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Chemical Engineering Science 57 (2002) 50215038
www.elsevier.com/locate/ces
Experimental investigation and modelling of continuous uidized beddrying under steady-state and dynamic conditions
J. Burgschweiger1, E. Tsotsas
Lehrstuhl fur Thermische Verfahrenstechnik, Otto-von-Guericke-Universitat, Universitatsplatz 2, D-39106 Magdeburg, Germany
Received 26 March 2002; received in revised form 23 August 2002; accepted 29 August 2002
Abstract
In a lab-scale device, continuous uidized bed drying has been investigated experimentally under both steady-state and dynamic
conditions. The mixing behaviour and residence time distribution of particles in the dryer have been shown to be that of a continuous
stirred tank reactor. Particle mass ow rate and inlet moisture content, gas mass ow rate, air heater capacity and gas inlet temperature
have been varied systematically. The average moisture content of outlet solids has been determined by means of microwave absorption.
In the course of the work, close reference to a previous investigation of batch uidized bed drying has been kept by using an adapted
version of the same equipment and the same material (water-moist -Al2O3 with an average particle diameter of 1:8 mm). Furthermore,
the model previously developed and successfully validated for batch operation has been the starting point of the actual theoretical
analysis. This model has been extended in order to account for continuous and dynamic conditions. Additionally, population balances
have been introduced. In spite of the fact that no other adaptations have been undertaken, and though the extended model does not contain
adjustable parameters, a very satisfactory agreement between calculated and measured results could be achieved. In this way, it could
be demonstrated that it is possible to treat all dierent modi of uidized bed drying (batch, steady continuous, dynamic continuous) in
a unied, successful and applicable manner. Two aspects are considered essential for the good nal performance: The use of separately
determined, product-specic single-particle drying kinetics as a basis for every scale-up duty, and a stepwise methodology of model
development with detailed experimental validation of every individual step.
? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Fluidized bed; Drying; Continuous operation; Dynamic modelling; Population balances; Microwave absorption
1. Introduction and scope
Drying of solids in bubbling uidized beds is, because of
good performance, low investment and maintenance costs,
and robustness of the respective equipment, a very com-
mon industrial separation process. Many dierent types of
materials, from chemicals to foodstus and from plastics to
fertilizers, are treated in this way, usually with large through-puts in the continuous mode of operation. From many points
of view (the modern requirements for quick process devel-
opment and implementation, the high energy consumption
related, in general, to drying, see e.g. Bond (1980),the need
to preserve product quality, the fact that dryers are often
Corresponding author. Tel.: +49-391-67-18784;
fax: +49-391-67-11160.
E-mail address: evangelos.tsotsas@vst.uni-magdeburg.de
(E. Tsotsas).1Now with: Berliner Wasserbetriebe, Neue Judenstrasse 1, D-10179
Berlin, Germany.
promising and rewarding candidates for de-bottlenecking)
ecient and reliable modelling tools for uidized bed dry-
ers are desirable and required. However, their development
presupposes several steps of fundamental research and test-
ing, including:
(a) The use of a realistic basic model for the distribution,
ow and mixing of particles and the drying gas in thebubbling uidized bed, as well as for heat and mass
transfer phenomena between the phases.
(b) Combination of this model with drying kinetic data for
the single particle, gained by direct measurement or
from separateand separately validated!models for
intraparticle transport kinetics.
(c) Validation by means of experimental data from batch
uidized bed dryers, i.e. with data from a type of equip-
ment that is usual and widely available in laboratory
practice.
(d) Extendibility and applicability of the model to the con-
tinuous modus of operation without readjustment of
0009-2509/02/$ - see front matter? 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 0 9 - 2 5 0 9 ( 0 2 ) 0 0 4 2 4 - 4
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its assumptions, structure and parameters, however, in
clear recognition of the fact that residence time distri-
bution of the solids and population dynamics may, now,
be of importance.
(e) Validation by means of experimental data from contin-
uous uidized bed dryers.
(f) Applicability to dynamic conditions, in regard of taskslike start-up simulation and the training of operators,
and with the perspective of model-based automatic con-
trol.
(g) Again, validation with experimental data for dynamic
continuous operation.
Many diculties and shortcomings may undermine and
break the above chain of steps, including the use of over-
simplied uid bed models (which is typical for early
work, see e.g.Zabeschek (1977))and the almost complete
lack of systematic and well-documented data on continuous
steady-state and dynamic dryer operation in the open liter-
ature (seeZahed, Zhu, & Grace, 1995). In regard of such
diculties a very popular approach has been to bridge the
gap between batch and continuous operations by attempt-
ing to transform data from the former to predictions for
the latter. In this way,Vanacek, Picka, and Najmr (1964)
calculate the average solids moisture content at the outlet
of a continuous dryer by integration of the batch uid bed
drying curve under consideration of the residence time dis-
tribution of the solids. More sophisticated versions of this
model have been presented by Chadran, Rao, and Varma
(1990) and Kannan, Thomas, and Varma (1995), while
some empirical comparisons are provided by Kannan and
Subramanian (1998).Other authors concentrate their atten-tion not on the RTD behaviour, but rather on dierences in
operating conditions within the dryer between the batch and
the continuous modi. In this way, Reay and Allen (1982,
1983),McKenzie and Bahu (1990)as well asBahu (1994)
tend to identify the continuous modus by a constant bed
temperature and try to develop methods for the transfor-
mation of batch drying data to this kind of situation. The
operation of batch lab devices at a constant bed temperature
by control of the gas inlet temperature is also discussed by
these authors. Combinations of the two outlined aspects
(dierent residence times and dierent state variables, i.e.
temperatures and moisture contents, of the gas and thesolids) have been proposed byViswanathan (1986),Liedy
and Hilligardt (1991) andZahed et al. (1995).In spite of
considerable progresses in several particular aspects, most
of these contributions still overestimate the value and ap-
plicability of batch data for the continuous operation. To
give an example, Liedy and Hilligardt still use batch drying
data in order to derive a combined kinetic parameter for
both the gas and the particle-side mass transfer. In this way,
the inuences of adjustable process parameters and product
parameters are mixed up, which is the main disadvantage
of their approach from both the theoretical and the practi-
cal point of view. In this sense closer to the present work
appears to be the steady-state, two-phase model ofZahed
et al. (1995), who, however, do not provide any experi-
mental support for their results.
It becomes evident from the above discussion that the dis-
tinction between gas-side and particle-side phenomena by
treatment of the latter at the level of one single particle (as
already suggested and underlined by, e.g., Schlunder (1976),Tsotsas (1994)and Kerkhof (1994)) has to be an essential
element and feature of uidized bed drying modelling. It re-
moves almost any importance from the batch-to-continuous
transformation, focusing on more fundamental and more re-
warding problems, namely the scale-up from the single par-
ticle to the batch dryer, the scale-up from the single particle
to the continuous dryer (depending on the task and the scale
of production), or even and in some cases (seeGroenewold,
Groenewold, & Tsotsas (2000)) the scale-down from the
batch dryer to the single particle. If consequently applied,
it lets the uidized bed drying research of the past three to
four decades converge to the epigrammatically stated steps
(a)(g) of model development and validation. Therefore,
it has been decided to follow this line of attack, step by
step.
