complete adjustment to a change
in its surface concentration. When Tm <<
ideally mixed.
the solids are
T can be estimated from the Fourier-number for a particle.
when Fo = ID T /R 2 = 0.4, the particle has almost reached p p
equilibrium with its surface concentration [101]. The timeT is more difficult to estimate, since the mecha-
m nism of particle mixing in a fluidized bed is not yet quite
well understood (see chapter 2).
Two different approaches will be considered here for an es
timation of T in practice. m
If solids mixing can be desribed with an axial mixing coef-
ficient Es' the time t needed to establish a stationary con
centration profile in a bed is given by:
Fo 1/4
- 54 -
L being the total bed height [101]. It seems reasonable
to use the same criterion for estimating the time needed
for complete solids mixing. Es will be determined according
to a relation presented by Potter [29]:
although use of that equation may lead to serious mistakes,
since the bed diameter is not included in it, while that
diameter has considerable influence on solids mixing {see
[28]). It follows:
If solids mixing is a result of solids transport caused by
bubbles, ' can be estimated from L/u , where u is the m p p
circulation velocity of the particles. i~ccording to Hoelen
[108], who based his calculations on the model of Bayens
and Geldart [31],
u p
up equals:
fw 0 b{uo - umf)
1-£ - ob0+2 f l w w
in which ob is the bubble hold-up, and fw the wake fraction
of bubbles. Rowe e.a. [12] found that f is about 0.25. w
A comparison between ' and ' will be made here only for m p data that refer to experiments to be described in chapter 5.
Assuming umf = 5 cm/s, u0
= 2 umf' db = 1 em, ob = 0.15 and
L = 30 em it follows
for axial mixing: ~s = 5 cm2;s, 'm = 45 seconds
for solids circulation u = 0.36 cm/s, ' = 84 seconds. p m
For a moisture diffusivity lll -11 2 10 m /s and a particle
radius R = 150~, a value ' 900 seconds is found, which p p means that ideal mixing of solids is a good approximation.
- 55 -
Mass transfer between particles and gas will be considered
now for the situation that the solids are ideally mixed.
Any influence of increasing moisture concentration on gas
velocities will be neglected.
The following superficial gas velocities are ascribed to the
dense phase, the bubble phase and the cloud phase respec
tively*:
dense phase (1-o -a Ju c b d
bubble phase ub + 3 ud
cloud phase e:(U -ud cb
3- rl b E: c
Here ob is the bubble hold-up; the cloud hold-up oc is re
lated to the cloud and bubble radius via:
The use of the foregoing velocities implies that the gas
flow in the bed is not influenced by solids movement. If
solids movement in the dense phase takes place at a dis
tinct velocity (as may occur in case of solids recircula
tion) the gas velocities mentioned above should be correct
ed for such effect, which will become important when the
number of bubbles in the bed is large.
When all phases pass the bed in plug flow, the following
balances describe the concentration changes:
(1-o -o >u d cgd c b d dl
c Kg S ( sR - C )- K
s m gd c s ( c d-e )( 3 . 13 > c g gc
*rn appendix C these three velocities are derived. It is shown there furthermore that combination of the velocities above with the total gas flow balance yields the same cloud as derived by Davidson and Harrison [14], however without the need to assume potential flow of solids around the bubble.
- 56 -
3 u d) cS £ d c gc £ c dl Kc Sc(Cgd-cgc)- n Qg(Cin-cgb)+
d cgb dl
c + o K' S ( sR - C )
c og s m gc
= n Q (C. - cgb) g ~n
(3 .14)
(3 .15)
cgc is the mean concentration of the gas in a cloud at
height l in the bed, and K~g is a mean overall mass trans
fer coefficient between particles and gas in a cloud. The
number of bubbles per unit bed volume is n = 3 ob/4n ~3·
All other symbols were introduced already in descr1bing
the mass transfer in a single cloud (section 3.1). From
that description relations can be found for C. , C and ~n gc
K' • og According to equation 3.9 C. is related to the sur-~n .
face concentration of the particles and the bubble concen-
tration;
c - c. sR/m ~n C /m - C sR gb
Although C Rand C. do not corresponc to the same level in s ~n
the bed, the use of equation 3.16 is allowed nevertheless
if CsR does not vary strongly with the height in the bed.
As stated earlier the influence of the gas concentration
on gas flow is neglected; Qg is taken constant. Since the
right-hand side of equation 3.16 is independent of the
height coordinate ·l it can be put equal to a constant y.
The mean concentration in a cloud, Cgc' is found by inte
grating the local gas concentration Cg in a cloud over the
cloud volume:
_ 1 (3n/4 1/2 V2j cg sin Sde
n/4
- 57 -
When relation 3.8 is inserted in the foregoing equation
it is found that egc is related to esR/m and egb
esR/m - egc
esR/m - egb !3~/4 R 3_~3 K' -0.4(1-s) [ 1+5~ c Q r(~/2-cose >]
g p ~/4
exp
f
The right-hand side of 3.17 is constant and set equal to f.
The mean mass transfer coefficient K' is derived from og equation 3.7:
K' = og W2 2 r·~· Kog sin ede
1T/4
R Q9 l K' R 3 - ~3 ] STI72 ~n 1+51T V2 a! c (3.18)
R 3 R 3 Qg c b
d esc d e Because of = f ~as follows from 3.17, the
d~ differential eq~ations3.14 and 3.15 can be combined. When
equations 3.16 and 3.17 are substituted into the differen
tial equations the following result is obtained in dimen
sionless form:
d cb = f(Nc+Nb)cb- N cd (3 .19) dO c
d cd = (Nd+S Nc)cd - fS N cb (3.20) dO c
The definitions of the dimensionless quantities are given
below.
Variables:
esR/m - egb
esR/m- ego
esR/m - egd
esR/m - ego
~ a = L
- 58 -
Parameters:
N = c
K s L c c
(E 0 f + oblub-ob(1-f)3 c
(1-o -o )u c b d
ud Nb
s
o K' S L c og s
(e: oc f+ob)ub-ob(1-f) 3 t
(1-oc -ob) ud
C 0 is the moisture concentration of the gas fed to the bed, g
and L is the total bed height.
The solution of the equations 3.19 and 3.20 is laborious but
straight-forward. The dense phase concentration and the bub
ble concentration as function of the height in the bed can
be expressed by:
(3.21)
(3.22)
The coefficients of these equations, which depend on the
dimensionlessparameters above, have been summarized in
table 3 .1.
The equations 3.21 and 3.22 are applicable for calculation
of the total concentration of gas leaving the bed (in rela
tion to the surface moisture concentration of the solids)
in both continuous and batch drying:
c e
Cge - CsR/m
Cgo- CsR/m (3.23)
(The cloud gas is assumed to remain in the bed when bubbles
pass the upper bed level).
Figures 3.11 - 3.13 show some results that were calculated
from equations 3.21 and 3.22. The basic data were obtained
in the following way:
- 59 -
Table 3.1
Constants in the equations 3.21 and 3.22.
al -(E - D - >..2)/(1..2 - >.. 1)
a2 +(E - D - >..1)/(1..2 - >..1)
>..1 - 1/2(D+A) + 1/2 V<D+A) 2 - 4 (AD-BE)
"z - 1/2(D+A) - 1/2 V<D+A) 2 - 4(AD-BE)
A = f (Nc +Nb)
B Nc
...
•••
0.0
...
1.1
...
.Figure 3.11
C 9 b-C•0
c.,..,.,.-c,o
t
Figure 3 • .1.2
c ... -c,0
c.,.,/,.-c8 o
t
Figure 3.13
- 60-
fixed data:
-t .om
2D Bubble concentration as function of the
II
in the bed; influence of the supergas velocity u
0•
u 0
~ 26 cm/s
6 ~ b
0.15
d = 300u p - t • Cll'i
21
Bubble concentration as function of in the .bed; influence of the bubble
u0 "" 26 m/s
ob = 0. 15 db s em
__....,_ z. ~ern
to
Bubble concentration as function of the height in the bed influence of the particle size
ID
It
- 61 -
the superficial gas velocity in the dense phase was
taken equal to the minimum fluidization velocity, and
calculated from the Ergun-equation, assuming a poro
sity €=0.45 and a solids density pp = 1350 kg;m3 •
the mass transfer coefficient K was taken from the c relation of Chibah and Kobayashi [86]:
particle-to gas transfer coefficients were calculated
from the work:of Nelson and Galloway [40], and based
on the dense phase velocity and the slip velocity in
the cloud for the respective processes;
the bubble hold-up was taken constant and equal to
0.15, while the linear bubble velocity was calculated
from ub = (u0-ud)/ob;
transfer in the cloud was considered only for the sim
plified cloud-model of Davidson and Harrison, that has
been described in section 3.1.
The results in figures 3.11 - 3.13 concern dimensionless
bubble concentrations(CsR/m-Cgb)/(CsR/m -cg0) only.
Under all circumstances the dense phase gas reached equili
brium with the surface of the particles from a few milimeters
above the distribution plate on, which is not surprising at
all in view of the conclusions in section 2.4; for this rea
son the state of mixing of the dense phase gas is unimportant.
Bubble concentrations were plotted as function of the height
l above the distribution plate; the influence of gas veloci
ty u , bubble diameter db and particle size d is illustrat-o p ed respectively in figures 3.11, 3.12, and 3.13.
3 .3.3 Batch drying of the bed
The following analysis of the moisture losses by the so
lids in a bed applies to batch drying only.
The amount of moisture released by the solids is:
d cs .... L(l-o ) (1-E:)--b dt u (C - C
0) o ge g
C R u (1-c ) (-s- -
o e m (3.24)
- 62 -
where ce follows from 3.23.
The moisture removed by the gas must be transported from
the interior of all particles to their surface. If, as
assumed earlier, ideal mixing of solids occurs, the mean
drying rate of particles can also be expressed as:
m (3.25)
K is a partial mass transfer coefficient inside the par-s ticles, which may depend on time; an ~stimate of K~ will
be given later.
When CsR is eliminated from equations 3.24 and 3.25 it
follows:
(c - m c 0)
s s g s
(3.26) L(l-ob}m K
1 + s u (1-c } o e
which relation can be presented in a dimensionless form:
d c s dF
0
- 3 Sh
s (3.27}
- 0 0 is defined as: c = (C - m c }/(C o_ m c ) c o s s g s g's
being the initial solids concentration. The Sherwood number,
already defined in section 3.2.2 is Sh = K R ;m. s s p
is some kind of a Fourier-number, uased on a mean con-
tact time of the gas in the bed.
Ill L(1-ob} (1-E)m T =
b R 2 uo(1-ce) p
For constant , 'b determines whether or not diffusion
resistance inside the particles is an important factor for
the drying process. For large 'b relation 3.27 reduces to:
d c s
dt
u ( 1-c ) o e
- 63 -
which means that diffusion resistance inside the particles
plays no role in the drying process. The differential equa
tion can be integrated with initial condition t = o, cs = 1.
It follows that:
c s ( 3. 28)
Diffusion resistance dominates the mass transfer process
completely when 'b becomes very small. Equation 3.27 can
be transformed for that case into:
d c s <It c
s
Under certain conditions, to be specified below, K can be s
considered as constant, from which may be derived:
exp [- 3 ( 3. 29)
The exponent Ks t/Rp is comparable with a Fourier-number
Fo Ill t/R 2 • p
Relations 3.27 - 3.29 cannot be compared with experimental
results from the literature, since published data are not
complete enough. However qualitative support for the use
fulness of the relations is found in mass transfer studies
mentioned in chapter 2.
Vanecek e.a. [26], drying fertilizers in a fluidized bed,
showed experimentally that the dimensionless solids concen
tration c is a function of t/R 2 only for results obtained s p with different mean particle sizes. This indicates that dif-
fusion limitation controls the drying process, as described
by equation 3.29.
Richardson and Bakhtiar [56] studied adsorption of some or
ganic compounds by alumina catalyst in a fluidized bed.
They assumed complete equilibrium (ce = o) between solids
and gas at the upper bed level, and calculated the solids
- 64 -
concentration from the measured gas concentration.
Diffusion limitation inside the particles did not occur,
since they found that the adsorption process was a func
tion of u t/L k d b d only, which corresponds with o pac e e equation 3.28. Richardson and Szekely [57]as well as Hsu and Molstad
[71] report on adsorption of carbontetrachloride by ac
tive carbon. They observed that a mass transfer coeffi
cient, calculated by assuming ideal solids mixing and
plug flow of gas, decreased in time, due to an increasing
diffusion resistance inside the particles.Such a situation
is to be expected when 3 Shs Tb has an order of magnitude
unity (equation 3.27).
An estimate of the time-depending value of Shs or Ks
can be made from the long term response of a single dry
ing particle, which was discussed in section 3.2.2. When
applying that analysis, the parameter Ng should be based
on the total superficial gas velocity U0
in the fluidized
bed and on a gas phase transfer coefficient Kg that cor
responds with the same velocity u0
; as shown in
3 .2.2 the drying rate of a particle may be limited by
the moisture removing capacity of the gas phase, which de
pends (apart from the bed temperature) on the gas velocity
u·o· Using Shs from relation 3.11 in a heterogeneously
fluidized bed may introduce some deviation when bubbles
and dense phase gas leave the bed with different concen
tration; however this deviation is expected to be very
small, since the time-dependin~~ values of Shs are influ
enced only slightly by the parameters Bi and N (see fi-g g
gure 3.10). The same figure shows that Sh becomes more s
or less constant from Fo = 0.2 on; this means that equa-
tion 3.29 can be applied when the loss of moisture by the
particles is negligible in the period Fo < 0.2.
In figures 3.14 - 3.17 the dimensionless solids concentra
tion cs is plotted versus time for a number of drying con
ditions, that are mentioned in each figure. Calculations
- 65 -
were made by integrating relation 3.27 numerically. Each
of the figures also presents l-ee' which is the degree
of equilibrium that is reached between exit gas and par
ticle surface. lloreover csR = CsR - m ego is given,
C: - m c s go
which expresses to which extent diffusion limitation in
side the particles occurs. This latter value holds only
forthe period in which the Sherwood-number Sh has rea-s
ched its limiting value (usually Fo > 0.2).
Figure 3.14 illustrates the influence of the gas velocity.
Increasingthe gas flow results in a higher drying rate and
lower exit gas concentrations. At very high gas flows the
drying rate will not increase anymore because of diffusion
resistance inside the particles and ineffective gas-solid
contact. In general the gas flow will be limited by the
carry-over of solids. When the bubble diameter increases
(figure 3.15) smaller drying rates are to be expected be
cause of a less effective contact between solids and gas.
For the conditions mentioned figure 3.16 shows that drying
proceeds quicker when the particle size becomes larger.
This should be ascribed to both a higher concentration of
the bubbles which leave the bed and a decrease of the bub
ble flow because of a larger minimum fluidization velocity.
