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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: [email protected]

    (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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    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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