Drying 2004 – Proceedings of the 14th International Drying Symposium (IDS 2004)
São Paulo, Brazil, 22-25 August 2004, vol. A, pp. 129-136


            Mirko Peglow¹, Stefan Heinrich¹, Evangelos Tsotsas2 and Lothar Mörl¹

     Otto-von-Guericke-University, Universitätsplatz 2, 39106 Magdeburg, Germany
             1.    Institute of Process Equipment and Environmental Technology
   E-mail: stefan.heinrich@vst.uni-magdeburg.de, mirko.peglow@vst.uni-magdeburg.de
                           2.     Institute of Process Engineering

               Keywords: fluidised bed drying, dispersion, sherwood number, nusselt number


         A model for fluidised bed (FB) drying is presented. The model is based on a simple two
         phase approach which distinguishes between a suspension phase and a bypass phase. To
         investigate the influence of backmixing effects in the gas phase caused by turbulences,
         dispersion coefficients are used. Therewith the complete range from ideal plug flow to
         total backmixing behaviour can be simulated and the influence of non ideal backmixing
         for different simulation scenarios can be analysed.


   While bubbling fluidised beds provide favourable conditions for a number of separation or reactive
processes, the performance of such processes is influenced and may be limited by the distribution and
mixing of fluid and solids in the bed, and by transport phenomena. Particle-to-fluid heat and mass transfer
coefficients which may appear to be much smaller than respective values for the single particle are one
well-known expression of this influence. Indeed, it could be demonstrated by Groenewold and Tsotsas
(1999) that too low transfer coefficients can be explained on the basis of suspension gas backmixing, and
the performance of fluidised bed evaporators and dryers can be predicted accurately, without any fitting.
However, the approach of these authors is semi-empirical, it implements – directly or in effective transfer
coefficients – only limiting cases of dispersion. On the other hand, models for spray fluidised bed
processes like granulation, coating or agglomeration – as those developed by Heinrich, Mörl and co-
workers (2002) – use finite dispersion coefficients for the solids, and need such dispersion coefficients in
order to properly describe the spatial distribution of particles with a wet surface. In the same models, plug
flow is assumed for the gas. Successful fluidised bed reactor models by Artlich et al. (1998) take finite
dispersion of both, the solids and the fluid, into account, but neglect fluid-to-particle mass and heat
transfer resistances.

The aim of this study is to develop a generic model which takes into account dispersion in all phases as
well as mass transfer resistance. Due to the model assumptions of normalised drying curve after van
Meel, (1958) heat transfer will be neglected.


Previous approaches
   To investigate backmixing effects in FB drying, Groenewold et al. (1999) introduced an apparent
Sherwood number. From the equivalence of efficiency for perfect backmixing and ideal plug flow
                                                               NTU CSTR                                 (1)
                                    1 − exp ( − NTU PFTR ) =
                                                             1 + NTU CSTR
an apparent Sherwood number was derived
                                            Re0Sc           Sh      A H                                 (2)
                                  Sh app =           ln 1 + PFTR V Bed
                                           A V H Bed             Re0Sc
applying the definition of the number of transfer units (NTU) to equation (1)
                                                      Sh A V H Bed                                      (3)
                                             NTU =                 .
The result of this transformation is that a backmixing effect is introduced into the drying kinetics. Using
this Sherwood number in a plug flow model, a perfect backmixing can be obtained, which can be used to
predict the first and second drying period at low Reynolds numbers (see Groenewold et al., 1997).

    The new model is applied for batch drying processes. It is
based on a simple two-phase-approach with an active bypass as
it is shown in Figure 1 (see Groenewold et al., 1997).
The assumptions for the model are:
• The bypass fraction is independent of bed height.
• A number of transfer units ( NTU SB ) describes the mass
     transfer between suspension and bypass phase.
• The mass transfer between particle and gas bulk is given by
     the real Sherwood number, which tends to a finite value at
     low Reynolds numbers. This Sherwood number is calculated
     after Gnielinski (1980).
• Backmixing effects in all phases caused by turbulences are
     considered by axial dispersion coefficients. The radial
     dispersion is neglected.
• The concept of the normalized drying curve (van Meel et al.,
     1958) is used to describe the falling drying rate in the 2nd                Figure 1. Two-phase-model.
     drying period.