An increase of particle size may result in lower drying
rates when the transfer resistance is mainly in the solid
phase, as expressed by equation 3.29. ~he value of the
moisture diffusivity inside the solids (figure 3.17) has
no influence on the gas-solid-contacting, but will lower
the particle's surface concentration noticeably below a
critical value of W; that value depends only on the total
amount of moisture that is removed via the gas phase. The
influence of the bed height and the partition coefficient
(depending on the bed temperature) are not shown, but can
easily be understood. As long as bubbles are saturated
when leaving the bed, the drying rate gets larger when the
bed height decreases, the effect becoming less pronounced
0
66
100 zoo
Influence of the gas velocity in uacth drying
100 200
Fixed data:
L 30 em
3001J
m 10000
ro 10-ll m2 /s
.sb 0.15
db 3 em
Calculated:
uo cm/s I-c c e s
15 0.99 0.99
26 0.93 0.98
37 0.()3 0.97
Fixed data:
L 30 em
u 0
26 cm/s
d 300w p
Ill 1 o-u m2 /s
m 10000
ab 0.15
Calculated:
0.98
3 0.93 0.98
5 0.&3 0.98
Influence of the bubble diameter in batci1 drying
0
Figure 3.17
- 67 -
100 !00
Influence of the particle size in batch drying
100 200
Influence of the moisture diffusivity inside the particles in batch drying
Fixed data:
L 30 em
uo 26 cm/s
II 10-ll m2/s
m 10000
6b 0.15
db 3 em
Calculated:
d p il 1-c e
200 0.73 0.99
300 0.95 0.98
400 1.00 0.96
Fixed data:
:::. 30 em
uo 26 cmjs
dp 30011
m 10000
ob 0.15
db 3 em
Calculated:
D m2 /s 1-c sR
10-!2 0.93 0.82
10-ll 0.93 0.98
10-10 0.93 1.00
- 68 -
when the exit gas is not saturated anymore. A decreasing
partition coefficient makes the moisture removing capa
city of the gas flow larger and will finally result in
moving the transfer resistance towards the solids phase.
- 69 -
4. Heat transfer aspects
The discussion about heat transfer between particles
and gas in a fluidized bed will be restricted to those
systems in which the main heat transfer resistance is con
centrated in the gas phase. This situation occurs mostly
in practice as shown below.
Suppose a fluidized bed drier is heated via the fluidizing
gas. Due to solids mixing a particle from the bulk of the
bed can arrive suddenly near the distribution plate, where
it comes into contact with dry and hot gas (moisture con
centration Cg0, temperature Tg). Whether or not a signifi
cant temperature gradient inside the particle arises is
estimated roughly on the basis of equation 4.1.
d Ts a (T -T ) - A (--) p g sR s dr r=R
p (4.1)
ap and Kg are the heat- respectively mass transfer coeffi
cient, As is the solids heat conductivity and Rp the par
ticle radius. ~s is the solids temperature, TsR the tempe
rature at the particle surface. 'J:he equilibrium moisture
concentration of the gas is c* and the total heat of de-g
sorption is b.Hv.
When drying of solids does not occur, equation 4.1 leads
to the well-known Biot-criterion, which says that gradients
inside the particle are negligeable when Bih = Cl.p Rp << 11 A s this situation usually holds in fluidization.
Equation 4.1 will be worked-out now for fluidized bed dry
ing. The maximum value of T -T R that can be reached is g s TF-Tp' with TF the temperature of the feed gas, and Tp the
temperature of particles in the bulk of the bed. TF and Tp
are related via an enthalpy-balance over the bed. Assuming
·comglete' equilibrium at the upper bed level it follows:
T -T F p <c* g
( 4. 2)
- 70 -
where pg and Cpg are the density and the specific heat
of the gas.
From 4.1 and 4.2 it can be derived that:
d T s (dr)r=R
p
which equation can be transformed into:
The Nusselt- and Sherwood number are defined as:
Nu Sh
with Ag the heat conductivity of gas, and ~ the moisture
diffusivity in the gas phase.
Some practical values are inserted into equation 4.3:
l\Hv 2.3 10 6 J/kg 3
pg 1 kg/m
As 1 W/m° C cpg 1050 J/kg°C
A 0.025 W/m0
g c Sh 2
m -5 2 2,10 m /s Nu 2
The chosen values of Sh and Nu are fairly high, in view
of the experimental results obtained in packed beds of
fine particles (see chapter 2) .
It follows from equation 4.3:
+ 8.76 c* - c 0
g g R
> 0
p
which means that the particle gets heated-up.
For the temperature gl~adient it will hold (see
d T T - T ( s) > 2 sR so dr r=R :/ R
p p
4 .1) :
I TaR - - -- j--- -
0
Figure
One finds now
I I I I
I ----~
Rp/2 Rp
~ r
4.1
that:
- 71 -
Tso is the temperature in
the center of the particle.
The factor 2 is justified
by considering that heat
transport takes place mainly
in an outer shell of the particle.
- T ~ 4.38(c; - c 0) so g
Taking C 0 = o and c* = 0.15 kg/m3 (which corresponds
g 0 g with about 60 C) the temperature difference inside the par-
ticle is found to be 0.7°C; this value should be compared
with the gas-phase driving force which, when calculated for
the same conditions from equation 4.3, appears to be 345°c.
The foregoing fully justifies the assumption that the tem
perature inside particles in a fluidized bed drier is homo
geneous.
?he fluidized bed itself is famous for its nearly homogene
ous bed temperature in heat transfer operations. '.!:his phe
nomenon is ascribed to the intensive solids mixing in the
bed due to bubble motion. As a consequence the heat trans
port resistance for heat transfer between the bed and the
vessel wall is concentrated in a thin layer near the wall,
the layer thickness being a few particle diameters upmost.
~part from solid and gas properties the heat transfer coef
ficient between bed and wall depends on the residence time
of particles or particle packets near the wall [105].
'.i'he influence of solids mixing on the heat exchange between
particlesand gas near the distribution plate was never con
sidered in the literature, although the work of Heertjes
e.a. [25,53])already showed that heat transfer between the
- 72 -
distribution plate and the particles directly might be
considerable .
In the following discussion heat transfer between par
ticles and gas will be combined with transfer between
the distribution plate and the particles directly.
'.L·he distribution plate is considered to be a porous pla
te, consisting of sintered granular material; the mean
diameter of the granules is dd, and the plate porosity
is Ed. The superficial gas velocity through the plate
isu • The plate is considered as a packed bed, and a heat 0
transfercoefficient a1
between granules and gas in the
plate is estimated from the theory of Nelson and Galloway
[40], discussed in chapter 2.
Asswning some practical values:
dd 10011 uo 1 m/s
Ed = 0.35 pg l kg/m3
Pr 1 )l -s 2.10 Ns/m 2
the Nusselt number is found to be 0.16 from which the
heat transfer coefficient a 1 is calculated as 40 w;m2 0 c,
when the gas conductivity Ag is taken 0.025 W/m 0 c.
The height of a heat transfer unit inside the plate is:
HTUd 6.7 10-4 m
0 when cpg is taken 1050 J/kg C and sd = 6(1-Ed)/Sd.
The low value of HTUd means that gas and plate reach com
plete equilibrium when the plate thickness is at least
3 mm. Such thickness will usually be met in practice,
whichallows the conclusion that the gas leaving the plate
and entering the bed, has the same temperature as the
plate at the bed-side. This temperature will be referred
to as T0 , which may deviate from the feed gas temperature
TF; the difference will depend on the conductivity of
- 73 -
the plate material, and on the heat transfer rate between
the plate and the bed.
The amount of heat, that is transferred to the bed via the
distribution plate is (suppose TF > T0
):
(4.4)
where u0 is the superficial gas velocity in the bed, and ~
is expressed as the heat flow per unit distributor area.
If ideal mixing of particles in the bed is assumed, a heat
transfer coefficient ad between the plate and the bed may
be defined by:
(4. 5)
where T s denotes the temperature of the particles •
From equations 4.4 and 4.5 it is easily derived that:
_E_ l+p ( 4. 6)
The solids temperature T wll depend on the total amount s of heat that is transferred to the bed by the gas and the
distribution plate together. Heat transfer by the gas oc
curs via the dense phase gas and via the bubble gas.
For the dense phase equilibrium between gas and solids is
easily reached, since the height of a heat transfer unit
equals a few particle diameters at most(see chapter 2).
For the exchange between bubbles and the dense phase a
model might be developped that is completely analogous to
themass transfer model presented in chapter 3. When heat
exchange between particles and gas in the cloud around a
bubbleis analyzed, it is seen that the gas that reenters
the bubble from the cloud has reached complete equilibrium
with the solids in the cloud; the solids temperature it
self remains very nearly constant when particles pass a
- 74 -
cloud (compare the mass transfer process described in
section 3. 1. 1) •
This means that temperature changes of a rising bubble
are described by:
with boundary condition Z = o, Tb = T0 •
For ideally mixed solids it follows:
( 4. 7)
Equation 4.4 is valid when the solids in the bed are heated
or cooled under quasi-steady conditions. ;Jhen the solids
are drying, equation 4.7 holds only when humidity changes
of the bubble are small, as will be the case at moderate
bed temperatues. ><hen the bubble's humidity changes consi
derably during the rising up, the bubble temperature should
be calculated from an enthalpy balance as has been done in
appendix B.
The temperature
the temperatures
bubble gas at the
Te of the gas leaving the bed follows from
Ts of the dense phase gas and T of the b l=L
upper bed level:
( 4. 8)
From equations 4.6-4.8 the total amount of heat that is
transferred to the bed, is found:
T -T F e T=T F s
( 4. 9)
The fraction of heat that is transported via the distribu
tion plate is:
T -T F o T -T F e - Nh l+p - s e
(4.10)
- 75 -
Thecontribution of the distribution plate to the total
heat transfer is difficult to estimate, since the lite
rature provides no data on the heat transfer coefficient ad.
According to the results of e.a. [53] (TF-T0)/
(TF-'£e) amounts 0.25 to 0.50, depending on the gas velocity
and the type of distributor, Lllt it may be much larger sin
ce the experiments were done in a small diameter bed (5 em)
in which a large probe was inserted; the latter will cer
tainly have decreased solids mixing. Our own experiments,
to be discussed in chapter 6, show (TF-T0
)/(TF-Te) is
0.5 to 0.7.
If the residence time of particles at a vertical bed wall
approaches to zero the heat transfer resistance between the
bedand the wall is said [50,105,107] to reduce to a so-
called contact resistance, which equals:
R = contact w
where Aeff w is the effective heat conductivity of the gas
solid suspension near the wall •. According to Hoelen [ 108]
Aeff w amounts approximately 70% of the effective conducti
vity Aeff of the dense phase, due to a higher bed porosity
at the wall. Aeff can be determined from the Schumann-Vas
relation'provided that As and Ag are known [SO].
Assuming As = 1 W/m 0 c, Ag 0.025 W/m °C and a bed porosity
0.45 the effective conductivity according to Schumann and
Vos is about Aeff = 4 Ag = 0.1 W/m 0 c.
For a particle size dp = 300~ a contact resistance is found:
R contact 2 X 0.7 X 0.1
The physical meaning of the contact resistance is somewhat
doubtful. It has been introduced because of the experimental
observation that the heat transfer coefficient does not be
come infinitely large, when the contact time between particles
and the bed wall approaches to zero. An infinite heat trans-
- 76 -
fer coefficient between bed and wall however may very well
exist, but is not likely to be observed normally since the
heat transfer resistance in such situation will be concen
trated elsewhere, for instance in the vessel wall.
Thecontact resistance will be used to estimate heat trans
fer between particles and the distribution plate, since con
siderable solids mixing might be expected at the distribu
tion plate, due to the upward gas flow and due to bubble
formation. Accordingly the residence time of particles near
the distributor will be very small, and the heat transfer
resistance will approach the contact resistance to some
extent. It follows that:
1 = 500 W/m 2 0 c Rcontact
As an illustration only figure 4.2 shows relation 4.10,
assuming p = 1 kgjm 3 , C = 10 3 J/kg 0 c, ad 500 2 0 g pg
W/m C, and furthermore assuming that equilibrium exists
between solids and gas at the upper bed level.
T F- T 0
1 r--r-..,...-r--r-.,--, TF-Te
I o.a
o.e
0.4
0.2
0.4
--...::>,._ u 0 • m/s
4.2
- 77 -
5. Experimental equipment and solids properties
The experiments to be described in this chapter con
cern the batch-drying of silicagel in a fluidized bed.
The course of drying has been measured by sampling the
solid material and analyzing the samples for their
moisture content; in some experiments gas concentrations
were measured inside the bed.
Bubble diameters and velocities were determined in sepa
rate experiments.
5. 1 Equipment
5.1.1 The fluidized bed driers
Two fluidized bed driers 1 each having a diameter of 30 em
and a height of 100 cm 1 have been used for experimental
work. The driers differed essentially in the way of sup
plying the heat necessary for the drying process. One
drier was heated via the bed wallt around which electri
cal heating wire was wound (figure 5. 1A) ; for the other
electrical heating of the fluidizing gas occurred before
feeding the gas to the bed (figure 5.1B).
Fi..:ure S.lA
\·,'all-neated drier
F'igure J.lB
Gas-heated drier
1 dr inlot
2 a.1r cutlet
3 .t'Otameter-
4 distr1.but1oa plate
5 el•ct.ric: b.•ater
6 th•rmocoupl•
7 O"'"l'lin« pol.nt
'l'he wall-he a ted
equipped with a
material called
- 78 -
drier was made of stainless steel, and
distribution plate of porous plastic
Flexolith * . The gas-heated drier was
built-up from Quick-fit elements and its porous distri
bution plate consisted of sintered stainless steel.
Both columns were isolated with glass-wool. Air was
used as the fluidizing and drying agent, the gas flow
being metered with rotameters.
samples of the solids in the bed were taken via sampling
pipes in the bed wall; the sampling positions were at
3,18 and 33 em above the distribution plate for the wall
heated drier, and at 2 and 30 em above the plate for the
gas-heated drier. Just before taking a sample 10 to 15
grams of the solids in the bed were removed via the sam
pling pipe to avoid any possible accumulation of material
in the pipe; these solids were put back into the bed im-
mediately after taking the sample. The s~mple weight was
about 5 grams, and the total weight of all samples was
negligeable compared to the bed con.tent.
5.1.2 Temperature measurement and control in the driers
The bed temperature was measured with a thermocouple which
could be moved up and down through the bed. A thermocouple
in the empty space below the distribution plate was used to
measure the feed gas temperature.
The wall-iteated drier was equipped with thermocouples in
the bed wall at three different heights to determine the
wall temperature; the thermocouples and 15 em of wire were
embedded in a groove in the wall, ·which was made flush
again by plugging the groove with solder. In the same way
a thermocouple and 15 em of wire were mounted in the dis
tribution plate of the gas-heated drier at the bed side of the plate.
During the drying experiments the bed temperature was kept
constant by adjusting the heat supply from the heating
elements. Eurother.m temperature controllers were applied
for this purpose. Fluctuations of the bed temperature were
* manufactured by Schuler Filtertechnik, Eisenberg
- 79 -
within 1°C on each side of the set point temperature.
Thecombination of bed temperature control and rather
large heat capacity of the heating elements caused some
times large fluctuations in the feed gas temperature
(gas-heated drier) instead of the gradual decrease as a
result of the proceeding drying process.