Doing so, we may and will refer to the previous suc-
cessful analysis of steps (a) (c) which is recapitulated
by Burgschweiger, Groenewold, Hirschmann, and Tsotsas
(1999a). It is not our purpose to modify this work, but to
extend it in order to address and include points (d)(g)
without any essential adaptation or readjustment. First, the
extended modelthen valid for continuous operation in the
steady as well as in the transient statewill be presented in
Section2, with reference to balances, population balances
and kinetics. Then, sorptive equilibrium, single-particlekinetics and other properties of the material used in the ex-
periments (-Al2O3) will be recapitulated in Section 3. In
Section4,a brief outline of the applied experimental facili-
ties and measuring techniques (including the determination
of outlet solids moisture content by means of microwave
absorption) will be given. Experimental data for uidized
bed drying under steady-state and dynamic conditions will
be presented, discussed and compared with the predictions
of the model in Section 5, including a brief assessment of
the impact of population dynamics. Conclusions are sum-
marized in Section6, along with a short outlook. In order
not to disturb the main text, a number of model parametersare treated in the appendix, or, in some cases, by reference
toBurgschweiger (2000).
While the presentation of a comprehensive, consistent,
systematically validated and industrially applicable treat-
ment of dierent operation modi of uidized bed dryers
is the purpose of the present paper, other important top-
ics are not within its scope. To this category belong: (i)
the abundant literature on details of ow mechanics in
udized beds, including CFD approaches from the last
years; (ii) investigations on uidized bed reactors and on
processes like agglomeration, granulation and coating; (iii)
product quality aspects and opportunities arising from the
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Fig. 1. Scheme of the uid bed model.
implementation of population dynamics; (iv) a detailed
study of parametric sensitivity of the present model; and (v)
the transition from dynamic modelling to automatic control.
While some of these topics will not be addressed at all, hints,
short remarks and references to actual work, which is going
to be communicated separately, will be given about some
others.
In regard of the model and experimental data that will
be presented in the following, the main restrictions concern:
(i) the assumption and realization of well-mixed conditions
of the solids; and (ii) the use of one-parametric population
dynamics. The rst can be in practice overcome to a certainextent by simulating lengthy uidized bed devices by a se-
ries of continuous stirred tank reactor (CSTRs). Otherwise,
specialized models considering axial dispersion of the solids
during their movement along the dryer would be necessary,
which are not within the scope of the present investigation.
Due to the second restriction, the present population mod-
elling can be only approximate when applied to cascades,
either ctitious or real (multistage CST-dryers). A more ac-
curate consideration of such cases would require a multidi-
mensional denition of populations and is, again, outside of
our present scope.
2. The model
By means of schematic representation, the main features,
notation and assumptions of the model for continuous u-
idized bed drying are depicted in Fig.1.This graph is very
similar to the scheme given byBurgschweiger et al. (1999a)
for the case of batch drying, underlining the continuity and
consistency between that and the present approach. Actually,
the graphs are identical, up to the mass ow rates for incom-
ing and outcoming solids, Mp; in and Mp; out, in the present
plot. Important assumptions and features of the model are
still the following:
Distinction between a particle-free bubble phase and a
suspension phase.
Plug ow in the bubble phase.
Perfect backmixing of the particles in the suspension.
Plug ow of the gas in the suspension phase. Notice,however, that backmixing nds implicit consideration in
the kinetics of mass and heat transfer between particles
and suspension gas, according toGroenewold and Tsotsas
(1997).
Consideration of mass and heat transfer between suspen-
sion gas and bubbles.
Heat transfer between the wall, which is assumed to be
isothermal, and the environment, the particles, the sus-
pension gas and the bubble gas.
Consideration of product-specic, particle-side drying
kinetics at the level of a single particle and by means of
normalization, i.e. by a characteristic drying curve, ().
Concerning perfect backmixing of the particles in the sus-
pension, it should be noticed that it now comprises both the
vertical (height) and the horizontal (conveying) direction.
Perfect backmixing in the vertical direction is an assump-
tion, subject to the overall validation of the model. Back-
mixing in the conveying direction of the solids is, as already
stated in the Introduction, a restriction that can be realized
by respective construction of the apparatus, or not. It has to
and will be validated separately (see SectionA.3). This re-
striction is simplifying in the sense of consideration of only
onethe verticalspatial coordinate.
The equations corresponding to the scheme of Fig. 1and expressing the model are given in the following
subsections.
2.1. Balance equations for the gas
Mass (referring to the evaporating component) and energy
balances are
(1 ) Mg@Ys
@ =
@
@( Mps Msb); (1)
Mg@Yb
@
=@ Msb
@
; (2)
(1 ) Mg@hs
@ =
@
@( Hps Qsp+ Qbs Hsb Qsw); (3)
Mg@hb
@ =
@
@( Hsb Qbs Qbw) (4)
for the suspension and the bubble gas, respectively, with the
enthalpies dened as
hs=cgTs+Ys(cw;g Ts+ hv); (5)
hb=cgTb+Yb(cw;g Ts+ hv): (6)
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for heat transfer between the suspension, respectively, the
bubble gas, and the wall,
@Qpw
@ =n()pwAw[Tp Tw] (25)
for heat transfer between the particles and the wall, and
@2 Qpp
@@=n()n()pwAps[Tp(
) Tp()] (26)
for heat transfer between particles belonging to dierent
populations. Enthalpy ow rates referring to the evaporating
component are calculated according to
@2 Hps
@@ = [cw;g (Tp)Tp+ hv(Tp)]
@2 Mps
@@ ; (27)
@ Hsb
@ = [cw;g (Tsb)Tsb+ hv(Tsb)]
@ Msb
@ ; (28)
with, approximately,Tsb= (Ts+ Tb)=2.
The heat ow rate from the apparatus wall to the envi-
ronment is
Qwe=kweAwe[Tw Te]: (29)
Eq. (26) has to be integrated over all populations
@Qpp
@ =
t=0
n()pwn()Aps[T(
) T()] d; (30)
in order to calculate the resulting heat ow rate to the popu-
lation with residence time , needed in Eq. (14). Assuming
a constant heat transfer coecient for all populations, the
somewhat simplied expression
@Qpp
@ =n()pwAps[Tp Tp()] (31)
is obtained.
2.5. Coecients and remarks
To apply the model, various informations must be avail-
able and parameters known, specically:
Fluidization parameters, like the expanded bed porosity,
, and the ratio of bubble to total gas ow rate,.
An equation allowing to calculate the mass ow rate of
solids at the bed outlet, Mp; out.
The residence time distribution of solids in the device.
Product-specic material properties, above all the normal-
ized single-particle drying curve, (), and sorption equi-
libria, necessary for the calculation ofYeq in Eq. (19).
The coecients of mass and heat transfer between particle
surface and suspension gas, ps, resp.,ps.
The coecients of mass and heat transfer between sus-
pension gas and bubbles,sb, resp.,sb.
Coecients of heat transfer between gas and wall, parti-
cles and wall, particles and particles, wall and the envi-
ronment.
In some cases, models for items of the periphery of the
dryer.
Fluidization parameters, the outow equation, the RTD ofsolids, particle-to-suspension-gas transfer coecients and
suspension-to-bubble-gas transfer coecients are specied
in Sections A.1A.5. The remaining, less important, coef-
cients of heat transfer are discussed in Section A.6. And,
product-specic properties are given in Section3.Concern-
ing the periphery of the dryer, models for the air heater and
the duct connecting this device with the dryer (see Section
4) may be necessary for some types of simulation. This is the
case in dynamic simulations involving changes of heaters
capacity, Q. (Notice that the capacity of the heater is a typ-
ical manipulated variable in automatic control of convec-
tive dryers.) Such models are available and can be found in
Burgschweiger (2000),which will, however, not be detailedhere.