   To describe the drying process for the first and second drying period, three states have to be taken into
account. The three following equations can be derived by balance in a differential volume element.
    • Air humidity in suspension phase
   ∂YS       ∂ 2 YS m DryAir     ∂Y        1                           dz     1                      dz   (4)
       = DS         +           − S dz +       NTU PS ( YSat − YS ) υ      −      NTUSB ( YS − YB )
    ∂t        ∂z  2
                      dm DryAir   ∂z     1− ν                         H Bed 1 − ν                   H Bed

    •   Air humidity in bypass phase
                   ∂YB       ∂ 2 YB m DryAir     ∂Y       1                       dz                    (5)
                        = DB        +           − B dz +      NTU SB ( YS − YB )
                    ∂t        ∂z  2
                                      dm DryAir   ∂z     1− ν                    H Bed
    • Particle moisture content
                         ∂X         ∂ 2X m          1                            dz                     (6)
                               = D P 2 − DryAir         NTU PS ( YSat − YS ) υ
                          ∂t         ∂z     dm P 1 − ν                         H Bed
Boundary conditions of first kind at the inlet and at the outlet
                      ∂YS         ∂Y                       ∂YS           ∂Y                             (7)
                                = B        =0                           = B           =0
                       ∂z z =0     ∂z z =0                  ∂z z = HBed   ∂z z = HBed
are used. The humidity of suspension gas and bypass gas weighed with bypass fraction gives the humidity
of air at the outlet
                                        Y out = (1 − ν ) YSout + νYB .
                                                                   out                             (8)
The equations to calculate the fluid dynamic and mass transfer are given in the appendix.


Numerical Solution
   The Method-of-Lines transforms the partial differential equations (4)-(6) into ordinary equations using
finite differences in space domain. This system was embedded in a Runge-Kutta-procedure of 4th order
using an adaptive step size control in form of a full/half step strategy. Due to the fact, that in the first
drying period the states of suspension and bypass humidity are in a steady state, equations (4) and (5)
were set to zero and the particle moisture balance was neglected. The obtained non-linear equations, used
for parameter studies for the first drying period, were solved using the Levenberg-Marquardt-Method.

Analytical Solution
   To get an analytical solution the drying efficiency is introduced defined as
                                                    Y out − Y in                                        (9)
                                               η=                 .
                                                    YSat − Y in

The total efficiency of a two-phase-model is obtained by summation of the single efficiency of suspension
and bypass, weighed with bypass fraction
                                           η = (1 − ν ) ηS + νηB .                                     (10)
To obtain an analytical solution the model of inactive bypass has to be assumed, where the efficiency of
the bypass is always zero. Therefore the following equations are obtained to describe ideal plug flow resp.
total backmixing behaviour
                                          NTU                                         NTU             (11)
               ηPFTR = (1 − ν ) 1 − exp −           ,            ηCSTR = (1 − ν )             .
                                          1− ν                                    1 − ν + NTU
To represent the effects of backmixing, bubble behaviour as well mass transfer from suspension to bubble
gas an additional plot is used. Therefore the determined efficiency can always be applied on a simple
(homogeneous) plug flow model, as it is written in equation (12)
                                           η = 1 − exp ( − NTU ) .                                    (12)
Out of this consideration a Sherwood number can be calculated, which fits the mass transfer to the
previously given drying efficiency in the simple plug flow model. This Sherwood number is defined as
                                                 Re0Sc                                            (13)
                                     Sh Plug = −           ln (1 − η) .
                                                 A V H Bed