'Ihe temperature in both fluidized beds appeared to be
homogeneousduring the experiments except in a layer of
aboutl em thickness above the distribution plate. In the
wall-heateddrier a large temperature gradient existed in
a layer of about 2 mm thickness at the bed wall. The exit
gas temperature was always equal to the temperature of the
bulk of the bed. ''~oreover the temperature of gas and so
lids in the bulk of the bed was the same since a thermo
couple in the fluidized mass did not reveal a temperature
difference before and after shutting-off the gas flow.
5.1.3 Gas humidity measurements
The humidity of the feed gas was measured with a wet- and
dry bulb thermometer (Aschmann psychrometer) .
A Panametrix humidity meter, that uses a capacity probe
sensitive for humidity changes, measured the humidity of
the exit gas during some experiments. The probe was cc:re-
fullycalibrated using a system in which air was recircu
lated through a trickle-column. The response-time of the
meter was also determined; after a stepwise change of the
gas-humidity the Panametrix meter needed 60 seconds to in
dicate the new humidity provided that the gas velocity
along the probe was high enough. As an example in figure
5.2 the exit gas humidity obtained from the Panametrix
meter is compared with the humidity calculated from the
solids sampling data via a mass balance.
During some runs gas concentrations were measured at dif
ferent levels in the bed. i'li th a water-jet pump a continu
ousgas flow was sucked out of the bed through a pipe of
5 mm diameter which was inserted from above into the bed.
20
10
- 80 -
9.6 kg'
• 15 cm/s
40 oc
• 't*' *'** .. + * * humidity
g/kg
~ time, minutes
Figure 5.2
Comparison of the exit gas Humidity obtained via the mass balance ( *) and via the humidity meter ( e ) .
The humidity in this gas flow was measured with the
Panametrixmeter. The pipe was connected in the bed with
a gas sensor which was constructed in such a way that it
sampled dense phase gas mainly. The construction of the
sensor had to fulfil the following demands in this respect:
- The suction opening of the sensor should be covered with
a fine clothing to avoid particles being sucked-up with
the gas flow.
The plane of the suction opening should be parallel to
the direction of the gas flow in the bed, and the suction
velocity should be as low as possible [113]. In such si
tuation sampling of individual bubbles will hardly occur
because of their high velocity. 'l'he amount of bubble gas
present in the sample flow depends only on the bubble
hold-up at the suction opening: that hold-up usually is
small (10 to 15%) which means that sample gas is with
drawn mainly from the dense phase. The sampling method
- 81 -
is favoured moreover by the fact that small bubbles tend
to avoid objects inserted in a fluidized bed.
The vertical dimension of the suction opening should be
kept as small as possible. Because of the static pres
sure difference over the height of the suction opening
the empty space behind the clothing acts as a short
circuit for the upward flowing fluidizing gas, which will
flow into the empty space through the lower half of the
suction opening and out again through the upper half when
suction does not occur. In case of low suction velocity
the sample flow will be withdrawn mainly through the lower
half of the suction opening, and will contain some gas
that is sucked from a level below the position of the
sensor. A small vertical dimension of the suction opening
is desired oncemore to approximate gas sampling in a horizon
tal plane as good as possible.
The sensor's form should be streamlined to avoid distur
bance of the flow pattern in the bed.
Figure 5.3 shows the final construction of the sensor.
fi /
0
Sensor for gas sampling 1 nylon gauze clothing 2 tube for suction
fi
- 82 -
A rectangular suction opening 10x1 em was chosen. The area
of 10 cm2 of the opening was based on a suction velocity
of 1 cm/s (corresponding with 10% of the linear dense phase
velocity} and a minimum suction flow required to obtain a
fast response to gas humidity changes by the Panametrix me
ter.
The gas sensor was tested in the wall-heated fluidized bed
drier during a run under the following conditions: dry
solidsweight 18.8 kg, gas velocity 15 cm/s, bed temperature
40°C.
Firstly the influence of the suction gas velocity on the
humidity reading from the Panametrix meter was studied.
Scale readings were made with the sensor in a fixed position
in the bed; the suction velocity was increased in two steps
and decreased afterwards. The experiment was repeated with
the sensor at another level in the bed. 'l'he results in tabll
5.1 show that the observed gas humidity is hardly influencei
by the suction velocity.
Table 5.1 Influence of suction velocity on the observed gas humidity
observed gas humidity*
sensor position§ 2.4 em 31.4 em
suction velocity
1.2 cmjs 10.2 g/kg 9.7 g/kg
2.4 10.0 9.6
3.6 9.9 9.4 2.4 9.9 9.6
1.2 9.9 9.5
§expressed as distance from the sensor's center to the distribution plate
*measurements at 31.4 em were takensome time after those at 2. 4 em
A second test concerned the question whether or not the sen·
sor samples gas from the dense phase only. During the run
the superficial gas velocity in the bed was lowered for a
- 83 -
short while to the minimum fluidization velocity. If the
sensorsamples a mixture of dense phase gas and bubble gas
a higher humidity is likely to be observed at the minimum
fluidization velocity because of the disappearance of bub
bles; the foregoing holds under the condition that the bub
ble humidity is lower than the dense phase humidity.
Table 5. 2 presents' results. Humidity readings were done two
minutes after decreasing the gas flow.
Table 5.2 Influence of bubble flow on the observed gas humidit
observed gas humidity
sensor position§ 0.5 em
superficial gas velocity
15 cm/s 2.3 g/kg
5.6 cm/s 1.7
2.4 em
2.6 g/kg
2.6
18.8 em
2.6 g/kg
2.6
§expressed as distance from the sensor's center to the distribution plate
Except for the level 0.5 em no effect was observed, indicat
ing either that the sensor samples dense phase gas only or
that bubbles and dense phase gas have the same humidity. At
the level 0.5 em a decrease of the gas humidity was found af
ter lowering the gas velocity. 'I'he value 1. 70 gjkg mentioned
is the reading after two minutes, but it continued to de
crease afterwards. This can be explained from the fact that
the intensive solids mixing in the bed stops at the minimum
fluidization velocity; as a consequence the surface concen
tration of particles at the distribution plate drops to a
very low value, because those particles are in contact with
dry gas all the time.
5.1.4 Experimental procedure during drying experiments
A weighed amount of solids was brought into the drier. The
solid material was moistened in the apparatus by fluidizing
with moist air. For this purpose steam was injected via a ven
turi-jet in the air before it entered the bed. The steam
flow rate was adjusted during the course of the adsorption
- 84 -
process to keep the relative inlet gas humidity more or
lessconstant, since the bed temperature increased somewhat
due to the development of the heat of adsorption. After
moistening the gas flow through the bed was stopped for at
least 15 hours to allow levelling out of any concentration
profile inside the particles.
A drying experiment started by heating the fluidized bed a~
fast; as possible; the time required for heating usually wa~
in between 10 and 20 minutes. All temperature measurements
wereregistrated on a multi-point recorder. Samples of the
bed content were taken after certain time intervals. '.t'he
inlet gas humidity was determined with the Aschmann Psychrc
meter. The time duration of a run usually was 4 hours. When
the experiment was stopped all samples were weighed. i\.fter
drying in a stove at 110°c for 36 hours, the samples were
weighedagain. When dense phase gas h~~idities were measured
during a run, the gas sensor was fixed at a chosen level in
the bed and the Panametrix meter was read about 70 seconds
afterwards. Measurements occurred usually at 7 levels in th
bed, starting from the bottom on. ~fterwards the exit gas h
miditv was measured, and it was checked that the humidity at
the lowest level had not changed much during the 10 minutes
period that was needed to complete this sequence of measure
ments.
5.2 Solid material
5.2.1 General properties
All experiments were carried out with silicagel as the flui
dized material. 'rhe silicagel, manufactured by Grace GmbH i
Hamburg, was a so-called sub-microporous kind of gel; the ma
jority op pores inside the particles has a diameter smaller
than 100 g, Extensive data on this type of gel were publist
[109-111] , such as equilibrium and diffusion data.
From sieve analysis the mean diameter of the get particles
was found to be 330~. The bulk density of the material was
743 kg/m3 •
- 85 -
Equilibrium and diffusion data were measured on a Cahn
Electrobalance. A silicagel sample of about 100 mg was
suspended in a pan in the thermostated weighing chamber;
the chamber was connected with a water containing vessel.
The water temperature was kept constant and below the tem
perature of the weighing chamber. By evacuating the whole
systemthe sample was exposed to pure water vapour. Diffe
rent relative humidities were set by adjustment of the
water temperature.
The connection between water vessel and weighing chamber
could be opened or closed via a valve. hfter the whole sys
tem had come to equilibrium the valve was closed and the
water temperature was made lower a few degrees Centigrade.
Whenthe valve was opened again desorption of moisture from
the sample occurred. The sample weight was recorded until
it became constant. The course of weight with time provided
data on the diffusion process inside the particles, vJhile
the constant value, v.!hich was read usually after about ten
hours,supplied the equilibrium data.
This method has been described extensively by Dengler and
Krlickels [109] who studied diffusion of moisture in adsorben
tia, among which silicagel from the same manufacturer as
thesilicagel used in the present experiments.
Figure 5.4 shows the sorption equilibrium curve, plotted
as the relative humidity RH versus the solids moisture con
tent X (Kg water/Kg dry gel) • In this form the equilibrium
curvedepends hardly on the solids temperature [112].
The data in figure 5.4 concern desorption measurements only
to avoid a possible hysteresis between ad- and desorption.
The different symbols in the figure refer to samples of
differentage, to show that deterioration of the silicagel
due to frequently adsorbing and desorbing moisture during
,the fluidized bed experiments did not occur; according to
:Dengler and Blenke [ 111] the sorption capacity of this type
'lof gel decreases to a constant value after many cycles of
adsorption and desorption.
3o X%
t
20
10
- 86 -
Figure 5.4
Equilibrium for tl:e system silicagel-moist air
-----')1- R H %
Data on moisture diffusion inside the particles will be
discussed in appendix D. In this thesis the diffusivity ID
will be taken constant and equal to 2.lo-l2 m2/s.
5.2.2 Basic fluidization data of the solid material
The dry particle density and the external porosity of a
looselypacked bed of were determined via pres
suredrop measurements over the packed bed. (The porosity
mentioned above is often referred to as the quiescent t:·ed
porosity, and is obtained in a packed bed after fluidizing
it for a while). This method has been described and tested
by Arthur e.a. [115]and is based on the Ergun relation. The
method is useful especially for determining the density of
porous particles. 'I'he particle density was found to be
1350 kg/m3
and the porosity of the quiescent bed 0.45.
16
8
4
0
- 87 -
C!P, em H20 .sb
~ f ~* *
* * *--*--*--*--*--*--
*
-/ .--.---.,..--
/.~ * • .......
/ /. ,....
u0 , cm/s • 10 20
Figure 5.5 Fluidization characteristics of the silicagel material
0.24
O.HI
o.os
Figure 5.5 shows the pressure drop and the bubble hold-up
of a bed of fluidized silicagel particles. The material is
well-£luidizing as long as its moisture content is kept be
low 34%; at higher moisture contents the particles were not
free-flowing anymore. :J.'he minimum fluidization velocity, de
termined from figure 5.5 is 5 cm/s.
Since the material is rather coarse, all bed expansion is
due to bubbles, and the part of the gas flow, that exceeds
the requirement for minimum fluidization, passes the bed in
the form of bubbles (provided that the bubble hold-up in the
bed remains small).
- 88 -
5.2.3 Data on bubble size and bubble velocity
- method of measurement
Werther and Molerus [10,114] developped a method for mea
surement of bubble size and bubble velocity in heterogene
ously fluidizing beds. Their method has been applied for
determination of these bubble properties in the fluidized
bed driers.
Bubble detection occurs with a very small device which can
be placed at different positions in the bed. The device
contains two miniaturized electrical condensers at a short
distance z above each other (see figure 5.6). Each conden
ser consists of a wire, clothed with an electrically isolat
ing material, which is mounted coaxially in a tube with an
internal diameter of 1 mm. The wire extends 3 milimeters
out of the tube. Wire and tube together form one condenser.
2
1
~J
Probe Signal
1------- Signal 1 I
I I ·n : : '------Signal 2
~1 I -dt2 time
Figure 5.7 Time-depending probe signals due to arrival of a bubble
Figure 5.6 Capacity probe
1 probe 2 pipe for cables 3 connection for coaxial cables
- 89 -
I The capacity of the condenser changes when the dielec
tricum around the extended wire changes due to the ar
rival of a bubble at the probe: the dielectricum of the
gas-solid suspension changes into the dielectricum air.
Measurement of the pulse-like capacity change caused by
• an individual bubble provides information about the re
sidence time ~t 1of the probe in the bubble. By using two
probesat a distance z above each other the bubble veloci
, ty ub is found from the delay time ~£2 between the two
;signals, Db can be used for calculation of the observed
·vertical bubble size Sb (see figure 5.7):
'It is obvious that the value of Sb observed depends on
the position relative to the bubble center, at which the
bubble and the probe meet each other. Other factors that
may cause scatter in the observed values of Ub and Sb are
zig zag movement of the bubble and changes of the bubble
shapeduring the rising-up [94]. In a heterogeneously flui
dized bed a spread of Ub and Sb is found moreover due to
an existing distribution of bubble sizes. I'Jerther [ 114]
has shown that auto- and cross correlation of the time
depending capacity signals yields mean values of Sb and
Ub at the point of measurement in a heterogeneous bed.
Extra-polation of the initial slope of the auto correlation
function of one signal gives the mean 6t1 of ~t1 , while
the maximum in the cross correlation function corresponds
~th the mean Et2 of ~t2 (s~e figure 5.8); from ~t1 and
~t 2 a mean bubble velocity Ub and mean vertical bubble si
:_e sb can be calculated. Sb can be considered as characteristic for the bubble size,
and is related to any definition of the bubble diameter if
theshape of the bubble and the bubble size distribution
areknown. Werther [114] assumed the bubble's shape to be
an ellipsoid of rotation,and a logarithmic normal distri
bution of bubble sizes. He showed that Sb was proporti-
- 90 -
ACF
t
Figure 5. 8A ;_utocorrelation function (F.CF) Figure 5. 8B Crosscorrelation function {CCF)
onal to the mean length of the vertical axis of the ellip
soids of rotation; the proportionality constant includes
the spread of the size distribution and a correction factor
but does not deviate much from unity.
Here Sb will be interpreted as being the diameter of sphe
rical bubbles. Deviations from this assumption may arise
(as stated in chapter 1 a bubble resembles a spherical cap
with indented base), but are not important in view of the
results of the drying experiments.
- experimental results
Measurements of bubble size and velocity were carried out
in the so-called gas-heated fluidized bed drier; the expe
riments occurred at ambient temperature (approximately 20°C
withoutusing the heater. Two Van Reysen capacity meters
measured the capacity changes of the probes and a Honeywell
Saicorcorrelating instrument computed auto- and cross cor
relation functions from the capacity signals. In contrast
withthe work of Werther [114]an electrical circuit for the
separationof bubble pulses from porosity fluctuations in
the dense phase was not used; since dense phase expansion
in the rather coarse silicagel powder does not occur, any
porosity fluctuation in the dense phase must be considered
- 91 -
as being a bubble.
