Epstein N., Grace J.R. Spouted and Spout-Fluid Beds: Fundamentals and Applications
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Drying of particulate solids 197
W
p
, is constant in all bed segments. The upward gas superficial velocity in each annular
segment, U
a
(i) = G
a
(i)/ρ, must be determined by an appropriate gas–solid flow model
(discussed in Chapter 3), as must also the minimum spouting conditions (U
ms
and P
ms
),
the spout diameter or width along the axial coordinate, D
s
(z)orL
s
(z), the fractional void
volume in the three regions, ε
a
(generally close to ε
mf
), ε
s
, and ε
f
, the fountain height,
H
f
, and finally the particle circulation rate, W
p
.
The interaction between the annulus and spout regions occurs at the interface in
two different ways: air crosses this interface from the spout to the annulus, or possibly
vice versa when there is annulus aeration, and heat transfers between the spout air and
particles that descend at this spout–annulus interface. The airflow rate passing through
this interface is calculated by a mass balance for each segment: G
it
(i)A
it
(i) =G
a
(i) A
a
(i)–
G
a
(i –1)A(i – 1). When G
it
(i) is positive, air flows through this interface toward the
annular region, then Y
g|it
(i) = Y
g|s
(i) in Eqs. (11.1) and (11.3). Conversely, when G
it
(i)is
negative, air flows through this interface toward the spout region; then Y
g|it
(i) = Y
g|a
(i)
in Eqs. (11.1) and (11.3). The heat transfer rate across the interface, q
i
, for each segment
can be described by a convective heat transfer coefficient between descending particles
and ascending air, an effective contact area between them, and a temperature difference
between the spout air and particles in the annular segment.
51
If the dryer is not thermally
insulated, a significant amount of heat is lost to the surroundings through the column
wall. This rate, q
lost
, can also be specified for each segment by a convective heat transfer
coefficient between the bed of particles and ambient air, an effective contact wall area,
and the temperature difference. Most 2A models in the literature neglect these heat
losses.
29,31,49
However, as pointed out by Freitas,
7
heat losses can be higher than those
estimated by the convective heat transfer coefficient formulation, because of flanges,
screws, connections, and metallic supports attached to the column acting as extended
surfaces, and also because of radiant heat transfer. Hence, it is better to consider this
heat transfer coefficient as an adjustable parameter to be determined experimentally for
each SB column.
Because particles also undergo drying in the fountain region (depending on their
moisture content), the fountain region can be represented by an additional finite segment
with a paraboloidal shape and height H
f
, as shown in Table 11.2. The overall mass and
energy balances in this segment can be formulated by the equations in this table.
51
The specific enthalpy for evaporating water, H
v
, is determined at the reference
temperature (T
R
= 273 K), comprising the latent heat of water vaporization added to
the heat of water desorption from the solids. This heat of desorption, obtained from the
sorption curves,
7,18
is taken into account only in the falling-rate drying period.
To solve these mass and energy balance equations in each segment, the following
complementary equations are needed to couple gas and particle temperatures in the
annulus, spout, and fountain regions, respectively:
±W
p
c
p−p|j
(i −1){T
p|j
(i −1) − T
R
}±H
v
W
p
{Y
p|j
(i) − Y
p|j
(i −1)}
=±W
p
c
p−p|j
(i){T
p|j
(i) − T
R
}+h
p|j
(i)A
p|j
(i)
×
[T
g|j
(i) − T
p|j
(i)] + [T
g|j
(i −1) − T
p|j
(i −1)]
2
(11.7)
198 Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar
and
W
p
c
p−p|a
(N ){T
p|a
(N ) − T
R
}+H
v
W
p
{Y
p|s
(N ) − Y
p|a
(N )}
=W
p
c
p−p|s
(N ){T
p|s
(N ) − T
R
}+h
p| f
A
p| f
×
T
g|ou t
−
T
p|s
(
N
)
+ T
p|a
(
N
)
2
−
T
g|f
−
T
p|s
(
N
)
+ T
p|a
(
N
)
2
ln
⎧
⎪
⎨
⎪
⎩
T
g|ou t
−
T
p|s
(
N
)
+ T
p|a
(
N
)
2
T
g|f
−
T
p|s
(
N
)
+ T
p|a
(
N
)
2
⎫
⎪
⎬
⎪
⎭
. (11.8)
In Eq. (11.7), the positive sign of “±” r epresents the annulus segment with j = a,
whereas the negative sign represents the spout segment with j = s. In Eqs. (11.7)
and (11.8), the convective heat transfer coefficients, h
p|j
, in the annulus ( j = a), spout
( j = s), and fountain ( j = f ) regions are calculated using empirical equations pre-
sented elsewhere.
14
The effective area for heat transfer is expressed as A
p|j
(
i
)
=
S
p
[1 − ε
j
(i)]A
j
(H/n)
V
p
, with j = a, s, f for the annulus, spout, and fountain regions,
respectively.