The analytical solutions for both limiting cases are plotted in Figure 2 for a bypass fraction of ν = 0 and
ν = 0.2 . It is assumed, that the coefficient (AV/Hbed)/Sc in equation (13) is 20 and the “real“ Sherwood
number, following the requirements mentioned above, is independent of the Reynolds number and has a
constant value of Sh = 20 . The plots show that a system with total backmixing has a lower efficiency
than the ideal plug flow. For plug flow the efficiency of suspensions phase achieves at high NTU
numbers (>10) a value of 1, wherewith the total efficiency is decreased by bypass fraction. For high
Reynolds numbers (small NTU numbers) a plug flow system and a system with total backmixing have the
same efficiency. For systems with high efficiency (close to 1) the difference between different models
comes out by conversion of efficiency to the Sherwood number.

                                   Figure 2. Numerical solution for PFTR and CSTR .

Testing the model – Analytical solution vs. numerical solution
   In order to check the quality of the numerical solution, simulation                 Table 1: Simulation Parameter.
results for a system with ideal plug flow (DS = 0) and a system with              Parameter
total backmixing (DS → ∞ ) were compared with the analytical                      mBed             [ kg ]        1
solutions given in equation (11). The mass flux was changed in a
range from the minimal fluidisation (Re0 = 33) up to the discharge
                                                                                  dP               [ mm ]        1
point (Re0 = 275). Further simulation parameters are given in Table               ρP               [ kg / m³]    2000
1. The comparison of the results is shown in Figure 3. It comes out               d App            [ mm ]        200
that the model can predict CSTR behaviour correctly by choosing an
adequate value for the dispersion coefficient. To analyse the                     X0               [ kg / kg ]   0.2
influence of the dispersion coefficient in the suspension phase,                  X crit           [ kg / kg ]   0.1
further simulations were done by changing the dispersion in a range
of DS = 0 – 50 m²/s keeping the fluidisation velocity constant. The               X hyg            [ kg / kg ]   0
results for minimal fluidisation and discharge point are shown in                 Y in             [ kg / kg ]   0.01
Figure 4. At low Reynolds numbers a dispersion of DS = 1 m²/s
results in a total backmixing, while at high Reynolds numbers close               ϑ in             [°C ]         50
to the discharge point a dispersion of DS = 20 m²/s is necessary to               DP                m2 / s       0.5
produce ideal CSTR behaviour.
Based on these results the following simulations were done with the               DB               [ −]          0
dispersion coefficients given in Table 2.                                         NTU SB           [ −]          0

             Table 2: Variation of Dispersion.
                                                                                  ν                [ −]          0.2
                                                                                  mDryAir                        0.02-
DS m 2 / s    0   0.1 0.5      1    2     5      10 20 50                                          [ kg / s ]    0.16

                        Figure 3. Numerical solution vs. analytical solution for PFTR and CSTR.

                     Figure 4. Variation of dispersion coefficient at fluidisation and discharge point.

Parameter variation – Standard case

                                    Figure 5. Standard case (parameters see Table 1).

   The simulation results with the parameters given in Table 1 are plotted in Figure 5. For all simulations
the total efficiency vs. NTU number, the Reynolds number vs. Sherwood number for simple plug flow
and the Reynolds number vs. the relative change of Sherwood are shown. For low Reynolds numbers at
fluidisation point the influence of backmixing on the mass transfer is low. The value of Sherwood is in a
range of 2 to 4 and has a maximal relative change of less then 15 %, while the efficiency differs about 5
%. At high Reynolds numbers (low NTU) the difference between ideal backmixing and plug flow is
evident. Small dispersion coefficients have no effect on Sherwood. Only for a dispersion coefficient

higher than 2 m²/s the behaviour changes articulately from ideal plug flow, while the mass transfer
worsens about 40 %.