Table 5.3 and figures 5.9 and 5.10 present the results of
theabove mentioned measurements, including the measuring
conditions.
In a horizontal plane in the bed the mean vertical bubble
sizeSb is constant (see table 5.3) ~the mean bubble velo
city ub decreases when going from the bed centre to the
bed wall. 'ihe former result is not surprising, while the
latter must be ascribed to the tendency of bubbles to move
towardsthe bubble center. Both results agree with the work
of Werther [144]. i<ccording to his observations most bub
bles rise in an annulus when they are near to the distri
butor, but in the upper regions of the bed the bubble fre-
quency becomes highest in the center core of the bed. Since
bubbles in a swarm rise at much higher velocity than more
or less isolated bubbles, a higher bubble velocity should
be expected in the bed center.
Figure 5.9 demonstrates the growth of bubbles with increas
ing height above the distributor for three different super
ficial gas velocities u0
and two different bed heights;fi
gure 5.10 presents the corresponding bubble velocities,
measured in the center of the bed.
It is seen from figure 5.9 that the bubble size increases
linearly with the height in the bed. Bubble growth which is
dueto bubble coalescence, becomes more serious at higher
gas velocities, vlhich has to be expected because the
bubbleflow itself becomes larger; the latter means that
either the bubbles themselves must be larger or the bubble
frequency must be larger with inherently more bubble coales-
cence. I'igure 5.10 shows that the mean bubble velocity al
so increases linearly with the height above the distributor. du
The derivative dlb is more or less constant, but the ini-
tial velocity of bubbles leaving the distributor increases
stronglyat higher superficial gas velocity.
•
•
4
z
• •
- 92 -
·~.·- * t 26 cm/s
* */./
*~·/ / . .
/ /. ~ om • 10 zo
I
Figure 5.9 iiean bubble size as function of the height above the distributor (packed bed height:. e14 em;* 31 em; 03 I. em duplo)
30
100 t
50
•
----::)1.- l , em
)lo l ,em 0
0 10 20 30 50
U em/a uo 9 cm/s b•
t ·- * ___,_---* ... ........-• ,... Z ,em
0 ~o---L---L---L---L--~1~o--~--~--~--~--2~o~~--~---L---L--.J3o Figure 5.10 11ean bubble velocity as function of the height
above the distributor (packed bed height: e14 em; *31 em; 031 em duplo)
- 94 -
Both figures indicate that the results are influenced
by the packed bed height. ~t low gas velocities that in
fluence is negligeable, but at higher gas velocities both
bubble size and velocity at an arbitrary level in the bed
are smaller when the packed bed height is smaller. This
effect should be ascribed to an overall circulation of
solids in the bed; usually circulation of solids becomes
more pronounced in higher beds and at higher gas velocities
which is in agreement with the observations. In fact a down
flow of solids at the bed wall was observed during the experiments.
For application of these results to fluidized bed drying
it is of interest to note that bubbles near the distributor
are quite small, which is favourable'for the gas-solids con
tacting in the bed. The large bubble velocities on the
other hand make gas-solid contacting less effective, and
may indicate that possible low exit gas humidities should
beascribed to a short residence time of bubbles in the bed
rather than to small exchange rates between bubbles and the
dense phase. These phenomena will be considered more exten-
sively in chapter 6. It is stated here oncemore that the re
sults of figures 5.9 and 5.10 were obtained with a finely
dividing· porous distribution plate. For other types of dis
tributors {for instance sieve plates) the development of
bubble size and velocity with height in the bed will be
different; in general larger bubbles will be found near the
distributorwhen the gas distribution is less even.
Table 5.3 Variation of bubble size and velocity in a horizontal plane
radial position {in em from bed center) 13 7 0 4 8
sb em 2.2 1.7 2.5 2.1 2.7
ub cm/s 44 50 55 48 46
- 95 -
6. Experimental results
Batch drying experiments have been carried out under
the conditions mentioned in table 6.1 A (wall-heated drier)
and table 6.1 B (gas-heated drier).
Mass transfer results and heat transfer results will
be discussed separately in the following sections.
6.1 Mass transfer aspects of fluidized bed drying
Experimental data concerning the mean solids moisture
content X as function of the drying time t are presented
graphically in Appendix E;figure 6.1 is presented here as
an example. The drying curve of run 12 is not shown, since
the data points gave a very large spread.
As can be seen from figure 6.1 solids sampling at different
levels in the bed but at the same time yieldsthe same mean
solids moisture content. This indicates that the particles
can be considered as ideally mixed from a viewpoint of so
lids drying (later on in section 6.2 it will be shown that
from a view point of heat transfer to the solids, this con
clusion should be relativated). In the experiment at a gas
velocity of 9 cm/s a slight spread in the moisture content
at different levels is found. Although solids mixing will
be less intensive at lower gas velocities, ideal mixing is
very nearly approached all the same, since the results do
not reveal any trend of solids moisture content being a
function of the level in the bed.
The agreement in figure 5.2 between measured exit gas humi
dity (via the Panametrix meter) and that one calculated
(via the mass balance from solids sampling data) is also an
indication that the particles are ideally mixed; otherwise
the drying rate of the solids would not be uniform in the
whole bed.
24
<l>
:~--~ . ""'~ "~·-----~. 0 t-·-2
411... '~ 11-he.ceJ'~-,_0-..._ •; !'!to
18
0 0
~ .. . ---.----% ---~-
X :'e,o
t * ~ d drier ~~~~··•<•
"-..."-...... *~Iii......._
............. 31 kg
~------~ om{o ted drier *..........._ gas-hea ..........._
®~*~31kg 28 em/• ®~
0
50 100 150
IJ:)
0'1
- 97 -
Table 6.1 A
Conditions of experiments in the wall-heated drier
run L,cm wd,Kg u0
,cm/s T °C xo,% H0
g/kg figures nr. s'
1 19.5 9.6 9 40 25.6 2 6.3
2 20.2 9.6 12 40 24.6 1.5 6.3
3 20.9 9.6 15 40 26.4 1 6.3
4 21.4 9.6 18 40 24.5 2 6.3 6.10
5 21.4 9.6 18 40 24.0 2 6.10
6 22.1 9.2 28 40 24.4 1 6.3
7 40.6 19.2 12 40 25.7 1.5 6.4 B
8 43.0 19.2 18 40 26.2 1 6.4 B
9 79.3 37.9 9 40 23.6 1 6.5
LO 41.6 19.0 15 50 21.8 5 6.8
1 41.6 19.0 15 60 21.8 3.5 6.8
rable 6.1 B
Conditions of experiments in the gas-heated drier
run L,cm wd,Kg u ,cm/s T °C xo,% H g/kg figures 0 s' 0 nr
.2 10.3 5.1 9 40 26.0 2
L3 10.9 5.0 15 40 25.4 2 6.2,6.7
L4 11.8 5.0 26 40 25.1 2 6.2,6.6
L5 38.1 18.8 9 40 25.1 1 6.4 A
L6 42.0 19.2 15 40 26.4 2 6.4A,6.7
17 45.7 19.2 26 40 22.0 3 6.4A 16.6
18 67.9 31.0 15 40 21.8 3 6.5,6.7
19 73.9 31.0 26 40 22.7 2 6.5,6.6
20 11.6 5.3 15 40 14.4 2 6.9
21 43.8 20.0 15 40 14.9 2 6.9
:l dry solids weight xo initial solids
moisture content
J superficial gas velocity
"o = inlet gas humidity
3 bed temperature
= expanded bed height
- 98 -
The drying curves as presented in figure 6.1 and Appen
dix E have been differentiated graphically, using the
mirror method. From the drying rate - ~~ the exit gas hu-
midity H was calculated via the mass balance: . e
dX dt
u p ~ D 2 (H - H0 ) o g 4 b e
( 6 .1)
in which Wd is the dry weight of the solids in the bed,
and Db the bed diameter.
Exit gas humidities were expressed as relative gas humidi
ties RHe with reference to the bed temperature. For seve
ral process conditions RHe as a function of X is shown in
the figures 6.2 to 6.9. For each experiment the correspon
ding initial solids moisture content is indicated on the
horizontal x-axis, and the relative humidity of the inlet
gas on the vertical axis. The drawn curve in the figures
represents the equilibrium curve, shown already in figure
5 • 4.
Figures 6.2 to 6.5 show the influence of the superficial
gas velocity on the drying process; each figure corresponds
to a fixed dry weight of solids, and the data refer to the
driers as indicated with G (gas-heated drier) or W (wall
heated drier). Cross-plotting of results gives the effect
of the bed height at constant gas velocity (figures 6.6 and
6.7). Finally figures 6.8 and 6.9. express the influence of
bed temperature and initial solids moisture content respec
tively.
Most results indicate that complete equilibrium between
exit gas and solids is not reached usually; nevertheless
the equilibrium is approached to such extent that the in
fluence of process parameters (bed height, bed temperature,
gas velocity) can only be marginal, which makes an interpre
tation of the results somewhat difficult.
50
40
30
20
10
0 0
40
30
20
10
- 99 -
RHe% ••• c G
t 5 kg
¥26 cm;a 015 cm;a
0
dl ~
cP•if;o
//
5 10
Fig:ure 6.2
¥ 28 Cmfa 0 18 • 15 0 12
* 9 9.6 kg w 40. (;::
5
Figure 6.3
••
10
• o
• 0
• oo
~/ ~
.i/ 0 •
~x% 15 20 25
--~• X%
15 20 25
10
•
50
40
30
20
10
RHe%
I
Figure
RHe%
t
- 100 -
u•c G 1.1 kg
..,. I emja
• II emja 0 U emja
6.4A
* 12 em/a e 18 cmla
40°C w 19 kg
•• • • • •
... . • .... . . . ..... ,.. . ..,.,. ...... .. .....
Figure 6.4B
ll
$0
RHe%
40 r 30
20
10
20
- 101 -
u• c G
31 kg
¥28 cm;s 0115cm;s
..
6.5
0 31 kg
• 19 kg 0 5 kg
.. .. •
10
0 0
10
~ 0
£• .... .. .. 0
oO 0~
.. ..
,.. 15 20
1S 20 2S
- 102 -
sor-------.-------~-------r------~------~
40
30
20
G u•c .:11 em/•
0 31 kg
• It kg
0 i kg
0
•
---lii-X% 0o~------~5------~10~----~1+5------~20~_LL-~
Figure 6.7
50
RHe'~> 1i em/• t II kg
40 Ge 40° c w. 50° c wo u• c 0
• 0 • 0 fP• • 30 ~,. •• ~0~ •
/ .. 'f.
./q, 20 •
/~ • +
10 • /0
0
~X'~>
5 10 15 20 25
Fisure 6.8
- 103 -
G u•c 15 em/•
30
20
18
Figure 6.9
[Estimation of errors
The exit gas humidity He was calculated via the mass
balance:
H e dX dt
The relative error in the dry solids weight amounts about
1% and is mainly due to the change of the bed content be
cause of withdrawing the samples. The mass flow of air in
thedrier contains a relative error of about 2%.
The absolute error in H0
is about 0.3 g/Kg at an average
value of H 0
2 g/Kg.
The error in ~: is estimated on the basis of the figure
below in which the symbols represent the data of three
samples that were taken subsequently.
-- 104 -
x, dX in the point (X2,t2) dt x2
is approached by: l Xs --r
x3 - xl l dX
t, h ts dt t3 - tl
The absolute error in the solids moisture content X as a
result of the moisture analysis is 0.01%. The absolute
error in the time t is about 0.25 minutes which is the time
necessary to take a sample. x 3 - X 1 and t 3 - t 1 may be es
timated as 1% and 25 minutes respectively (see figure 6.1
and Appendix E). From the foregoing the relative error in
~~ is calculated to be 4%, which means that the relative
error in H -B is 7%. For -H e o o 10 g/kg the absolute erro
in He is 1 g/kg, corresponding with a relative error of
8.3%.~he equilibrium gas humidity H* at 40~ l°C is 49~ 2 g/kg, which causes a relative error of 4% in H*.
It follows that the relative error in the relative humidity
of the exit gas is about 12%.
dX Forsome drying curves the determination of dt may contain
a larger error than the 4% indicated above, if the curve
cannot: be drawn in a completely unambiguous way, because
of some spread in the moisture content of the silicagel
samples. This spread is not a result of the moisture ana
lysis, but, as mentioned before, must be ascribed to less
intensemixing(see figure 6.1, the curve for a gas velocity
of 9 cm/s) •
Figure 6.10 presents the results of an experiment in duplo
to demonstrate that the reproducebility of an individual
experiment is fairly good.
- 105 -
50
RH 8 %
40
t duplo
*run 4
run 5
18 em/a Jf.
w 9.8 kg • • 30
40 °C ~ Jf.
'bb o>P o• o•
20 ~· ~~ •
• t 10
X% 0
0 5 10 15 20 25 30
n~ure 6.10
Measurementsof the dense phase gas humidity as function of
theheight in the bed are summarized in tables 6.2 A*and
6 .2 B. Because of the response-time of the Panametr ix meter
determinationof a humidity profile took 10 to 20 minutes,
depending on the number of levels at which the humidity was
measured. During that period the solids moisture content de
creases less than 1%, as can be seen from the corresponding
drying curves in figures 6.11* and 6.12; this means that
thegas humidity may decrease in that period 0.5 g/kg when
calculated from the equilibrium curve, assuming a bed tem
perature of 40°C. The period in which a humidity profile has
beenmeasured is indicated in the tables by means of the ti-
mes and te; ts refers to the start, te to the end of the
*The data in table 6.2A and in figure 6.11 refer to a run that has been interrupted twice. 'l'his will be discussed later on.
- 106 -
period, with the start of the batch drying experiment ta
ken as zero-time. For the gas-heated drier the temperatures
of the distribution T0 and the bed Ts at times ts
and te are also mentioned.
It is seen from table 6.2 A that the sensor indicates a
uniform gas humidity in the bulk of the bed. An increase
of the humidity is found in a layer of about 2 em thickness
just above the distributor. Since the sensor is supposed
to measure the dense phase gas humidity the observations
just below and above the bed level (at positions 40.4 and
45 em respectively) demonstrate that both bubbles and den
se phase gas have the same humidity when leaving the bed.
The data obtained in the gas-heated drier (table 6.2 B)
are influenced by the fluctuations of the temperature near
the distributor, which temperature varies very slowly in
time because of the control of the bed temperature. After
completing a sequence of measurements at different levels
in the bed (starting from the bottom on) the reading at a
heightof l em above the distributor was repeated, and ap
pearedto differ from the first reading at that location.
The deviations are related with changes of the distributor
temperature T0
, as can be seen in the table. vmen T0
in
creased during the measurement of a gas humidity profile,
the second reading at l em above the distributor was higher
than the first one, and vice versa. The results in table
6 .2 B allow the conclusion that the dense phase humidity is
uniform in the whole bed.