In general, the following features of the model are impor-
tant and should be stressed:
The model is capable of treating both continuous
steady-state and dynamic operations.
Intrinsic dynamics are considered by population balances.
Neglecting of the latter would simplify the model consid-
erably. Its impact on the result will be discussed later on
in some detail.
In the case of batch operation, the model is reduced to
exactly that form, which has been presented and validatedbyBurgschweiger et al. (1999a).
Both for the batch and continuous operations, scale-up
from the single particle to the uidized bed is essential
and characteristic for the model. Single-particle drying
kinetics and hygroscopic equilibria have to be determined
by separate experiment.
The model is free of adjustable parameters and, thus, pre-
dictive at the level of the uidized bed dryer.
Checking the predictive performance of the model by com-
parison with experimental results has been the major task
of the work to be presented. In contrary, a detailed analysis
of parametric sensitivity is outside the scope of the presentcommunication, so that only a few hints and short remarks
will be given in this respect.
2.6. Mathematical resolution
For the numerical solution of the system of model equa-
tions the following strategy has been applied:
Dimension reduction of the population balance equations
with the method of characteristics (seeRamkrishna, 1985,
Rhee, Aris, & Amundson, 1982).
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Table 1
Sampling points for the stepwise linear approximation of sorptive equilibrium
(dimensionless) 0.000 0.050 0.100 0.650 0.750 0.805 0.930 1.000Xeq (dimensionless) 0.000 0.027 0.040 0.090 0.120 0.200 0.670 0.800
Discretization with respect to the residence time coordi-
nate,.
Discretization with respect to the height coordinate,, and
integration by a fourth-order RungeKutta-algorithm.
Integration with respect to the time coordinate, t, by an
explicit Euler method.
Details of the numerical solution can be found in
Burgschweiger (2000). Convergence and stability, as well
as acceleration of the solution by the use of dierent step
sizes in time and residence time or by coagulation of spe-
cic populations, are discussed in the same work. A Mat-
lab/Simulink environment has been used for combining thedryer model with the models for peripheral devices (heater,
duct).
Macroscopically signicant and/or observable quantities
like the moisture content, the mean caloric temperature, and
the mean temperature of outlet air (Yout; Tg; out, resp., Tg; out),
as well as the mean solids moisture content and temperature
(X, resp., Tp), are derived from the results of the numerical
solution by simple mixing rules between bubble and sus-
pension gas, or simple integrations over the various popula-
tions. A mean caloric particle temperature, Tp; cal, can also
be derived, see Burgschweiger (2000). Due to the CSTR
behaviour of the device with respect to solids RTD (Sec-tionA.3), the mean outlet moisture content of the particles,Xout, is equal to X.
3. Materials and properties
For all experiments alumina (-Al2O3) spheres have been
used. Their mean diameter ofdp= 1:8 mm has been mea-
sured by image analysis. The particle size distribution was
very narrow, with a standard deviation of less than 0:02 mm,
justifying the treatment of the bed as a monodispersed
packing. From measured bulk densities, the density of dryparticles has been determined to p = 1040 kg=m3. The
bed porosity at minimal uidization has been mf = 0:40.
The heat capacity of the dry material has been measured by
dierential scanning calorimetry as a function of tempera-
ture, leading to values of, e.g., cp= 944 J=(kg K) at 40C.
According to the Geldart classication, the solids be-
long to group D. It is exactly the same solids as used by
Burgschweiger et al. (1999a)for batch uidized bed drying
experiments. Water and air have been the moisture and the
drying agent, respectively.
Alumina of the -type is highly hygroscopic to water.
Desorption equilibrium data have been determined exper-
imentally by Burgschweiger et al. (1999a) and are taken
over without change for the calculations of the present work.
Specically, the value pairs of relative air humidity, , and
solids equilibrium moisture content, Xeq, according to Ta-
ble1have been used in combination with linear approxima-
tions for the intervals in-between. Checks at temperatures
between 25C and 50C justify the use of the same isotherm
in the simulation of all experimental conditions. Mod-
elling in terms of a combination of Langmuir-adsorption
and capillary condensation provides additional support for
this simplied consideration (Burgschweiger, & Tsotsas,
2000). Furthermore, it delivers the values of adsorption
enthalpy as a function of moisture content, hs(X), whichare necessary for implementing Eq. (15). According to
Burgschweiger (2000), the maximal, monolayer value of
adsorption enthalpy is hs; mo = 497 kJ=kg, still consider-
ably smaller than the evaporation enthalpy of free water.
From the sorption isotherm and the relationship
Yeq=MwMg
(X; Tp)psat(Tp)
P(X; Tp)psat(Tp); (32)
the value ofYeq which is necessary in order to calculate the
evaporation ow rate from Eq. (19) is obtained. This type
of approach is reasonable for highly hygroscopic materials.
It separates the inuence of hygroscopic vapour pressure
reduction from the impact of intraparticle mass and heat
transport kinetics, so that only the latter is considered in
the normalized single-particle drying curve (). For ()
the function plotted in Fig. 2 has been used throughout
the present work, which again, has been taken over from
Burgschweiger et al. (1999a),without any change or adap-
tation. According to the common denition, the normalized
solids moisture content is
= X Xeq
XcrXeq(33)
and the critical moisture content has been determined to be
Xcr= 0:20. All theoretical implications, including the in-
troduction of a rst drying period with slightly falling dry-
ing rate and the atypical denition of Xcr, as well as the
derivation of () from experimental data have been dis-
cussed thoroughly by Burgschweiger et al. (1999a). Here,
it should only be stressed that the choice of the model
for single-particle drying kinetics is not restrictive with re-
spect to the overall modelling of the dryer. Other than the
present description, including classical normalization after
van Meel (1958) and diusion models (for a review see,
e.g.,Tsotsas, 1992)could also have been used. In this sense,
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1
0.8
0.6
0.4
0.2
00 0.2 0.4 0.6 0.8 1
Approximation
Measurement
Fig. 2. Normalized single-particle drying curve of -alumina (spherical,
dp = 1:8 mm) after Burgschweiger et al. (1999a), derived from mea-
surements on a microbalance (Xcr
= 0:20). The solid line is used in the
calculations.
particle-side kinetics constitutes only one module of the sim-
ulation, which is, in principle and after respective experi-
mental validation, product-specic and exchangeable. What
must in any case be strictly guaranteed, is the reference of
() to the single particle, by exclusion of any integral ki-
netic inuence. From this point of view, the methods for
modelling of dryers and chemical reactors are fundamen-
tally the same, since reactor analysis should also be based
on dierential, gradient-free chemical kinetics.
4. Experimental
A owsheet of the experimental set-up is depicted
in Fig. 3. Its core is a uidized bed with circular
cross-section of 0:15 m inner diameter (cross sectional area:
Fbed = 174:2 cm2). Glass with 5 mm thickness was the
wall material (heat capacity of the wall: Cw = 2902 J=K).