Parameter variation – Influence of bed mass

                       Figure 6. Variation of bed mass ( mBed = 2 kg, parameters see Table 1).

   In this simulation the bed mass was increased to mBed = 2 kg. Therefore a two times higher NTU
number was obtained. The results are plotted in Figure 6. The results show, that for low Reynolds
numbers and large values for NTU the backmixing has a low influence on Sherwood. In this range the
relative change of Sherwood is less than 10 %. This is caused by the large mass transfer surface, which
leads to a total saturation of suspension gas for plug flow as well as total backmixing. The variation of
dispersion has a low effect on Sherwood, which has a value of about 2. Finally the abatement of
Sherwood is caused by the inactive bypass, which dominates the total efficiency of the system. For high
Reynolds numbers an evident effect of non-ideal backmixing exists analogue to the results of standard

Parameter variation – Influence of NTU suspension – bypass

                Figure 7. Variation of NTU suspension – bypass ( NTU SB = 1 , parameters see Table 1).

   In this simulation the model of inactive bypass was transformed into an active bypass model by setting
NTU for mass transport between suspension gas and bubbles to 1. For high NTU numbers the total
efficiency of the system is almost 100 %, which means, that the influence of bypass is neglect able or in
both phases the humidity of air is equal. It is obvious, that a small value of the backmixing coefficient
decreases the Sherwood number for plug flow at 40 %.

Parameter variation – Influence of bypass fraction
   In this simulation the bypass fraction was decreased from 20 % to 10 %. In consequence the efficiency
rises for ideal plug flow and high NTU numbers to 90 %. These results are very similar to the standard
simulation, however Sherwood values are a bit higher.

                         Figure 8. Variation of bypass fraction ( ν = 0.1 , parameters see Table 1)

Simulation of first and second drying period
   For the set of parameters given in Table 1 under variation of dispersion coefficients the complete
drying process was simulated. Mass flux of dry fluidisation gas was set to m DryAir = 0.1 kg/s. The second
drying period is characterised by a constant falling normalized drying rate. The results of this simulation
are plotted in two different manners. At first the simulated drying curve vs. process time and in a second
diagram the specific evaporation rate vs. particle moisture content is plotted. Increasing backmixing
causes a decreasing of drying rate. For the complete drying process no further simulations were done.

                  Figure 9. First and second drying period ( mDryAir = 0.1 kg/s, parameters see Table 1).


   Based on previous approaches a generic model was derived to investigate the influence of backmixing
effects in gas phase on fluidised bed drying. With the help of the model, regarding the simplifying model
assumptions mentioned above, the limiting cases of ideal plug flow and total backmixing can be
reproduced as well as simulation scenarios of non-ideal backmixing. Especially the influence of
backmixing effects in suspension phase were investigated in the framework of this study. For active
bypass the influence of backmixing becomes more important due to the higher efficiencies achieved.
Experimental data has to be compared of the simulation results to check the model.


 A           surface area                                                         m²
 D           dispersion coefficient                                               m2 / s

 dm           differential mass                                      kg
 H            height                                                 m
  m           mass flux                                              kg / s
 Re           Reynolds number                                        −
 Sc           Schmidt number                                         −
 Sh           Sherwood number                                        −
 t            time                                                   s
 X            particle moisture content                              kg / kg
 Y            air humidity                                           kg / kg
 z            length coordinate                                      m

 Greek Symbols
 ϑ         temperature                                              °C
 ν         bubble fraction                                          −
 υ         normalized drying rate of single particle                −

 Subscripts                                            Superscripts
 app          apparent                                 in           inlet
 B            bypass phase                             out          outlet
 Bed          fluidized bed
 Crit         critical
 CSTR         continuous stirred tank reactor
 hyg          hygroscopic
 P            particle
 PFTR         plug flow tubular reactor
 PS           particle-suspension
 S            suspension phase
 Sat          saturation
 SB           suspension-bypass
 V            Volume
 0            superficial


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