Since the results obtained just below and above the upper
bed level do not show a significant difference, toth dense
phase gas and bubble gas leave the bed with the same humidit~
the wall heated drier
(The data refer to a run that has been interrupted twice for a period of 16 hours after a drying time of 70 minutes. 'I he reason for this is explained in section 7. 2)
wd 18.8 kg, uo = 15 cmjs, T = s 38°C humidities in g/kg
t * te * s distance above the distributor, em min min 0.5 0.9 1.4 2.4 3.4 4.4 5.7 11.4 15.7 18.8 40.40f 451"
14 28 14.2 14.1 14.1 14.4 14.5
28 36 ll.8 13.4 13.5 13.4
36 46 10.3 10.6 12.4 12.6 12.8 12.7 12.9
46 59 10.3 11.2 11.9 12.4 12.6 13.1 13.1 12.7 12.7 ...... 0
12 22 9.5 11.2 11.3 11.3 11.4 11.6 --..!
22 35 9.7 11.1 11.2 11.3 11.2 11.1 ll. 0
37 46 9.3 9.8 10.5 11.0 10.8
47 57 8.5 9.3 10.2 10.4 10.4 10.2 10.2
18 30 9 .1 9.6 9.8 10.1 10.3 10.3 10.2 10.2
30 43 8.5 9.2 9.5 9.6 9.7 9.6 9.5 9.5
49 60 7.5 8.0 8.2 8.4 8.4 8.6 8.5 8.5
75 87 6.9 7.4 7.5 7.7 7.9 7.9 7.9 7.9
90 112 6.5 6.9 7.2 7.3 7.4 7.2 7.2
125 140 5.0 5.3 5.6 5.9 6.0 5.3 5.3
140 150 5.0 5.6 6.0 5.8 5.7 5.8
for notes, see table 6.2 B
Table 6.2 B Measurements of the concentration profile over the bed height
in the gas-heated drier
16.4 Kg, u0
= 15 cm/s, humidity in g/kg
t * t * T 0 °C T oc s e distance above distributor, em D
38t s
min min 0.5 1 1.5 2 2.5 3.5 4.5 33.5- 1 at at at at t te t t s s e
35 54 9.5 - 12.1 - 12 ll.8 ll.5 11.0 10.8 10.6 62 59 38.5 38
56 73 ll.O 11.3 11.1 11.2 11.3 - - 11.3 11.1 11.0 60 57 38 39.5
80 97 8.8 9.9 9.8 10.0 10.2 - - 10.5 10.8 11.0 56 61 38 39.5
112 127 7.8 9.0 9.0 8.9 8.9 - - 9.0 9.2 9.7 54 61 38 39
142 156 9.7 9.5 9.4 9.2 8.) - - 8.6 8.5 8.4 59 55 39.5 37
178 192 8.6 8.5 8.4 8.2 8.1 - - 8.0 8.0 8.1 55 55 39.5 38.5
206 221 7.1 7.8 7.8 7.6 7.5 - - 7.5 7.4 7.4 55 56 39.5 39
231 246 7.2 7.3 7.3 7.1 7.0 - - 7.0 6.9 7.0 55 55 39 138.5
* t and t refer s e to the start and the end of measuring a concentration
profile over the bed height; the time zero indicates the start of the drying
experiment.
& this position is just below the bed level.
t this position is just above the bed level.
.... 0 co
- 109 -
wd = 18.8 kg
t
---...... time. minute•
F~gure 6 ~ ll Drying curve, corr~sponding to the data ~n table 6. 2A. Arrows indicate the moments of interruption
10
~ time, mJnu'tea
Figure 6 ~ 12 .Gryinl) curve: correspondir,g to the Jata in 6. 2P
- 110 -
6.2 Heat transfer aspects of fluidized bed drying
Under all experimental conditions the temperature in tt
bulk of the bed was homogeneous, while the exit gas had
reached temperature equilibrium with the bed completely.
In a layer of about 1 em the thickness just above the dis
tributor the temperature indicated by a bare thermocouple,
differed from the temperature in the bulk of the bed. In tt wall-heated drier that layer temperature was lower than the
bulktemperature, while the opposite was found in the gas
heated drier. A large temperature gradient was observed in a layer of
about 2 mm thickness at the wall of the wall-heated drier.
Figure 6.13 shows the wall temperature T as function of w time for some experiments in the wall-heated drier. The bee
temperature, which became constant within a heating period
of 20 minutes, v;as 40°C. ':'he inlet gas temperature was 20c
Otherexperimental conditions are indicated in the figure.
•or-------.-------.-------~------,-------,-------,-------.,------~ Twoc • ·~ wd 9. kg, u 0 .. ···-... ~.
--·-·-·~· . --·-·-·--·--·
26 cn/s
~ time.minutew
•o,k-------L-------SLO-------L-------1~0&------~-------1~5-t------~------~ZO~t
t 80 ••
9. 6 j(.g, llt" 15 C!n/S
·-.-·-·--.......!!'-----• ----· --~- ·----·--· ~time,minu'fes
•o~----~~---~----L----L---~-----~----L------L 0 60 100 150 ao
~ time.minute•
·0~----~------~------L-----~-------L------~----~~----~ 50 100 15& HO
Figure 6.13 \vall-temperature as function of time for batch drying in the wall-heated drier temperature is 400C)
- 111 -
The shape of the curves corresponds with the course of
relative exit gas humidity as function of the mean solids
moisture content; the latter has already been shown in
figure 6.3 in section 6.1. At a gas velocity of 26 cm/s
the relative exit gas humidity decreases continuously,
which means that the heat consumption in the bed will al
so decrease; the driving force for heat transfer(T -T ) w s follows that trend. At the velocities of 15 and 9 cm/s
the drying rate of the bed is more or less constant during
a certain period, which corresponds with a nearly constant
wall temperature Tw. At the latter two velocities the dri
ving forces (Tw-'.Ls) do not differ very much, which can be
explainedby different wall-to-bed heat transfer coeffi
cients aw. These coefficients aw have been calculated from
the observed drying rate (see section 6.1) and the tempe
rature measurements. Calculations are based on the follo
wing total heat balance over the bed:
L is the expanded bed height, Db is the bed diameter and
~Hv is the heat of evaporation; the latter was taken from
the work of Dengler e • a. [ 10 9] (see also Appendix D) •
Figure 6.14 presents a as function of the mean solids moisw
ture content X under conditions that correspond to those in
figure 6.13. It is seen that a is constant during the cour-w
se of drying. Since the calculations were made with the mo-
mentary temperatures observed, the fluctuations in the wall
temperature (caused by the temperature control) result in
some spread of the calculated aw-values.
In table 6.3 heat transfer coefficients aware given for
different drying conditions.
Ul
Ul
111 I
- 112 -
* * * *-* * * * 0 0 * --o o_ 0 0
a Wj. 2 o w• m C
t 0
* * u cm/s wd kg 0
* * * 26 9.2 * * D 15 9.6
* 9 9.6 ... x.'Y• I II 11 :u II
Fi~ure 6.14 Influence of the mean solids moisture content on the wall-to-bed heat transfer coefficient
Wall-to-bed heat transfer coefficients
a, (W/m2 °C) w
gas velocity u0
, cm/s dry solids weight, kg
9 12 15 18 26
9.6
19.2
153
37.9 211
237
205
242 292
220
266
The results express the trend that is usually observed.
When the minimum fluidization velocity(umf = 5 cm/s) is exceededthe heat transfer coefficient increases strongly;
a maximum value is to be expected at a certain gas veloci
ty because an increasing part of the bed wall is in contac
with gas bubbles.
At larger bed heights the heat transfer coefficient is
likely to decrease if, according to van Heerden e.a.[l16]
- 113 -
an overall solids circulation occurs in the bed; such
·circulation could not be observed because of the untrans-
• parency of the bed wall, but was observed in the gas-
heated drier.
The value of aw at 37.9 kg dry solids weight and a gas
velocityof 9 cm/s does not correspond with the explanation
given above, but should be considered as very inaccurate;
the driving force for heat transfer during that run was
about 3 to 4°C, while it was in between 10 and 30°C for
the other runs.
For an experiment in the gas-heated drier figure 6.15
shows as function of time the temperatures:of the feed gas
(TF), of the distribution plate at the bed side (T ) and of 0
the bulk of the bed(Ts). An overall decrease of both and
T is seen in the figure (after the heating period of about 0
20minutes), but due to serious temperature fluctuations
that decrease is not monotonous.
~ time,minut••
Figure 6.15 Course of temperature with time in the gas-heated drier
The fluctuations must be ascribed to the temperature con
trol of the bed. The bed temperature responds rather slowly
tochanges in the feed gas temperature; besides the heat ca
pacity of the whole heating section causes a delay in adjus
ting the heating power to the heat requirements in the bed.
The ratio(TF-T0
)/(TF-Ts) is plotted versus time in figure
6.16for the same conditions as mentioned in figure 6.15.
0.5
uo: 28 cmJ•
wd: 18 ke
- 114 -
On an average(TF-T0
)/(TF-Ts) appears to be constant during
thecourse of drying, as was observed already by Heertjes
and coworkers[25,53] and is to be expected according to the
theory developped in chapter 4.
For different drying conditions time-averaged values of
(TF-T0
V(TF-Ts) were determined by averaging the momentary
value~ calculated at time intervals of about 20 minutes.
Al thouqh the length of the interval may affect the final
result, such influences are fairly small, as is illustrated
below with the data from figure 6.16.
time interval*
minutes
10
20
40
mean value of
(TF-T0
) I (TF-Ts)
0.645
0.649
0.669
*the first data point of figure 6.16 is not considered
Table 6.4 presents the mean values of (TF-T0l/(Tf-Tsl.
From these data the factor ctd being the heat trans-
fer coefficient between the distribution plate and the bed,
wascalculated with the aid of equation 4.6. Table 6.5 gives
the results.
- 115 -
Table 6.4 Mean values of(TF-T0
)/(TF-Ts) for different
drying conditions
dry solids gas velocity u o' cm/s weight,kg
9 15 26
5 0.53 0.82* 0.72
19.2 0.52 0. 77 & 0.65
31 0.57 0.62
Table 6.5 ad in W/m2 0 c for different
drying conditions
dry solids gas velocity u0
, cm/s weight,kg 9 15 26
5 107 718 702
19.2 102 527 507
31 209 445
When comparing the results of table 6.4 with the theore
ticalapproach given in chapter 4, it is seen that the con
cept of the contact resistance for heat transfer is not
applicablebecause of the observed influence of both gas
velocity and bed height; moreover the experimental data
I reveal values of ad that may exceed very much the es-
1 timated value of ad = 2 1 effw/dp = 500 iv/m2 oc (see chapter 4) •
* mean value of 0.84 (run 13) and 0.80 (run 20!
& mean value of 0.85 (run 16) and 0.69 (run 21)
- 116 -
Some doubts about the physical meaning of the contact re
sistance were expressed already in chapter 4. The influence
of gas velocity and bed height on the value of ad should be
explainedfrom a viewpoint of solids mixing in the bed. With
increasinggas velocity particle mixing becomes more inten
sive, which means that refreshment of particles near the
distribution plate occurs at higher frequencies,resulting
in larger transfer coefficients ad. At high gas velocities
the bubble hold-up near the distribution plate may become
so large that it starts limiting the heat transfer from the
plate to the bed. '.L'he influence of solids mixing becomes
lesswhen the bed height is increased. If solids mixing can
be described with an axial mixing coefficient Es' the influ
ence of the bed height should be discussed in terms of a
Fourier· number (see section 3.2.2):
E .t Fo = s ~
According to measurements of de Groot[117]E increases only' s · little with increasing bed height L which means that mi-
xing is less efficient in high beds. If overall circulation
of solids takes place mixing phenomena can be compared on
the basis L/up with up the circulation velocity of the par
ticles. According to Hoelen[108] (see also 3.2.2):
fw 0b (uo -umf) up 1-f -ob(1+2 f ) w w
in which fw is the wake fraction of bubbles.
The foregoing formula indicates a decrease of u with in-p
creasing bed height, as the bubble hold-up ob becomes smal-
ler; the latter is due to higher bubble velocities because
of a large mean bubble diameter. So also in this situation
solidsmixing is less intens when the bed becomes higher, and
theheat transfer coefficient between the distributor and
the bed becomes accordingly smaller.
- 117 -
7. Discussion of experimental results on mass transfer
7.1 Exchange between bubbles and the dense phase
From the experiments of which the results are reported
in tables 6.2 A and 6.2 B it is concluded that at the con
ditions of those experiments the dense phase humidity is
essentially constant throughout the bed. From those tables
it also follows that bubbles and dense phase gas have the
same humidity when leaving the bed.
The latter fact is to be expected on the basis of the the
ory developped in chapter 3. Since a constant bubble size
and velocity were assumed in chapter 3, the influence of
the observed changes of bubble size and velocity with the
height in the bed will be considered.
·For a uniform dense phase gas concentration, which is in
equilibrium with the surface concentration CsR of the par
ticles, the change of the bubble concentration with the
height in the bed is given by:
Q (1-y) g m
- c ) gb
in which y follows from equation 3.16. For a spherical bub
.ule it holds that:
since V = .!!. d 3 b 6 b
and Q = l TI u d 2 g 4 d b
( 7. 1)
Both Ub and db depend linearly on the height l in the bed,
as is concluded from the experimental results presented in
figures 5. 9 and 5.10:
lJ b = a l + b
db p l + q
After substitution of these relations into 7.1 and integra
tion of 7.1 1 it follows, under the assumption that bubble
growth is a result of coalescence only, that:
- 118 -
- 2. l.ud (1-y) ] 2 aq - bp
For the situation u0
26 cm/s and L = 31 em the constants
a, b, p, q were determined from figures 5.9 and 5.10.
Since under those conditions the observed bubble size and
velocity were the largest of all experiments carried out,
exchange between bubbles and the dense phase must be better
than according to those data. The following values are
found:
a = 2.3 1/s
b 0.57 m/s
p 0.26
q = 0.01 m
In figure 7.1 the dimensionless bubble concentration is
plotted versus the height in the bed, taking ud = 5 cm/s
and y o.
0.15
Cgb- cliO
c•R/m-cllo
L
26 cm/s
31 em
~l.om
Figure 7.1 Calculated bubble concentration as function of height for variable bubble size and velocity
- 119 -
It is seen from the figure that mass transfer to a bubble
occurs mainly (although not completely) in the lower part
of the bed: 70% of the maximum concentration CsR/m is
reached within 5 em, 80% within 10 em; for other conditions
nentioned in figures 5.9 and 5.10, this effect is even more
pronounced, and complete saturation of bubbles may occur
withina few centimeters. This effect, although partly ex-
pected because of the driving force near the distri-
butor, is caused especially by the presence of small bub
'bles near the distributor and large bubbles at higher levels
·in the bed. So the conclusion is justified that bubble gas
(and accordingly the total gas flow) teaches a high
equilibriumwith the surface concentration of the particles
in a rather low bed. 'this conclusion holds even better if
otherphenomena (as mass transfer during the formation of
bubblesor during coalescence of bubbles) contribute to the
exchange process.