For the purpose of visual observation, we have refrained
from insulating the uidization section (see also Section
A.6, wall surface area in contact with the environment:
Awe = 1759 cm
2
). The distributor was a 3 mm thick sin-tered plate (porosity = 0:34) made of brass particles with a
diameter of about 0:6 mm, as pressure drop measurements
have conrmed. Feed solids could be taken from one of
two small bins (B1, B2). A controllable rotating cell wheel
(H1) with linear characteristics was used in order to adjust
the mass ow rate, Mp; in, of feed material, which was left
to fall into the uidized bed after the wheel. For the output,
a centrally placed downcomer tube (inner diameter: 15 mm,
cross-sectional area: Fweir = 1:77 cm2, outer diameter:
18 mm; Lweir= 0 :2 m, see also Fig. 14, SectionA.2)was
used. After it, a second rotating cell device (H2), the equip-
ment for determination of the average moisture content of
outlet solids, Xout, and another bin (B3) are placed. The
drying agent (air) is conveyed by a frequency-controlled
radial channel compressor (V) and treated by a controllable
electrical heater (W) before entering the uidized bed. A
two-stage (condensation and adsorption) air dehumidier
is available, but has not been used in the majority of the
present experiments.The instrumentation provides various measurements of
temperature, pressure, pressure dierence, gas ow rate and
inlet gas moisture content. However, most important is the
measurement of outlet gas moisture content,Yout, and outlet
solids moisture content, Xout. The former is accomplished
on-line by infrared spectroscopy, a technique that has been
elaborated in detail and successfully applied in the past
(e.g.Burgschweiger et al., 1999a). For the latter, a tubular
microwave absorption device has been used. Operation is
semi-batch, involving a zero-level determination, lling-up
of the device, the actual measurement of microwave at-
tenuation and phase-shift, and emptying of the resonance
chamber. For the realized, relatively low solids through-
puts, in-line placement and operation have been possible
(not samples, but all outlet solids go through the microwave
equipment, MIC 111 in Fig.3). With a measuring circle du-
ration of 32 s, even quick transients could be followed with
a high temporal resolution. More information about the mi-
crowave technique, the calibration of the device, and the suc-
cessful control of the overall moisture balance can be found
in the work ofBurgschweiger (2000),along with some gen-
eral discussion of possibilities for the determination of solids
moisture content. It should be stressed that the microwave
method has performed very satisfactorily for the relatively
high levels of solids outlet moisture realized in the presentwork (see next section), but is not adequate for specica-
tion control of almost bone-dried industrial products (since
most materials contain a number of atoms interfering with
microwaves, the measuring signal of water inevitably dis-
appears into noise at some low moisture level). However,
the technique may be applicable and rewarding for the de-
termination of inlet or intermediate solids moisture contents
in multistage industrial dryers.
For the experiments, the material has been wetted with
demineralized water and, typically, treated in a centrifuge
before use. The moisture content of inlet solids has been
determined o-line, by gravimetry. The condition of drysolids has been dened by residence of 24 h at 130C in a
drying oven.
5. Results and discussion
5.1. Continuous uidized bed drying under steady-state
conditions
A total of 53 drying experiments have been conducted
under steady-state conditions by systematic variation of the
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granulate
B1
M I
T I
air
H2
B3
MIC
H1
109
102
111
air
EL.
V
W
T I C101
Fluidized bed
dryer
air
B2
MI
110
DN 80/DN 65
DN 80/DN 50
DN 80/DN 40DN 80/DN 40
DN 40/DN 20 DN 40/DN 20
DN 80/DN 65
FI
104
FI
103
P I D
107
P I D
106
105
P I D
PI
108
TI
100
DN 80/DN 50
air dryer
granulate 1 granulate 2
Fig. 3. Flowsheet of the experimental set-up.
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parameters:
Particle mass ow rate (dry-based), Mp, between 0:48
and 1:69 g=s;
Particle inlet moisture content, Xin, between 0.436 and
0:690 kg H2O=kg dry solids;
gas mass ow rate (dry-based), Mg, between 19 and58 g=s;
air heater capacity, Q, between 800 and 4356 W; and
inlet gas temperature, Tg; in, between 58:0 and 150:0C.
Notice that the last mentioned parameters, the air heater
capacity, Q, and the inlet gas temperature, Tg; in, (here, the
inlet of the uidized bed dryer is meant), are not independent
from each other. For steady-state operation of the heater
and negligible heat losses, they are coupled by the simple
relationship
Q=cg Mg(Tg; in Tg; e): (34)
In Eq. (34),Tg;e is the air temperature at the entrance of the
heater, which has for all experiments approximately been
Tg;e = 30C; (due to some heating of the air in the radial
channel compressor, see Fig. 3). With a specic heat ca-
pacity for the air ofcg = 1008:3 J=(kgK), and at constant
gas mass ow rate, Mg, the heater capacity, Q, and the gas
inlet temperature, Tg; in, can immediately be transformed to
each other. The reason for explicitly treating the air heater
capacity and using it in several plots is the perspective of
identication and development of automatic control strate-
gies, which has been one motivation for the presently re-
ported work. From this point of view, the directly adjustable
heater capacity is a more interesting variable than the gasinlet temperature.
The gas inlet moisture content,Yin, varied between 0.0027
and 0:0102 kg H2O=kg dry air in the experiments, while the
temperature of inlet particles,Tp; in, has always been ambient
(approximately 20C). These parameters are of rather minor
importance and will not be discussed further. All experimen-
tal results are available in tabulated form in Burgschweiger
(2000).Here, the most interesting trends in the inuence of
the main operating parameters on the average moisture con-
tent of outlet solids, Xout, as well as on the average caloric
outlet gas temperature, Tg; out, will be shown in Figs. 48.
In the same plots, experimental data will be compared withthe results of model calculations.
As expected and shown in Fig. 4, increasing heater ca-
pacities, Q, (which at a constant gas mass ow rate, Mg,
are equivalent to increasing gas inlet temperatures), lead to
decreasing moisture contents of outlet solids, Xout. In the
same time (lower plot of Fig. 4), an increase in the temper-
ature of outlet gas, Tg; out, is observed. Both the measured
values (open symbols) and the calculated results which are
depicted with the solid lines correspond to gas mass ow
rates of Mg= 39 g=s and show a very good agreement with
each other. In the same diagrams, calculations with another,
higher gas mass ow rate ( Mg = 56 g=s) are also plotted
500 1000 1500 2000 2500 3000 3500 4000
simulation:experiment:simulation:
90
80
70
60
40
30
20
50T
g,out
[C]
sgMg /39=
sgMg /39=
sgMg /39=
s/g56Mg =
s/g56Mg =simulation:simulation:experiment:
Xout
[-]
0.35
0.3
0.25
0.2
0.15
0.10.05
0500 1000 1500 2000 2500 3000 3500 4000
s/g39Mg =
]W[Q
]W[Q
.
.
.
.
.
.
.
.
Fig. 4. Inuence of heater capacity, Q, on steady-state, average moisture
content of outlet solids, Xout, and gas outlet temperature, Tg;out ( Mg= 39
resp. 56 g=s; Mp= 1:21 g=s; Xin= 0:61; Yin = 0:009).
simulation: = 1000 Wsimulation: = 2500 W
experiment: = 2500 W
Xout
[-]
0.4
0.35
0.25
0.2
0.15
0.1
0.05
030 35 40 45 50 55
0.3
30 35 40 45 50 55
80
60
50
40
30
20
70
Tg,out[C]
simulation: = 2500 W
experiment: = 2500 W
simulation: = 1000 W
Q.