Saturation of bubbles in a low bed should not be expected
of
in general. If bubbles are being formed near the dis-
tributor (as may occur when a sieve plate or a bubble-cap
distributor is used instead of a porous plate) , raoisture re
moval via the bubble phase is not efficient anymore. This
is illustrated with the results of a special experiment.
4.5 Kg silicagel (on dry weight basis) was dried in the gas
heateddrier at a gas velocity of 15 cm/s. After the bed had
reached the desired bed temperature, the gas flow through the
iistributor was made equal to the flow required for minimum
fluidization (u 5 cm/s) • l'~t the same moment bubbles were ' 0
formed by blowing extra gas (corresponding with a superfici-
al gas velocity of 10 cm/s) into the bed through four pipes,
whichwere inserted vertically from above into the bed down
to 4 em from the distributor. The four pipes were spread
evenly over the whole crossection of the bed. The bubble
gas was heated by a separate heating element, in such a way
that the temperature of the gas, when entering the bed,
equaled the bed temperature; further heat was supplied in
- 120 -
the normal way by the gas flow through the distributor.
Figure 7.2 shows the exit gas humidity (expressed as RH ) e versus the mean solids moisture content; an arrow indicates
the end of the heating period, at which moment the bubble
gaswas introduced via the four pipes. Run 13 is presented
in the same figure, since the experimental conditions of
thatrun were comparable. As large bubbles are very likely
formed by the flow through the pipes, the decrease of RHe
in comparison with run 13 should be ascribed to a decrease
of the mass exchange between bubbles and the dense phase.
11
11
RV., 0 /o
t * bubble gas via pipes
• run 13
• • * • * • * * •
• •••• ff.
/
• •• I
/. • * * * * *
l * * * • ** *
.. X, 0 /o
·~------~------L-------L-------L-------L-~ I II u
Figure 7.2
- 121 -
7.2 Exchange between particles and gas
As the exit gas reaches complete equilibrium with the
surfaceof the particles, which was concluded for the pre
sent experiments in the foregoing section, any difference
betweenactual exit gas relative humidity RHe and equilibri
um relative humidity RH* (based on the mean solids moisture
content) must be ascribed to diffusion resistance inside
theparticles. Figure 7.3 shows the relation between REe
and X that should be expected for batch drying of a solid
material for which the equilibrium curve is linear.
0.5
t
Figure 7.3 Relation between gasconcentration and mean solids concentration according to chapter 3
In this figure dimensionless gas and solid concentrations
havebeen plotted according to the results in figures 3.8 A
and 3.8 B by eliminating the Fourier-time. The curves are
explained as follows.
- 122 -
In the beginning the solids surface concentration decreases
more rapidly than the mean solids concentration due to in
creasing diffusion resistance inside the particles; this
cruses an increasing difference between actual and equili
briumexit gas concentration. After some time the concentra
tion profiles inside the particles become similar in time,
which means that (m C - m C 0) I (C - m C 0
) becomes con-g g s g stant,while also the Sherwood number Sh, has reached a con-
stant value (see figure 3.9).
The equilibrium curve of the silicagel material, used in
theexperimental work, is not one single straight line, but
might be approached by two straight lines (with an inter
section at X = 16%) in the range of solids moisture contents
that was met during the experiments. This fact is easily re
cognizable in the experimentally obtained relations between
RHeand X (see figures 6.2 to 6.10).
Equation 3.27 described the change of the mean solids con
centration with time in dimensionless terms:
d c s d Fo
3 Sh s 1 + 3 Shs Tb
( 7. 2)
in which cs (C - m C 0) I (C - m C 0
) • T represented s g so g somekind of a Fourier-number for the bed:
L(l-ob) (1-e:)m
u0
(l-ee) (7.3)
Themass balance given in equation 6.1 can be written in similar terms:
d c s - '(f""F"O
me -mc 0 ge g
(7.4)
since L(l-ob) (1-e:) is the volume of solids per unit area
of the bed's crossection, and C and C can be set equal s g
to Xps and Hp respectively. . 0
- 123 -
By combining equations 7.2 and 7.4 it is found that:
= (7. 5)
in which is defined as Tb for is zero.
If the exit gas reaches complete equilibrium with the sur
' face of the particles {as was the case in all experiments)
ce in equation 7.3 can be taken zero and it follows:
3 Shs Tbe 1+3 Shs Tbe
{7.6)
In table 7.1 the right-hand side of equation 7.6 is shown
for different experimental conditions. Shs has been taken
equal to its final value according to the calculation of the
1 ong term response of a single particle {section 3. 2. 2) • The
t 4
Big
1 + Ng
Limiting Sherwood-number, calculated from equation 3.11
- 124 -
limiting value of Shs has been plotted in figure 7.4,
whichwas calculated from equation 3.11. It is approached
if Fo > 0.06, which corresponds with a drying time of -12 2
27 minutes if W and Rp are taken as 10 m /s and 165v
respectively. hlthough a drying time of 27 minutes is not
negligible compared with the total duration of an expe
riment, the limiting Sherwood-number has been reached in
all experiments if the mean solids moisture content has
becomesmaller than 16%; this can be found from the experi
mentalresults presented in figure 6.1 and in Appendix E.
Table 7.1
run 'iid ,Kg nr.
2 9.6
3 9.6
4 9.6
6 9.2
7 19.2
8 19.2
9 37.9
10 19.0
11 19 .o 13 5.0
14 5.0
15 18.8
16 19.2
17 19.2
18 31.0
19 31.0
12
15
18
28
12
18
9
15
15
15
26
9
15
26
15
26
L, em
20.2
20.9
21.4
22.1
40.6
43.0
79.3
41.6
41.6
10.9
u. 8
38.1
42.0
45.7
67.9
73.9
Bi ___!L
l+N g
0.766
0.925
1. 08
1. 63
0.381
0.540
0.146
0.826
1. 42
1. 78
2. 84
0.304
0.460
0.734
0.285
0.454
Sh s
4.70
4.65
4.60
4.45
4.84
4. 77
4.94
4.69
4.49
4.41
4.20
4.87
4.81
4.71
4.88
4.82
3Sh 'be 'be 77~~5~~--
(X<l6%)1+3 Shs 'be
.784
.628
• 522
.322
1. 474
1.050
4.14
.702
. 410
.326
.186
2.06
1. 260
• 7 28
2.04
1.176
(X<l6%)
0.917
0.898
0.878 wall
0. 811 heat
0.955 drie
0.938
0.984
0.908
0.847
0.812
0.701
0.968
0.948
0.911 gas-0.968 heat 0.944 drie
- 125 -
For this reason the data in table 7.1 concern only that
period of the drying process, for which X is smaller than
16%. The equilibrium curve can than be approached by one
single straight line, the partition coefficient m being
1 .28 10 4 at 40°C.
The data on 3 Shs Tbe/(1+3 Shs Tbe) in table 7.1 reveal
that a high of equilibrium between exit gas humidity
and mean solids moisture content should be expected with
most of the experimental conditions. It is also seen that
the influence of the gas velocity at constant dry solids
weight, as well as the influence of the amount of solids
in the bed at constant gas velocity is marginal concerning
the degree of equilibrium at the upper bed level. These
calculations are confirmed by the experimental results pre
sentedin figures 6.2 to 6.10 for X< 16%. Only at Wd=5 kg
the observed of equilibrium is much larger than ex
pected according to table 7 .1. E'or most experiments the ob
served influence of gas velocity and bed height is even
smallerthan predicted in table 7.1: this might be caused by
inaccuracyof the measurements, as discussed already in
chapter 6).
The results of the drying experiment that was interrupted
twice (see table 6.2 A and figure 6.11) also indicate that
thedifference between surface concentration and mean con
centration of particles must be small. Otherwise the drying
rate after the period of interruption should have been lar
ger than immediately before that period due to levelling
out of the concentration profile inside the particles. Fi
gure 6.11 shows that the shape of the drying curve before
and after the interruption is almost the same, which I'',eans
that the profile inside the particles is very flat, and
that the exit gas can reach equilibrium with the mean solids
moisture content very nearly.
In table 7.2 the decrease of the mean solids moisture con
tentcalculated according to equation 7.2 is compared with
the results of some experiments.
2
3
4
6
10
11
13
14
17
18
19
- 126 -
Table 7.2 Comparison of experimental results
with equation 7.8 *
16 11.4
16 10.0
16 8.0
16 5.5
16 7.1
16 3.4
16 5.0
16 4.3
16 8.2
16 12.8
16 9.7
113
110
70
38
76
44
46
31
77
187
137
185
200
185
161
253
247
229
177
249
343
365
X
2.5
2
3.5
2
4
2
3.5
3.5
4
4
3.5
(x -x )/(x -x ) 1 2 1 calc.
0.341
0.429
0.640
0.750
0.742
0.900
0.880
0.936
0.650
0.267
0.504
meas.
0.310
0.433
0.574
0.745
0.635
0.843
0.866
0.912
0.613
0.278
0.554
ratio
calc/meas
1.10
0.99
1.12
1. 01
1.17
1.07
1.02
1.03
1. 06
0.96
0.91
*only those runs are compared for which the solids mois
ture content at the end of the process was significantly
below 16% (see graphs in Appendix E).
l:'or constant Sherwood-number equation 7. 2 may be integrated
0
C - m e: so cg
3 Sh Fe 1 1+3 ~hs 'be
Taking Xps and m Cg0 = x*p8
(x* being the moisture con
tent in equilibrium with the feed gas humidity) it can be
derived that:
[ 3 Sh ]
1-exp - l+3 S~ (Fo 2 - Fo1 ) s 'be
(7.8)
x1 and x 2 are the solids moisture content at Fo1 and Fo2
respectively: the latter correspond with drying times t 1 an
t 2.
x 1 has been taken 16%: for x 2 the moisture content at the e
of a run was chosen; t 1 and t 2 were determined from the
- 127 -
graphs in Appendix E, translated into Fourier-times, and
used to calculate (X1-x2 )/(X1-x*) from equation 7.8.
The agreement between calculated and measured decrease of
themean solids moisture content is fairly good.
Forx > 16% the influence of gas velocity and bed height
on the degree of equilibrium between exit gas humidity and
,,,ean solids moisture content is not very clear: no trends
can be discovered from the results presented in figures
6. 2 to 6.10. This might be ascribed partly to the inaccu
racy of the measurements, which is largest immediately af
ter the start of an experiment, partly to some other factors
as will be discussed in the following.
For X> 16% the partition coefficient m = (C /C )at equilibrium s g is decreasing all the time during the course of drying,
Hhich means that the moisture removing of the gas
flow becomes larger. Also the effect of diffusion limitation
inside the increases at smaller m. The latter leads
to a decrease of the surface concentration of the particles,
which is not necessarily followed by a proportional decrease
of the gas concentration since m becomes smaller. This fact
might explain why the exit gas humidity in some experiments
remains more or less constant during a period shortly after
the start of a run (see figures 6.3 and 6. 4 A and B for in
stance) although that effect may be due to the of the
equilibrium curve as well.
Another factor that might influence the degree of equilibrium
betweenexit gas and mean solids moisture content is solids
mixing in the bed. If solids mixing becomes more intens the
moisture withdrawal by the gas flow is spread more evenly
overall particles in the bed. Generally solids mixing is more
ideal at higher gas flows.
In this respect the assumption about a uniform surface con
centration of throughout the bed (made in chapter
3) should be verified more precisely.
i>s was concluded in section 7. 1 transfer between
and gas occurs mainly in the lower part of the bed.
- 128 -
Suppose therefore the fluidized bed drier is divided sche
matically into two regions L1 and L2 (see figure 7.5), each
of which is ideally mixed for both solids and gas. In the
region L1 the gas phase (superficial velocity u0
) is satu
rated completely with respect to the surface concentration
CsRl of the particles in L1 • In the region L2 the surface
concentration of the particles is CsR2 ; exchange between
particles and gas in the region L2 is completely neglected,
which is allowed if the difference between CsRl and CsR2 is
small. Solids are circulating between the regions L1 and
L 2 with a velocity~' which is so large that the mean so
lids concentration Cs is the same in both L1 and L2
•
During their residence time T1 in L1
the particles loose
moisture from an outer layer o while the surface concentra
tion decreases from CsR2 to CsRl' The concentration profile
in that layer is linear if o is small. If particles arrive
in L2 the relatively dry layer at the outer surface picks
upmoisture again because of mass exchange between the par
ticles mutual in L2 ,and because of diffusive transfer from
the interior of particles to their surface.
u C R1/m 0 I
-c,
c,R2
u : ~ P__t u0 lup
-c,
CIR1 t "• Figure 7.5
- 129 -
If the residence time of particles in L 2 is sufficiently
large, the dry layer will disappear completely.
The mass balance for the whole system is:
c L(1-E) = u ( sR1
o m (7.9)
E is the bed porosity, L = L1+L2 is the total bed height.
Transferbetween particles and gas occurs only in there
gion L1
, so it holds:
c u ( sR1 _ C o)
o m g 2 ( 7. 10)
Here 2 Ill 6 (CsR2 - CsR1 ) is considered as the mean flux for
a particle during its residence time 11
in the region L1
.
The amount of the moisture a particle looses per unit sur
f ace area during T 1 is:
2
from which is found:
o~w if the resi~ence time 1 1 is taken as 1 1
~ombination of 7.10 and 7.12 yields:
- .m c 0 g
u 0
m
{7 .11)
(7 .12)
(7.13)
Some data will be inserted into equations 7.12 and 7.13.
According to the calculation on solids circulation in the
bed (section 3.2.2) u equals 2.33 cm/s at u = 26 cmjs. p 0 From the data in figure 7.1 L1 might be estimated as 10 em.
For a diffusivity Ill = 10-12 m2;s and a specific surface of
the particles s = 10 4 m2;m3 it is found that o = 4~ while s
(CsR2-csR1)/(CsR1-m Cg0
) = 0.054 where mistaken as 104
•
- 130 -
It follows that the assumption of a uniform surface con
centration of the particles in the bed is quite reasonable,
especially if the process has lasted some time, since C a1 0 s. approaches tom Cg during the course of drying.
Howeverthe zone L1 , in which bubbles come near to satura
tion, depends strongly on the bubble size near the distri
butor (which size in turn is depending on the superficial
gas velocity and the bed height, as shown in figures 5.9
and 5.10). So L1 = 10 em might be an over-estimation, which 0 means that (CsR2-csR1)/(CsR1-m Cg) can be larger under some
experimentalconditions than calculated above. The latter
will occur especially at low superficial gas velocities,
as in such case L1 may be small, while moreover solids mi
xing is not intensive.
- 131 -
8. General conclusions
A model has been presented to describe the transfer
between particles and gas in fluidized bed driers. The
influenceof bubbles on the transfer process was taken in
to account, as well as the effect of a possible mass trans
fer resistance inside the particles.
Ourexperimental results obtained with silicagel of mean
particle diameter d 330~ do not allow a conclusion about p the validity of the model. It has been shown that nearly
complete equilibrium between exit gas and surface of the
particles is to be expected in even low beds because of
smallbubbles that are formed near the porous distribution
plata However in most industrial equipment such ideal gas
distributorsare not applied which means that the bubble
size will be larger in general. Under such conditions the
influence of bubbles on the drying process is more pronounced,
as was illustrated with a single experiment in which the
gas feed was not spread evenly over the crossection of the
bed. So even gas distribution may be very important in drying.
Due to a high degree of equilibrium between exit gas humidi
ty and mean solids moisture content, the influence of pro
cess parameters in the batch drying experiments could only
be marginal. This high degree of equilibrium was a result
both of small bubbles and of rather flat concentration pro
files inside the particles. 0n basis of the presented the
ory this could have been expected. Right after the start
of an experiment in the gas-heated drier (when the bed is
still being heated-up) the exit gas humidity often was lar
ger than according to equilibrium conditions. This is caus
ed by. the saturation of small bubbles near the distributor,
where the temperature will differ considerably from the tem
perature in the bulk of the bed if heat has to be supplied
bothfor drying and for heating. Due to bubble growth the
exchange between bubble gas and solids becomes less effi
cient and bubbles will leave the bed with a humidity that
- 132 -
exceeds the equilibrium humidity. This effect, which was not
observed in the wall-heated drier, underlines the importance
of bubbles in the drying process.
In a second period of the drying process the influence of
processparameters on the degree of equilibrium was rather
confusing. This might be caused by inaccuracy of the measure
ments although possibly some other factors are involved.
Among these factors are the shape of the equilibrium curve
(decreasing partition coefficient) and the question whether
the surface concentration of the particles throughout the
bed is uniform or not.
In the final period of the batch experiments the sorption
isothermcan be approached by a straight line through the
origin. Calculated and measured decrease of the mean solids
moisturecontent were found to agree reasonably well.
Because of intensive solids mixing the mean solids moisture
. content is uniform throughout the bed.
The influence of the particle size in fluidized bed dryinq
will be discussed from a theoretical viewpoint,as this parameter was not varied in the experiments.
For particle sizes larger than used in the present experimen
equilibrium between exit gas and surface of the particles
willbe better approached. This is caused both by larger
clouds around bubbles and by a decrease of the fractional
bubble flow. As shown in figure 3.16 large particles dry at
higher rate than small ones. This trend continues as long as
diffusion resistance inside the particles is not dominant.
So for very large particles the drying rate is not limited
anymore by transfer of moisture to bubbles, but by diffusior
inside the particles, and the drying rate will decrease witt
increasing partie le size.
When drying smaller particles than used in the present expe
riments the foregoing reasoning holds the other way round.
Diffusion limitation inside the particles becomes negligi
ble while the importance of bubble behaviour increases.
- 133 -
This is illustrated below with the height of a mass trans-
fer unit for the bubble, introduced in section 3.3.1:
3/2 1/2 vb ub 4 Rb g
HTUm Qa 9 ud
Here the relations v = 4 3 ub = 0.7yg.2 Rb' b 3 1T Rb I
Qa 31T ud R 2 b
have been substituted.
From equations 1.1 and 1.2 the dense phase velocity ud
might be estimated:
d 2 p
By combining the foregoing relations the number of mass
transferunits Nm for the bubble phase is found:
The
E
pp
7 213
N = L m HTUm
9 4
3 d 2
E pp p L r;;--=-!-~~;;.. __ ,i~
1-c 72 8 Jl v " 3 .b
following orders of magnitude will be inserted:
0.5 -5 2 ]1 2.10 Ns/m
kg/m 3 2 1350 g 10 m/s
150 L = 0.30 m
from which can be calculated:
5 Nm = 2.4 10
d 2 __p__
~ The table below shows the importance of bubbles when dry-
ing small particles I as seen from the value of N : m N Rb= 1 em ~= 2 em m
dp 330).1 26 9.2
dp 110).1 2.9 1.0
The foregoing shows that in our experiments with silicagel
particles of diameter 330).1 an influence of bubbles cannot be
- 134 -
found in case of even gas distribution. As diffusion limi
tationwas rather small, equilibrium between exit gas and
mean solids moisture content is likely.
From the heat transfer aspects of the present work it is
concluded that the heat transfer coefficient between the
bedand the bed wall is constant during the course of drying
the observed influence of gas velocity and bed height agree
with results from the literature.
Heat transfer between the feed gas and the bed occurs for
a large part via the distribution plate; this effGct will
depend on the construction and the material of £he plate.
For instance if sieve plates are applied as distributor
heat transport via the plate will be much less and the heat
for drying is supplied by transfer in the bed between gas
and particles directly; this may cause a large zone above
the distribution plate, in which the temperature differs
considerably from the temperature in the bulk of the bed.
The contribution of the plate to the heat transfer can be
describedwith a heat transfer coefficient, which shows a
similar behaviour as the transfer coefficient between the
bed and a vertical bed wall.
- 135 -
APPENDIX A
Transfer between a single sphere and stagnant gas;
a simplified approach
A.1 No gas phase transfer resistance
A single particle of uniform concentration C 0 and s radius Rp is in contact with stagnant gas ( volume v, concentration C = 0). The gas is ideally mixed and its
g concentration is in equilibrium with the surface of the
particle. If the partition coefficient m is very large,
the gas concentration has reached the equilibrium value
Cs0 /m very nearly before the diffusion process inside the
particle has properly begun. This means that the concen
tration profile inside the particle is concentrated in a
narrow layer o at the particle surface; in such a small
layer a linear profile may be assumed.
At any time it holds that:
2 C o + C R
4TI R o(C 0 - s s ) p s 2
(A1)
where CsR denotes the particle's surface concentration,
which is in equilibrium with the gas phase: CsR = m cg.
Thepartial mass transfer coefficient Ks is defined as:
~ v dt Ks 4TI R 2
(C 0 - m C ) p s g
(A2)
As Ks ID/o (ID is the diffusivity inside the particle),
the Sherwood number Sh is found:
K R R 21T R 3 c 0 - m c Sh ~=__£_
lt 0 v
Sh .!..:x X
m c where x ___L
c 0 and E =
s
Solving equation A2 after
f(x) = _x_ + Zn(I-x) 1-x i n which Fo == IDt/Rp 2 •
I2 s c g
v 4 3 31T R m p
elimination
g:
(A3)
of o, it follows:
(A4)
- 136 -
In figure A.1 the curve for Bi = oo presents ESb as func-2 g
tion of Fo/E as calculated from the combination of equa-
tions A.3 and A.4. A reasonable agreement with the exact
solutio~ presented in figure 3.5 is found. For small Fo/E2
it holds that ES8 ~ E~o, whereas ESh ~ E2/Fo holds for 2 large Fo/E .
A.2 Gas phase transfer resistance present
Since in the foregoing Sh becomes infinitely large when
Fo tends to zero, transfer resistance in the gas phase will
be limiting when Fo is very small.
Gas phase transfer is described with a mass transfer coef
ficient K • It is assumed that the volume of the gasfilm in g
whichthe resistance is concentrated, is negligeable. The gaf
volume V is, as under A.1, assumed to be ideally mixed.
If Cgi is the gas concentration at the interface particle
gas, the following relations hold:
d c R 2 !cc 0 v ____..:r 41T - m cgi> dt p 5 s (A5)
!(c o - m c .. ) = Kg(Cgi - c ) 0 s g~ g (A6)
4n R 2o(C 0 c 0 + m cg:i
v c = s g p s 2 (A7)
When in A5 o and C . are eliminated via A6 and A7, the gas g~
concentration C may be calculated as function of time t g
(boundary condition: t=o, C =o). In dimensionless terms it g is found that (B. = Kg R /m Ill):
~g p
x._
1-x
+
1 1/2
+ [a ~g E _x _{_1_-_x_< _1-~6;..__a=.i9:oz__E_>}l ... < 1-x> 2 J
+
!n f 1-x 11-i "tg E) +Vt •,2 E( x-x2 11-i- Bi2 E) l 1 ..----- (1-x)
2
[arcsin {2x (1- i Big E)-1} + I]= 9 Fo/E2
- 137 -
1 10
Parameter E·Bi 9
•Bi 0 g
t 0
10
-1 10
-3 10
0.05
-2 10
-1 10
0 1 10 10
--~ Fo/E2
2 10
Figure A2 'l.'ransfer between a single particle and gas, with gas
phase transfer resistance taken into account
- 138 -
In figure A.2 x is plotted versus Fo/E2 for different va
lues of Big E. An overall mass transfer coefficient K0
g
might be defined by:
v~ dt
In figure A.1 Bi K R /m S is plotted as E Bi versus og ogp og Fo/E2 for different B. E. It may be derived from the equaJ.g tions above that:
Bi og
On combining figures A.1 and A.2 it is seen that the gas ph2
becomes very nearly saturated during the time that the trans
fer resistance is in the gas phase, when E Big< 0.01.
- 139 -
APPENDIX B
Change of the temperature of a rising bubble
The change of the bubble temperature with height in the
bed is derived from the enthalpy balance of the bubble
(equation B.1):
(B .1)
HereC and C are the specific heats of air and water va-pa pw pour respectively. According to section 3.3.1 the following
d H expressions can be used for Hb' ____ b and Hin: dl
Hb = H *- (H* - Ho) exp (- l/HTUm)
When substituting these relations into B.1 it is found that:
+ C {H * - (H* -Ho) exp(-d [cEa 8 - l/HTU ) }] Tb Qa J2W m m (T - T ) err ub vb [cpa - (H* -Ho) exp (- l/HTUm)}] s b + C {H* pw
(B. 2)
By integration of B.2 with boundary condition l = o,Tb=T0
,
a relation for the bubble temperature as function of the
height l in the bed is found:
T - Tb c + c H*- C (H* -Ho) s Ea J2W J2W
T - T cpa + c H*- Cpw(H* -Ho) exp(- 1>/HTUm) s 0 pw
-exp [-Qa l l u vb b
(B. 3)
- 140 -
Equation B.3 reduces to equation 4.7, presented in chap
ter 4, if the humidity changes of the gas, when passing
the bed, are small.
- 141 -
APPENDIX C
:velocities of the three phases in a fluidized bed
When a spherical bubble rises up in a fluidized bed the
mean gas velocity Uv in the bubble (relative to the bubble
U b itself) equals three times the dense phase velocity u d
far away from the bubble [14,118]. Uv is related with a
downwardgas velocity Uc (again relative to the bubble ve
locity) in the cloud around the bubble:
u == c
in which ob and oc are the bubble and cloud hold-up respec
t ively, and E: is the porosity.
Applying the foregoing to bubbles rising in a heterogene~
ously fluidized bed gives the following linear gas veloci
ties in the bubble and the cloud phase, relative to fixed
c oordina tes +
bubbles: ub + 3 ud
clouds: ub - 3 ud ob - r E: c
The total flow* of bubble gas, respectively cloud gas pas
sing an arbitrary horizontal level in the bed,is:
(C .1)
cloud gas: (C. 2)
The dense phase flow through that same level is:
Equations C.l - C.3 have been applied in the material ba
lances 3.13- 3.15.
+the influence of an overall solids movement on the gas flow is neglected.
*expressed per unit crossectional area of the bed.
- 142 -
The total flow through a horizontal bed level must cor
respondto the superficial gas velocity u0
:
(C. 4)
Equation c.4 will be compared with the total gas balance
thatwas derived by Lochett e.a [119] and by Rietema e.a.
[118] • Their result was:
(1 + 2 obl ud + ob ub = u (c. sr 0
By combining equations c.4 and c.s it is found that:
oc 3 (C. 6)
ob a-1
in which a = ub t:/ud
Equation C.6 represents exactly the cloud to bubble ratio
as derived by Davidson and Harrison[14] under the assump
tion of potential flow of solids around the bubble. How
ever as the latter assumption has not been made in the
derivation of equations C.4 and C.5, equation C.6 is valid for any type of solids flow.
Thepresent result does not imply that the cloud is con
centric around the bubble.
- 143 -
APPENDIX D
D.1 Diffusion of moisture in silicagel particles
In section 5.2.1 the method developped by Dengler e.a.
[109-111] for measurement of moisture diffusivities in
solid particles was described already. The method was
based on a step-wise change of the water vapour pres-
sure in an evacuated weighing chamber in which a
of solid material was suspended in a pan on a Chan Elec
trobalance. height changes of the sample were recorded
until equilibrium was reached (which means constant sam
ple weight)~ cfterwards the water vapour pressure was
lowered again and weight recordings were
steps in lowering the vapour pressure were
order to achieve an isothermal sorption process.
• The
in
The changes of the sample weight during time were inter-
preted in the following way which was presented
by Dengler e.u.[109].
A particle is supposed to be a sphere, of a
homogeneous mixture of solid material and pores. Moisture
diffuses through the pores under influence of a concentra
tion gradient in the pores. :Equilibrium between the gas in
the pores and the local pore-wall is assumed to be reached
instantaneously: ps X = m Cg' with X the moisture content
of the solid material, ps the solid density, Cg the gas
phase concentration and m the partition coefficient of the
linear sorption isotherm. Transport inside the particle is
described by the following differential,equation (s is the
internal porosity of the particle):
.L { (1-s) p X + sC }=ID2 (r 2 (D .1) 3t s g r
By substitution of the equilibrium relation it follows:
ID d cg
~= e d (R2 (D.2) -2- ar r
- 144 -
Here IDe ID/(m(1-E)+ E) is an effective diffusivity which
is taken as constant. For constant surface concentration
cg* of the particle the mean concentration cg as function
of time can be approached by:
X c - c 0
g g =~ 3~ Fo - 3 Fo
c*- c 0
( D .3)
g g
as long as X < 0.928. C 0 is the initial concentration
ID t/R 2 . Fori linear sorption isotherm it also and Fo e P
holds x = (x - x ) I (x*- x ) , with x the solids moisture 0 0 0
content at t o and x* the moisture content when equili-
brium has been reached. From D.3 the diffusivity IDe can be
calculated if X has been measured as function of time:
lD e ( D. 4)
Figure 0.1 presents IDe as function of X as obtained by
Dengler e.a. with silicagel type G.127 of mean particle
radius RP = 1. 32 mm. '.Che values of x0
and x* are indicated
in the figure. The data refer to moisture adsorption, and
m~ 4 I I I • • • IDe s • • • * • • 2 .. D 0 D D g-
* 0 * 0 0 t _, 0 0 * * 0 10 - * 0 -
8 * 0 -
6 * * 0 --4 0
2 0
Figure D,1
I I
0,2 0.4
n * Xo o 12
X* 1 15
I
0.6
0 •
23 31 %
25 33 %
I
0.8
')o-
Measurements of ID
X
according to ~engler e.a.[109] (RP = 1. 32 mm)
- 145 --11
10 I I • I I I
ID m~ 8 I
e s 6 • -
t • • •
• • • •• • 4 8
• * * * * * * 2
ooooD :J 0 0* -
c '] 0 0 oo *
0 0 0
0 0
-12 Do 0 oo 10 r- iii 0 0
8 0 *0
* * -6 * * * * 4
* * •* * *
2
-13 10 l l l l l l l
0 0.2 0.4 0.6 0.8
)Ia X
* 0 D * • Xo 15.2 17.0 21,2 26.3 29.4 %
x* 12.1 14.3 17.4 24.2 27.4 %
Figure D.2 !·'.easurements of ID for silicagel e used in fluidized bed drying ex-periments (Rp 0.165 mm).
the change of the mean moisture content during each expe
riment was about 3%. 'l'he results of Kriickels [110] are al
most the s<une, although he reports a sharp decrease of De
in the region x < 0.1; Dengler e.a.present no data in this
region.