Q.
Q.
Mg [g/s].
Mg [g/s].
Q.Q.
Q.
Fig. 5. Inuence of gas mass ow rate, Mg, on steady-state, aver-
age moisture content of outlet solids, Xout, and gas outlet temperature,
Tg;out (Q =1000 resp. 2500 W; Mp= 1:21 g=s; Xin= 0:60; Yin= 0:007).
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0.45
0.4
0.35
0.3
0.25
0.2
0.1
0.05500 1000 1500 2000 2500 3000 3500 4000
simulation: 0.88 g/s
experiment: 0.88 g/s
simulation: 1.94 g/s
=pM
=pM
=pM
0.15
500 1000 1500 2000 2500 3000 3500 4000
80
70
60
40
30
20
50
simulation: 0.88 g/s
experiment: 0.88 g/s
simulation: 1.94 g/s
=pM
=pM
[ ]WQ
=pM
[ ]WQ
Xout
[-]
Tg,out[C]
.
..
.
.
.
.
.
Fig. 6. Inuence of heater capacity, Q, on steady-state, average moisture
content of outlet solids, Xout, and gas outlet temperature,Tg;out ( Mp =0:88
resp. 1:94 g=s; Mg= 57 g=s; Xin= 0:67; Yin= 0:005).
Fig. 7. Inuence of solids mass ow rate, Mp, on steady-state, av-
erage moisture content of outlet solids, Xout, and gas outlet tempera-
ture, Tg; out (Q = 1815 resp. 4356 W, corresponding to Tg; in = 80 resp.
150C; Mg= 36 g=s; Xin= 0:62; Yin= 0:004).
0.14
0.1
0.12
0.08
0.06
0.040.02
00.1 0.2 0.3 0.4 0.5 0.6 0.7
simul.: Tg,in= 80 Cexper.: Tg,in= 80 Csimul.: Tg,in= 150 Cexper.: Tg,in= 150 C
140
130
70
60
40
120
50
0.1 0.2 0.3 0.4 0.5 0.6 0.7
8090
100
110 simul.: Tg,in= 80 Cexper.: Tg,in= 80 C
simul.: Tg,in= 150 Cexper.: Tg,in= 150 C
Xin [-]
Xin [-]
Xout
[-]
Tg,out[C]
Fig. 8. Inuence of inlet solids moisture content, Xin, on steady-state,
average moisture content of outlet solids, Xout , and gas outlet temperature,
Tg;out (Tg; in =80 resp. 150C; Mp = 0:97 g=s; Mg =41 g=s; Yin = 0:002).
(broken lines). The change of gas mass ow rate has, ob-
viously, only a small inuence on the results of the simula-
tion. On the one hand, the capacity of the gas stream to take
over vapour is increased by an increase of the gas ow rate.On the other hand, the gas inlet temperature is decreased
at a constant heater capacity (see Eq. (34)). In the same
time, changes in the uidization parameters, the expanded
bed height, the holdup and the average residence time of
the solids, as well as in the gas-side kinetic coecients take
place. Competitive trends are mutually neutralized, so that
a slight increase of dryer capacity, a moderate decrease, or
no change at all, can be the outcome of a variation of gas
mass ow rate.
The same behaviour is illustrated in Fig.5,where the gas
mass ow rate is plotted on the abscissa of the diagrams, and
the heater capacity is the parameter. Both, the calculations(solid lines) and the experimental data (open symbols) show
a very moderate decrease of dryer capacity (that means an
increase of Xout) at increasing Mg for the heater capacity
ofQ= 2500 W. To the contrary, a slight increase of dryer
capacity (a decrease of Xout) with increasing Mgis observed
at Q = 1000 W (only simulation, broken lines). Somewhere
in between, the result would be completely indierent upon
a change of the gas mass ow rate, roughly corresponding
to the intersection point of the broken and solid lines of
Fig.4.
The inuence of heater capacity or (Fig. 7) gas inlet
temperature, and of solids mass ow rate on the outlet
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0.35
0.25
0.3
0.2
0.15
0.1
0.05
00.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
experiment: Tg,in=
population model:
without popul. dyn:80 Cexperiment: Tg,in= 150 Cpopulation model: 150 Cwithout popul. dyn: 50 C
80 C80 C
[ ]s/gMp.
Xout
[-]
Fig. 9. Calculations with and without consideration of population dynamics
in comparison with data from Fig. 7, upper plot.
conditions of the dryer is illustrated in Figs. 6 and 7. By an
increase of product throughput, Mp, decreasing gas outlet
temperatures are obtained (Fig. 7, lower plot), which in-
dicates a better use of the drying agent. In the same time,however, higher outlet solids moisture contents occur. It is
interesting to observe that the calculated dotted curve in
the upper plot of Fig. 7 has an inection point at aboutMp= 1 :2 g=s.
Finally, the inuence of solids inlet moisture content is
demonstrated in Fig. 8. Again, the gas inlet temperature
is indicated as the parameter, instead of the heater capac-
ity. As expected, changes in the feed are clearly, though
not direct proportionally, observable in the product of
the dryer. The sensitivity upon variations of Xin is higher
at low gas temperatures, leading to relatively steep and
non-linear Xout(Xin) curves (upper plot). In contrary, an al-most linear dependence ofTg; outonXin is observed for both
values of Tg; in (lower plot). Notice that the inlet solids
moisture contents of about 0.44 in Fig. 8 have been real-
ized by pre-drying the material, which is otherwise used
immediately after wetting and centrifugation (compare with
Section4).
Parity plots of measured and calculated values of XoutandTg; out for all continuous drying experiments, as well as
comparisons in tabular form are given by Burgschweiger
(2000), and will not be repeated here. They document a
very good predicting performance of the model. As al-
ready pointed out, this performance has been attained
without any kind of tting or manipulation of adjustable
parameters.
5.2. Impact of population dynamics
In Fig.9,the same experimental data as in the upper plot
of Fig.7 are presented. However, two types of calculations
are now plotted: calculations with population balances, as
in Fig.7,and calculations based on average values. As the
diagram shows, and due to various non-linearities of the
process, the result of working with distributed variables and
then averaging is not the same as the result of calculating
with averages from the very beginning. Specically, the ca-
pacity of the dryer is overestimated, when population dy-
namics are neglected. In the latter case, inhibitions due to
single-particle drying kinetics and sorptive equilibrium are
more directly discernible, so that, e.g., the large slope of the
curve forTg; in= 80C ( Q =1815 W) at about Mp = 1:4 g=s
correlates with a high gradient of the sorption isotherm inthe region of 0:1 X 0:6, in combination with the crit-
ical moisture content at Xcr= 0:2 (Section3). However, it
should be borne in mind that the deviation between calcula-
tions with and without population dynamics is of complex
nature and parametric behaviour, so that the present results
may not be generalized.
As to the absolute value of the eect, it certainly appears
to be signicant according to Fig. 9.However, the error is
considerably smaller in relative terms, that means in regard
of the dierences between the high inlet moisture content
ofXin= 0:62 and the calculated, rather low values of Xout.