Since IDe is not independent of X , application of equation
D.4 is not allowed. For this reason Dengler a.a. introduced
thename apparent diffusivity for the result calculated from
D.4. Kruckels[110] states that the apparent diffusivity is
depending on the concentration gradient inside the particle,
and tries to explain the influence of x from this viewpoint.
- 146 -
For constant x it can be seen from figure 0.1 that initially
m decreases with decreasing solids moisture content, but e
from X = 25% downwards the diffusivity increases slightly.
In Figure 0.2 our own results are shown which were obtained
withthe silicagel particles used in the batch drying experi
ments (R = 0.165 rom). There is a qualitative agreement with p the data of Dengler e.a. IDe also increases with increasing x, but for x > 0.8 a decrease of IDe was observed in most expe
riments. At constant x the same trend as found by Dengler e.a.
exists between IDe and mean solids moisture content X.
However the order of magnitude of Th for the present parti-.e cles is a factor 100 smaller than for the coarser material
Dengler e.a. used. From this it might be also be concluded
that the given interpretation of the measurements is not cor
rect, since for a proper interpretation IDe should be indepen
dent of the particle size.
0.2 Heat of adsorption
The heat of moisture adsorption inside silicagel particles
(apart from the heat of moisture evaporation) is given in
the following table as function of the solids moisture con
tent, according to the measurements of Dengler and Kruckels
[109].
X % 3 5 10 15 20 25
llH v kcal/kmol 4000 3500 2500 2000 1700 1700
- 147 -
APPENDIX E
Experimental data, concerning the batch drying
ex per imen ts •
""' l48 -G u•c 1 ke
0 n cun/e
* U om/a
0
0 .. 0 .. 0
.. 0
0 .. 0
.. .. 0
Figure El time, minutes
0 100 0 50 150 zoo lfil
II
~· w 9,6 kg * I Cft'l}s
·~·. 40°C 0 U cmjs 8 (il • 11 omt• .. 8 t!•
8 11 cm;e II .. 8 e>• .ft.).
* 21 om/a .. 8
~· ('jt .. 8 8 • 0* X 'I. * .. 8 • 0 11
t *
8 • 0 * .. * 8 • 0 .. • 0 * 8
11 .. 8 • 8 .. 8 .. " ..
time, minutes
150 zoo Zil
G •oltc 19 kg
• * 9 em/a .. • a em;• D !6 cm;a •
II ... 8 • ..
D ... ..
X 'I. 8 • .. 11
1 8 • ['j
8 8
8 11 8
8
~"'igure E3 ---~- time, minutes
- 149 -....... • • w 40°C 1t kg
• # * ll Ofl't/• •• ,. • 11 om I• • • • • 20 • • • • • X 'I. •
11
1 •
• •
10
a
Figure E4 time, minutes
0 0 so 100 150 200 250
Zl ••• G u•c U crn/e • * • 0e 1t ke
* 0* 1 ke X% • • H
t • *
• * • II • ·o00 * 0
0 0 0 * 0 0 0 0 0
11 0 * 0 0 0 0 0 0 * 0
0 * 0 *
I Figure E5 time,mlnutes
I I I •• 1DD 158 ·iao lit
25
w 15 Gffl/• ±19.1 ka
~. Clit '* • so 0 c 0 60 °C
20 °. 0 • X% 0 •
t • • • 11 0
0 • 0 •
10 c
• 0 • 0 •
0
I 0
0
Figure E6 -+-time,minutes
0 100 150 200 250 0 50
- 150 -
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1.
2.
3.
4.
5.
6.
7.
b.
9.
10.
11.
12.
13.
14.
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16.
17.
18.
19.
20.
21.
22.
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T.B.Anderson, R.Jackson, ibid, l• 12, 1968.
K.Rietema, S.M.P. Nutsers, Proc.Int.Symp. "Fluidization and its applications", Toulouse, 1973.
R.D.Oltrogge, Ph.D.thesis, University of Michigan, 1972. 1l
A.A.H.Drinkenburg, Ph.D.thesis, Eindhoven University,1970.
'l' .E.Broadhurst, H .A.Becker, Proc. Int. Symp. "Fluidization and its applications", Toulouse 1973.
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- 156 -
List of symbols
A
B
Bi g
Bi og
c
H
L
crossectional area of the bed
constant
Biot-number, Kg Rp/m •
overall Biot-number, K0
g Rp/m •
concentration of moisture
heat capacity of the gas
moisture diffusivity in solids
gas phase diffusivity
bed diameter
extraction factor, V/! 1T R 3 m 3 p
axial solids mixing coefficient
Fourier-number
absolute gas humidity
(in chapter 1: bed height
mass transfer coefficient between cloud and dense phase
mass transfer coefficient for the dense phase gas
mass transfer coefficient for the gas in the cloud
overall mass transfer coefficient
idem, mean value in a cloud
partial mass transfer coefficient for the particles
bed height
number of mass transfer units in a bubble's cloud
idem, between dense phase and cloud
idem, in the dense phase
kg/m3
J/kg oc
m2/s 2 m /s
m
m2/s
Kg/kg
m)
m/s
m/s
m/s
m/s
m/s
m/s
m
:tf g_
Nu
Pr
Q
R c
Rcontact
R p
Re
Sc
Sh
Sh_s
'f
- 157 -
idem, concerning one single particle
Nusselt-number, ad /A p p g
Prandtl-number• ~ Cpg/Ag
exchange flow between bubbles and the dense phase
circulating flow through a cloud on dry air basis
flow through one single orifice
circulating flow through a cloud
gas flow around one single particle
solids flow through a cloud
bubble radius
cloud radius
heat transfer resistance
particle radius
Reynolds number, p u d Ill g Q p
vertical bubble dimension
specific cloud surface
specific particle surface
Schmidt-number, ~/p ID
Sherwood number, Kg dp/ID
Sherwood number for a particle,Ks dp/m
temperature
linear bubble velocity
linear dense phase velocity
volume of gas around one single particle
bubble volume
volume per unit mass of moisture
3 m /s
m3/s
m3;s
3 m /s
m3/s
m
m
m
m
- 158 -
cloud volume in between 6=~/4 and 6=6
solids moisture content kg/kg
c dimensionless concentration
bubble diameter
cloud diameter
diameter of pores in distributor
equivalent diameter
particle diameter
dimensionless mean concentration in a cloud
( in chapter 2: bubble frequency
m
m
m
m
m
1/s)
fw wake fraction of a bubble
acceleration of gravity 2 m/s
jm Colburn factor
height coordinate m
m partition coefficient
n number of bubbles per unit bed volume
p heat transfer parameter
r radial coordinate m
s fractional bubble flow
t t~
u 0
superficial bubble velocity m/s
bubble point velocity m/s
gas velocity in cloud, relative to bubble m/s velocity
superficial dense phase velocity m/s
minimum fluidization velocity m/s
circulation velocity of particles in m/s the bed
total superficial gas velocity m/s
u v
- 159 -
gas velocity in bubble, relative to the bubble velocity
Ci. Ub E/Ud
m/s
a 1 heat transfer coefficient for a pore wall W/m2 0 c
idem, between distributor and bed
idem, between particle and gas
tortuosity
em parameter
y constant in equation 3.16
layer thickness
ob bubble hold-up
oc cloud hold-up
heat of evaporation
pressure drop
e: porosity
Ed dense phase porosity
e:0
packed bed porosity
e tangential coordinate
heat conductivity of gas
idem, of solids
m
J/kg
N/m2
effective conductivity of the dense phase W/m °C
A idem, near the bed wall eff,w
lJ viscosity
Pg gas density
p0
dry gas density
ps,pp solids density
o dimensionless height coordinate
W/m °C
Ns/m2
kg/m3
kg/m3
kg/m3
- 160 -
T time necessary to mix-up the bed's content s m
tp relaxation time of one particle s
$ heat flux w;m2
x dimensionless concentration
Subscripts
b bubble
c cloud
d dense phase
e exit of the bed
F feed gas
g gas
i interface
in refers to gas leaving a cloud and entering a bubble
o refers to l=o
p particle
r radial position
R surface of a particle
s solids
Superscripts
o refers to t=o (solids) or l=o (gas)
* refers to equilibrium conditions
Samenvatting
Bij het drogen van granulaire stoffen in een geflui
dizeerdbed is het dichte fase gas verzadigd met betrek
king tot het vochtgehalte aan het oppervlak van de deel
tjes, vanwege het zeer goede kontakt tussen deeltjes en
gas in de dichte fase. Bij gevolg zal stofoverdracht naar
de bellenfase een belangrijke invloed hebben op de droog
snelheid van het bed.
Een model is opgesteld om de vochtoverdracht naar een bel
in een fluidizatie-droger te beschrijven. Dit model is ge
baseerd op een vereenvoudiging van de cloud-theorie van
Davidson en Harrison, en neemt zowel stoftransportlimite
ring in de gasfase als diffusielimitering in de deeltjes
in beschouwing.
Door toepassing van het model op een heterogeen bed wordt
de afgasvochtigheid berekend, alsmede worden voorspellin
gen gedaan over het verloop van batch droogprocessen.
Het ligt voor de hand dat naarmate bellen grater zijn, zij
meer invloed uitoefenen op de droogsnelheid. Daarnaast
blijkt uit het model dat de rol van bellen belangrijker
wordt naarmate de deeltjes kleiner zijn, vanwege een drie
tal redenen:
- bij n poeder vindt het drogen van deeltjes in de cloud
random een bel nagenoeg niet plaats, omdat de cloud zeer
klein wordt~ - bij fijn poeder gaat hoegenaamd alle gas als bellen door
het bed~ - bij fijn poeder ligt de weerstand voor stoftransport over
het algemeen in de gasfase en kan diffusielimitering in
de deeltjes verwaarloosd worden.
Batchgewijze droogexperimenten zijn uitgevoerd met de mo
delstof silicagel (gemiddelde deeltjes diameter 330~) in
twee apparaten: in een apparaat werd het fluidizatie-gas
verwarmd voordat het in het bed werd geleid, in het andere
werd warmte toegevoerd via de bedwand.
Gemeten is de invloed op de droogsnelheid van de super
ficiale gassnelheid, de bedhoogte en de bedtemperatuur.
Tevens zijn de belgrootte en belsnelheid als funktie
van de hoogte in het bed gemeten.
Uit de experimenten blijkt dat het afgas van de droger
meestal in hoge mate evenwicht bereikt met het gemiddeld
vochtgehalte van de deeltjes. Op theoretische gronden is
dit te verwachten, maar het geeft geen uitsluitsel over
de juistheid van de theorie.
De hoge graad van evenwicht is een toevallige samenloop
van omstandigheden:
- Bij de poreuze bodemplaat die tijdens de experimenten
is gebruikt, ontstaan zeer kleine bellen, die zeer snel
evenwicht bereiken met het oppervlakte-vochtgehalte
van de deeltjes;
- Bij de gebruikte silicageldeeltjes blijkt amper diffu
sielimitering op te treden, waardoor het koncentratie
profiel in de deeltjes vrij vlak is.
De experimenten hebben ook gegevens betreffende de warmte
overdracht opgeleverd. Daarbij bleek dat in de wandver
warmde droger de warmteoverdrachtcoefficient tussen de
wand en het bed niet van het vochtgehalte van de vaste
stof afhangt. In de gasverwarmde droger bleek een groot
dee! van de warmte via de bodemplaat aan het bed te wor
den toegevoerd, voor welk verschijnsel ook een theoreti
sche benadering is gegeven.
Stellingen
1. Bij stofoverdracht tussen een enkele bel en de dichte
fase in een gefluidizeerd bed neemt de overdrachts
coefficient toe met toenemende beldiameter. Dit ver
schijnsel wordt veroorzaakt door konvektieve stromin
gen, die samenhangen met instabiele bewegingen van de
bel.
2. Bij lage waarden van het Reynoldsgetal (Re<lO) zijn
in een gepakt bed de kengetallen van Sherwood en
Nusselt veel kleiner dan de verwachte minimale waarde
Sh=Nu=2. Dit wordt vermoedelijk veroorzaakt door de
lage waarde van het Bodensteingetal bij Re<lO, waarbij
vooral de uniformiteit van de pakking een rol speelt.
P.A.Nelson, T.R.Galloway, Chem.Eng.So. E.A.Ebaoh, R.R.White, A.I.Ch.E.J.4, 1 J.A.Moulijn, W.P.M.van Swaay, Chei.
1, 1975. , 1958. .So.~,845, 1976.
3. De bewering dat de warmteoverdrachtscoefficient tussen
een gefluidizeerd bed en de bedwand niet afhangt van
de lengte van het verwarmend wandoppervlak is onjuist.
Wel wordt die afhankelijkheid minder merkbaar bij gra
te lengten.
N.I.Gelperin, V.G.Einstein in J.F.Davidson, D.Harrison, "Fluidization", Aoademio Press, Londen, 1971. J.H.N.Jaoobs, afstudeerrapport T.H.E., Januari 1976.
4. Indien in een homogeen gefluidizeerd bed enkele bellen
worden geinjekteerd met korte tussenpozen (die grater
mogen zijn dan de verblij van een bel in het bed)
ontstaat een voorkeurskanaal waardoor die bellen op
stijgen. Dit duidt op het bestaan van een mechanische
struktuur in de direkte fase.
J.F.J.Roes, afstudee T.H.E., Februari 1977.
5. Bij de stroming van poeders uit bunkers is de
uitstroomsnelheid mede afhankelijk van de viskositeit
van het aanwezige gas.
6. Bij een spherical cap bel opstijgend in vloeistof
zullen oppervlakte-aktieve stoffen zich koncentreren
op.het grensvlak tussen de bel en zijn zog, en niet
aan de onderzijde van het bolvormige gedeelte, zoals
Weber veronderstelt. Dientengevolge interpreteert
Weber zijn stofoverdrachtsmetingen aan spherical cap
bellen niet juist.
M.E.Weber, Chem.Eng.Sa.JO, 1507, 19?5.
7. In de door Davidson en Harrison gegeven berekening
van de grootte van de cloud rondom een bolvormige bel
in een gefluidizeerd bed is de aanname van potentiaal
stroming van de vaste stof onjuist en overbodig. Ook
zonder deze aanname kan hetzelfde resultaat gevonden
worden.
J.F.Davidson, D.Harriaon, "Fluidized particles", Cambridge University Press, 1962.
Dit proefsahrift, Appendix C.
8. Door de trage afhandeling van sollicitatie-procedures
bij veel bedrijven en instellingen is een sollicitant
genoodzaakt naar een groot aantal vakatures te dingen.
Dit leidt weer tot een extra belasting van personeelsafdelingen, en dientengevolge tot een nog tragere
afhandeling.
Eindhoven, 20 mei 1977 J.H.B.J. Hoebink