Furthermore, it must not be overseen that the investigated
dryer has the broadest possible RTD of solids, namely that
of a CSTR (see SectionA.3). The neglecting of population
dynamics will be less critical for types of lengthy dryers,
which are used in industry in order to approximate plug ow
behaviour of the solids (compare with the discussion in Sec-
tions1 and2). On the other hand, dryer design and scaling,
i.e. accuracy in the calculation of the average value Xout,
is not the only aspect related to the use, or not, of popula-
tion balances. The other, at least equally important aspect,
is product quality, which is very clearly inuenced not only
by the average value of outlet solids moisture content, but
also by the distribution of the same. From this point of view,
population balances are a presupposition of advanced qual-ity assessment. The same is true for the understanding of
particle formation processes, like uidized bed granulation
or coating.
Some preliminary hints about the quality aspect can be
found in literature, e.g. in the form of an example for the
distribution of outlet solids moisture content by Tsotsas
(1999). Burgschweiger (2000) calculates for specic op-
erating conditions, the time dependence of solids moisture
content and temperature in a batch dryer, and compares the
results with the moisture contents and temperatures of parti-
cle populations which have experienced the same residence
time in the continuous modus of operation. It should, how-ever, be very clear that such investigations are, still, intro-
ductory, and that the aspect of product quality will have
to be thoroughly studied in future work. The modelling
tools which are necessary for this type of analysis are now
available.
5.3. Continuous uidized bed drying under dynamic
conditions
Concerning the dynamic operation, start-up as well
as step-response experiments and simulations have been
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-1500 -1000 -500 0 500 1000 1500
inlet:
outlet, simulation:
outlet, experiment:
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
t [s]
X[-]
Fig. 10. Dynamic response of the average moisture content of out-
let solids, Xout, to a step change of inlet particle moisture content,
Xin ( Mp= 0:94 g=s; Mg= 41 g=s; Tg; in= 80C; Yin= 0:003).
conducted. Here, and with reference to Burgschweiger
(2000),we refrain from discussing dierent types of start-up
behaviour and concentrate on the transients initiated by
stepwise changes of operating parameters of the dryer.
Specically, responses to step changes of the inlet solids
moisture content, Xin, the heater capacity, Q, the particle
mass ow rate, Mp, and the gas mass ow rate, Mg, are
presented in Figs.1013,respectively.
For a new steady state to be reached at the outlet of the
dryer after a sudden change of inlet moisture content, Xin,
(which is realized by switching from one to the other small
hopper of Fig. 3), about 500 s are necessary, as shown in
Fig. 10. Duration and course of the transient are properly
predicted by the model. For the new steady state, what has
been said in Section5.1holds. It is, in this context, interest-ing to observe that the outlet step, i.e. the dierence between
new and old steady state, is considerably dumped in com-
parison to the inlet onean expression of non-linearity of
the system. The same behaviour can be observed in Fig. 8,
considering the dierent scales of the abscissa and the ordi-
nate of the plot.
With a duration of some thousands of seconds consider-
ably slower is the response to a number of stepwise changes
of the capacity of the heater, Q. In the respective plots of
Fig.11 not only the change of average outlet moisture con-
tent of the solids, Xout, but also of the gas inlet temperature,
Tg; in, and the gas outlet temperature, Tg; out, are plotted. Asthe diagrams show, the gas inlet temperature reacts faster
than the state variables at the gas or solids outlet of the dryer.
The response behaviour is obviously inuenced by the rel-
atively large thermal inertia of, on the one hand, the heater
and the duct to the dryer, and, on the other hand, the dryer
(holdup and wall) itself. Only the former is of importance
forTg; in, while the former and the latter have an impact on
Tg; out and Xout.
Two stepwise increases of inlet solids mass ow rate, Mp,
have been realized in the experiments of Fig. 12. It is in-
teresting to observe that both the average moisture content
of outlet solids (middle plot) and the average caloric out-
00 5000 10000 15000 20000 25000
1
23
4
5
6
7
8
9
Q[kW]
t [s]
Xout
[-]
t [s]
0 5000 10000 15000 20000 25000
experiment
simulation
0.6
0.5
0.4
0.3
0.2
0.1
0
0 5000 10000 15000 20000 25000
160
140
120
100
80
60
40
20
0
experiment: Tg, insimulation: Tg, in
Tg,outTg,out
Tg
[C]
t [s]
Fig. 11. System response (average moisture content of out-
let solids, Xout, inlet and outlet gas temperature, Tg; in resp.
Tg;out) to a series of step changes of air heaters capacity,
Q ( Mp= 1:21 g=s; Mg= 39 g=s; Xin = 0:61; Yin= 0:009).
let gas temperature (lower plot) react to these changes by,
rst, a rather quick response covering about 90% of the in-
terval between the new and the old steady states in a timecomparable to that also observed in Fig. 10. However, the
remaining parts of state change are much slower, which cor-
relates with the relatively long time necessary in order to
reach the next operating point of constant mass ow rate of
solids at the outlet of the dryer, Mp; out. The latter can be
seen in the upper plot of Fig.12.Here, Mp; in = Mp; outis the
condition for steady state, while every dierence betweenMp; in and Mp; out leads to a transient change of holdup. ForMp; in Mp; out the dryer is lled-up, until a new stable op-
erating point with a higher holdup has been attained. The
additional amount of solids corresponds to the area between
the broken and the solid curves in Fig. 12.
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t [s]
120
100
60
80
40
20
Tg,out
[C]
2000 3000 4000 5000 6000 7000
Xout
[-]
0.03
0.04
0.05
0.06
0.07
0.08
0.02
t [s]
2000 3000 4000 5000 6000 7000
t [s]2000 3000 4000 5000 6000 7000
1.8
1.6
1.4
1.2
1.0
0.8
0.6
. Mp,in
experiment:
calculation: Mp,out
.
.
Mp
[g/s]
Mp,out
Fig. 12. System response (mass ow rate of outlet solids, Mp;out, average
moisture content of outlet solids, Xout, gas outlet temperature, Tg; out) to
step changes of the mass ow rate of particles at the inlet of the dryer,Mp; in ( Mg= 35 g=s; Tg; in= 150
C; Xin= 0:62; Yin= 0:003).
From what we have learned about the very moderate in-
uence of gas ow rate in Section5.1,it may be anticipated
that stepwise changes of Mg will not produce anythingmore than small dynamic disturbances and uctuations
around a coarsely constant steady state. This expectation is
veried by the data of Fig. 13,specically by the behaviour
of Xout and Tg; out. Notice (lower plot) that the outlet gas
temperature remains approximately constant in spite of the
considerable changes in the respective inlet value, due to the
already discussed combined action of competitive eects.
As previously pointed out, the dynamic models for the air
heater and the duct to the dryer, that have not been presented
here, but can be extracted from the work ofBurgschweiger
(2000), are necessary in order to calculate transients
ofTg; in.
Fig. 13. System response (average moisture content of out-
let solids, Xout, gas inlet temperature, Tg; in, gas outlet tem-
perature, Tg;out), to step changes of gas mass ow rate,Mg ( Mp= 1:21 g=s; Q= 2500 W; Xin = 0:61; Yin = 0:007, notice that
the calculated curves can hardly be seen in the lowest plot, because of
overlapping with the experimental data).
It should be clear that even relatively small deviations be-tween measurement and calculation in the steady state be-
come, by the kind of the plots, quite obvious in Figs. 1013.
In the same time, these gures, taken together with the men-
tioned start-up results ofBurgschweiger (2000),document
a very satisfactory predictive performance of the model un-
der dynamic conditions. Since no adaptation of any kind has
been undertaken, the requirements formulated in the intro-
duction are fullled. Furthermore, the possibility is opened
to design, train and optimize automatic control algorithms
by application of the present model. Although some work
has already been done in this direction (Burgschweiger, Wu,
Tsotsas, & Doschner (1999b)), an in-depth treatment is
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outside of the scope of the present publication and will have
to be separately attempted.
6. Conclusions and outlook
Starting point of the present work has been recent inves-tigations (Burgschweiger et al., 1999a) that have success-
fully treated the problem of scaling-up from single particle
to batch uidized bed drying kinetics. In the present pa-
per the same material (-Al2O3, water-moist) was used and
the laboratory uidized bed apparatus was adapted in order
to enable continuous operation. In this way, a considerable
amount of experimental results on continuous uidized bed
drying under both steady-state and dynamic conditions has
been gained, and could be presented and discussed. In the
experimental part, infrared spectroscopy for the detection of
water vapour in the exhaust air has been accompanied by
microwave determination of the average moisture content
of outlet solids. The mixing behaviour and RTD of particles
in the dryer have been shown to be that of a CSTR. Parti-
cle mass ow rate and inlet moisture content, gas mass ow
rate, air heater capacity and gas inlet temperature have been
varied systematically.
Furthermore, the previous model has been extended in
order to account for continuous operation and enable dy-
namic simulations. Population balances are included in the
extended model, and their inuence in comparison to simpli-
ed, average-based approaches has been quantied. In spite
of the fact that no adaptations have been undertaken with
respect to the mentioned previous work byBurgschweiger
et al. (1999a), and though the model does not contain ad-justable parameters, a very satisfactory agreement between
calculated and measured results could be achieved. In this
way, it could be demonstrated that it is possible to treat all
dierent modi of uidized bed drying (batch, steady con-
tinuous, dynamic continuous) in a unied, successful and
applicable manner. Separately determined, product-specic
single-particle drying kinetics as a basis for every scale-up
duty, and a stepwise methodology of model development
and validation are considered to have been essential for the
nal performance.
Although the implementation of dynamic modelling and
of population balances clearly opens the way for ecient,model-based automatic control and an improved assessment
of product quality, these topics have not been treated in
the present communication. They are, together with a more
profound analysis of parametric sensitivity of the model,
subjects of actual work. Fluidized bed processes other than
drying (e.g. catalytic reaction, granulation) have also not
been addressed. Conditions of solids movement along the
dryer that are located between the investigated CSTR be-
haviour and ideal plug ow (which is not dicult to simu-
late) may require specic modelling. The same is true for the
implementation of population dynamics to CSTR cascades,
whichin principlerequires a multidimensional deni-
tion of populations. These topics will be subjects of future
work.
Notation
A surface area, m2
Ar Archimedes number (Ar= [gd3p=2g][(p g)=g])
b exponent of the expansion equation, dimensionless
c specic heat capacity, J=(kg K)
C heat capacity, J/K
d diameter, m
F cross sectional area, m2
g acceleration of gravity, m=s2
h specic enthalpy, J/kg
hs specic enthalpy of sorption, J/kg
hv specic enthalpy of evaporation, J/kgH enthalpy ow rate, J/kg
k heat transfer coecient through walls, W=(m2 K)
L height, mLe Lewis number [Le=g=(cggg)]
m exponent for the pulsation coecient, dimension-
less
M mass, kgM mass ow rate, kg/sM molecular mass, kg/mol
n residence time distribution density, 1=s
NTU number of transfer units, dimensionless
Nu Nusselt number (Nu=dp=g)
p partial pressure, Pa
P total pressure, Pa
Pr Prandtl number (Pr=gcgg=g)Q heat ow rate,W
Re Reynolds number (Re=udp=g)
Sc Schmidt number (Sc=g=g)
Sh Sherwood number (Sh=dp=g)
t time, s
T temperature, K or C
Tg;out caloric average temperature of outlet gas, K or C
u ow velocity, m/s
X particle moisture content (dry basis), dimensionlessXout average moisture content of outlet solids, dimen-
sionless
Y gas moisture content (dry basis), dimensionless
z bed height coorinate, m
Greek letters
heat transfer coecient, W=(m2 K)
mass transfer coecient, m/s
diusion coecient, m2=s
normalized bed height, dimensionless
normalized particle moisture content, dimension-
less
pulsation coecient, dimensionless
thermal conductivity, W=(m K)
kinematic viscosity, m2=s
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normalized single particle drying rate, dimension-
less
inow coecient for downcomer tube, dimension-
less
density, kg=m3
residence time, s
ratio of bubble to total gas ow rate, dimensionlessr dimensionless visible bubble ow rate, dimension-
less
relative humidity, dimensionless
porosity, dimensionless
Subscripts
0 supercial
b bubble gas
bed bed
cal caloric
cr critical
e environmente inlet of air heater
elu elutriation
eq equilibrium
g gas
in inlet
mf minimal uidization
mo monolayer
out outlet
p particle
s suspension gas
sat saturation
w wall
weir weir w; g gaseous water
w; l liquid water
- average value
Abbreviations
CSTR continuous stirred tank reactor
RTD residence time distribution
Acknowledgements
The authors gratefully acknowledge nancial support bythe Volkswagen Stiftung.
Appendix A
A.1. Fluidization parameters
The expanded bed porosity, , is calculated after
Richardson and Zaki (1954),
= Re0
Reelu1=b
; (A.1)
with the exponent given byMartin (1997),
b=ln(Remf=Reelu)
ln mf: (A.2)
All Reynolds numbers are dened with the particle diameter
dp and the supercial gas velocity
u0=Mg
gFbed: (A.3)
At minimal uidization, application of the Ergun formula
according toMartin (1997)leads to
Remf= 42:9(1 mf)
1 +3mf
(1 mf)2Ar
32141
;
(A.4)
while the Reynolds number at the elutriation point is ob-
tained afterReh (1968):
Reelu=
4
3Ar : (A.5)
The ratio of bubble to total gas ow rate is expressed in the
form
=rRe0 Remf
Re0: (A.6)
The parameter rdepends on the Geldart classication of
the particles and on bed dimensions. For the experiments
presented here, solids of group D were used and r varied
between 0.26 and 0.50 according to the equation given byHilligardt and Werther (1986) for bed diameters dbed be-
tween 0:1 and 1 m:
r=
0:26; Lbeddbed6 0:5;
0:35(Lbeddbed
)1=2; 0:556 Lbeddbed6 8;
1; Lbeddbed
8:
(A.7)
A.2. Outow equation
A centrally placed downcomer pipe has been used for the
particle outlet (Fig.14). Anticipating a Toricelli relationshipand correcting it by an inow parameter, , and a pulsa-
tion factor,, the outcoming mass ow rate, Mp; out, can be
calculated from holdup, Mp, and the expanded bed height,
Lbed, to
Mp; out
=
0; Lbed6Lweir;
FweirFbed
Mp
2g
Lbed
Lweir
Lbed
; Lbed Lweir:
(A.8)
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.
p,in
Lweir L
bed
bed
M
L
Mp,out.
Fig. 14. Schematic representation of solids outlet.
The pulsation factor allows for an outcome of particles even
at expanded bed heights that are, in the average, somewhat
smaller than the height of the weir. Since the pulsation am-
plitude vanishes at minimal uidization, the simple relation-
ship
= Re0
Remfm
(A.9)
has been used for the calculation of . Both the inow pa-
rameter, , and the exponent, m, of Eq. (A.9) have been
derived from measurements of steady-state holdup to
= 0:0316; (A.10)
m= 0:2605: (A.11)
By this tting, a very good reproduction of experimental
data could be attained (for more details seeBurgschweiger,
2000).
A.3. Residence time distribution
The density function of the residence time distribution
has been calculated by means of the equations valid for a
CSTR to
n() =1
exp
(A.12)
with
=Mp= Mp: (A.13)
0.0025
0.0015
0.002
0.001
0.0005
00 500 1000 1500 2000 2500
CSTR
experiment
[s]
n[1/s]
Fig. 15. Measured and calculated (CSTR) residence time distribution of
solids ( Mp = 1:91 g=s; Mg = 43:3 g=s; Tg; in = 61C; Mp = 0:90 kg;
Xin =Xout = 0:013 (dry particles); similar results are obtained for wet
product).
The assumption of perfect backmixing of the solids has been
checked by tracer experiments. In these, a pulse of slightlycoloured particles is inserted in the apparatus. Outcoming
particles are collected within several time intervals, sepa-
rated and weighed. An example of the excellent agreement
of measured densities with the CSTR behaviour after Eqs.
(A.12) and (A.13) is given in Fig.15.
A.4. Particle-to-suspensionn-gas transfer coecients
The mass transfer coecient between particle surface and
the suspension gas, ps, is calculated from the respective
Sherwood number, Shps, by the application of the model
developed byGroenewold and Tsotsas (1997),that meansfrom the relationship
Shps= Re0Sc
Aps=Fbedln
1 +
Sh Aps=Fbed
Re0Sc
: (A.14)
In Eq. (A.14) Sh is the single-particle Sherwood number
after the classical formulae ofGnielinski (1997).For spher-
ical particles, the ratio of mass transfer area to the cross
sectional area of the bed is
Aps
Fbed= 6(1 )
Lbed
dp; (A.15)
while the absolute value of Aps (the total particle surface
area) can be obtained from the holdup, Mp, to
Aps= 6Mp
dpp: (A.16)
This treatment is the same as for batch drying by
Burgschweiger et al. (1999a). It is important to remember
that Eq. (A.14) is a transformation of real to appar-
ent Sherwood numbers. While only the laws of momen-
tum and mass transfer around a single particle determine
Sh, the inuence of backmixing of the suspension gas
bed has been worked into Shps, which is an apparent ki-
netic coecient. Eq. (A.14) has been shown to resolve
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J. Burgschweiger, E. Tsotsas/ Chemical Engineering Science 57 (2002) 5021 5038 5037
the low-Sherwood-number (or Nusselt-number) problem,
which has been often described in literature (e.g. Kunii
& Levenspiel, 1991). Its accuracy has been successfully
tested by Groenewold and Tsotsas (1999) on more than
700 experimental runs from literature, including aver-
age particle diameters between 0:125 and 4:3 mm, i.e.
solids belonging to Geldart groups A, B, and D. A sig-nicant inuence of the Geldart group on the predictive
performance could not be detected. Since the mentioned
analysis by Groenewold and Tsotsas refers to evapora-
tion and sublimation experiments, as well as to drying of
non-hygroscopic materials within the rst period, it consti-
tutes one more limiting case of the present modelling. All
other parameters have been the same, so that consistency is
guaranteed.
Using the Prandtl number,Pr, and the single-particle Nus-
selt number,Nu, in Eq. (A.14) an apparent Nusselt number,
Nups, and the heat transfer coecient ps between suspen-
sion gas and the particle surface are calculated.
A signicant merit of the outlined procedure is that no
tting has to be done at the level of heat and mass transfer
coecients, in contrary to what has for a long time been
common in literature and in industrial practice. Without this
type of approach it would not have been possible to keep
the model free of adjustable parameters. On the other hand,
some empiricism in the development by Groenewold and
Tsotsas, condensed in the consideration of axial dispersion in
an apparent kinetic coecient, should not be overseen. The
use of dispersion coecients or of CFD simulations would
be preferable from the theoretical point of view, which has,
however, by far not yet reached the level of development
that would be necessary for an equally good predictive per-formance and for applicability to, e.g., the task of uidized
bed dryer design.
A.5. Suspension-to-bubbles transfer coecients
The product sbAsb (Eq. (21)) is calculated as by
Burgschweiger et al. (1999a),i.e. from the number of mass
transfer units between suspension and bubbles
NTUsb=gsbAsb
Mg=
Lbed
0:05m: (A.17)
By analogy, the productsbAsb (Eq. (22)) is obtained from
sbAsb
cg Mg= NTUsbLe
2=3: (A.18)
Obviously, Eq. (A.17) is empirical and a rather coarse
approximation. However, as Groenewold and Tsotsas
(1999) point out, the model is, especially for large par-
ticles of group D, not very sensitive to errors in NTUsb.
In any case, suspension-to-bubbles transfer must not be
neglected. Previously proposed models which consider the
bubble phase as an inactive bypass (e.g. Subramanian,
Martin, & Schlunder, 1977) do not perform satisfactorily,
unless by unrealistic, compensative tting of other model
parameters.
A.6. Remaining heat transfer coecients
The coecient for heat transfer between gas and wall,gw, is calculated after Baskakov et al. (1973), see also
VDI-Warmeatlas (Martin, 1997). The wall area is derived
from the expanded bed height Lbed to
Aw =dbedLbed: (A.19)
It is assumed that the eective heat transfer areas for the
bubble and the suspension phase are proportional to , re-
spectively (1 ), in Eqs. (24) and (23).
The heat transfer coecient between particles and the
wall, pw, the coecient for the so-called particle convec-
tion, is derived afterMartin (1984, 1997).This approach isconsistent with the above-mentioned equation of Baskakov
for gas convection. Again, we refrain from repeating here the
well-known formulae. Although the average particle tem-
perature, Tp, and the average solids moisture content, X,
are used in the calculation, the inuence of evaporation in
the sense of a latent heat sink within the drying particles is
not really accounted for in the model of Martin. No signi-
cant error is caused by this simplication in the case of large
particles, as recent, detailed investigations by Groenewold
and Tsotsas (2001)have shown. The same is, however, not
true for ne-grained products.
Since similar heat transfer mechanisms are valid during
the collision between a particle and the wall, and the collisionbetween two particles, the calculation of particle-to-particle
heat transfer in Eqs. (30) and (31) is also based on the
coecientpw after Martin. Assumptions in the respective
derivation are discussed byBurgschweiger (2000).
Finally, the heat transfer coecient to the environment
may be calculated from a series combination of insulation
and free convection to
kwe=
1
we+
siso
iso
1
: (A.20)
Equations byChurchill and Chu (1975)have been used forwe, see Burgschweiger (2000).To enable visual observa-
tion, the experiments were conducted without insulation.
With values of typically kwe = we = 5 W=(m2K), this pa-
rameter has not been important for the simulations.
In general, the sensitivity upon all heat transfer parame-
ters of this subsection has been relatively lowa fact that,
however, should not be deliberately extrapolated. As already
mentioned, a detailed sensitivity analysis is outside the scope
of the present communication.
With reference toBurgschweiger (2000),the models for
peripheral elements, specically for the air heater and the
duct to the dryer, will not be given here.
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