11.3.2 Drying kinetics models
To solve this set of equations (from 11.1 to 11.8), the rate of evaporation must be
determined, knowing that the evaporated water transfers from the particle surface to the
bulk of the gas. There are, generally, two periods of drying: the constant- and falling-
rate drying periods (Figure 11.1b). The first, or constant-rate, period is characterized
by evaporation of free or unbound water that covers the particle surface, owing to the
heat supplied by hot air. During this period, the thin air layer surrounding the particle
surface becomes saturated with water vapor, the particle temperature equals the wet-bulb
temperature, and the drying rate is constant. Consequently, if drying of particulate solids
occurs in this period, the decrease of the solids moisture content in the i-segment of the
bed can be described by the convective vapor transfer from the thin saturated gas layer
surrounding the particle to the bulk of t he gas:
±W
p
{Y
p|j
(i) − Y
p|j
(i −1)}=β
p|j
(i)A
p|j
(i)[Y
g|j
(i)]
M
(11.9)
with
[Y
g|j
(i)]
M
=
{Y
sat
g|j
(i) − Y
g|j
(i)}−{Y
sat
g|j
(i −1) − Y
g|j
(i −1)}
ln
"
Y
sat
g|j
(i) − Y
g|j
(i)
Y
sat
g|j
(i −1) − Y
g|j
(i −1)
+
. (11.9a)
In Eq. (11.9), j = a for the annular segment with the first term of this equation (on the
left side) being positive; j = s for the spout segment with the first term of this equation
Drying of particulate solids 199
(on the left side) being negative. The convective mass transfer coefficient, β
p|j
, can be
estimated by empirical correlations discussed elsewhere.
14
A similar formulation can be
developed for the fountain segment.
51
In the course of this first drying period, particulate solids can shrink, decreasing their
volume and, sometimes, changing their shape. The volume reduction, when considerable,
must be correlated to the amount of evaporated water and incorporated into the model. In
the specific case of isotropic particle shrinkage with unchanging density, the decrease in
the mean volumetric sphere-equivalent diameter of a particle, d
v
, can be formulated as
d
v
= d
v0
1 −
(Y
p0
− Y
p
)ρ
p0
(1 + Y
p0
)ρ
w
1/3
, (11.10)
with the subscript 0 denoting the initial drying time. Eq. (11.10) or a similar equation
must be added to the set of model equations for determining changes in the particle
properties and their effect on the SB operation in each segment of the bed during this
initial drying period.
The constant drying rate period lasts until the appearance of the first nonsaturated
zones on the particle surface, signifying that the particle moisture content has attained
its critical value, Y
p
cr
. From this point on, the water to be evaporated is adsorbed on or
below the particle surface, and there is therefore an additional mass transfer resistance
to removal of this bound water. Consequently, the drying rate starts to decrease, and the
particle temperature begins to increase, as the heat supplied by the hot air no longer
dictates the amount of water evaporated. The drying kinetics of nonporous or partially
porous solids in this falling rate period can be formulated adequately by liquid diffusion
theory.
7,18,49
According to this theory, water moves from the particle interior to the
particle surface owing to a concentration gradient. Water evaporates at the particle
surface only, following the liquid–vapor thermodynamic equilibrium. The resistance
offered by the particle to this water diffusion is expressed by an effective diffusion
coefficient, D, a property of the particulate solid. In this theory, other factors that can
affect the water movement inside the s olid structure, such as the capillarity, t emperature,
or pressure gradient, are neglected.
55
Even though this liquid-diffusion theory is a
simplified formulation of the actual drying kinetics in the falling-rate period, errors
are minimized by the experimental determination of D in a given application. This
coefficient should be evaluated using a corrected procedure to represent the SB drying
operation.
7,18,49
The theory has been commonly applied for drying particulate solids in
SB dryers during the falling-rate period.
6,7,13–19,22,27–29,31,34,41–42,49–50
The three-dimensional Fick diffusion model
56
of moisture transfer can be written as:
∂Y
p
∂t
=∇·(D∇Y
p
). (11.11)
Eq. (11.9) must be linked to Eq. (11.11) to complete the simulation of the drying
operation for all types of particulate solids (shrinkable or not). At Y
p
= Y
p
cr
, Eq. (11.9)
is replaced by Eq. (11.11). When Y
p
cr
is unknown as a function of temperature, its value
can be estimated by the sorption curve of the material at the given particle temperature
and an air relative humidity of 95 percent. This estimate is verified if the drying rate
200 Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar
in the falling-rate period equals that in the constant-rate drying period at Y
p
= Y
p
cr
.
Eq. (11.11) must be solved numerically to describe the falling-rate drying period, with
appropriate boundary and initial conditions for particles in each segment.
Niamnuy et al.
9
successfully used Eq. (11.11) together with the energy equation to
describe the drying of shrimp, a case in which there is considerable volume shrinkage.
Bia
˙
lobrzewski et al.
8
used the same approach to simulate the drying of carrot cubes in
a SFB.
Improvements in the simulation of particle drying by using computer tools favor the
analysis of the product quality, supplying information to define criteria for preserving
the particle quality during drying.
8,9
This is an open area to be pursued in future years.
11.3.3 Additional considerations
Additional information about the drying operation mode must be incorporated in the
model equations. For batch drying of particulate solids, the number of cycles particles
require to reach their final moisture content establishes their mean residence time in the
dryer. Therefore, this number specifies the final drying time. Initial conditions of the
particulate solids, such as Y
p0
, T
p0
, and M, need to be added to the input data.
Extension of the model formulation given previously to continuous drying of par-
ticulate solids requires knowledge of the RTD for the specific SB configuration. An
RTD relationship must be incorporated in the model to provide the age of each particle
in the i-segments of the bed and, consequently, its actual moisture content, as well as
other properties that depend on its age (actual time in the dryer). As particles form the
discrete phase, their overall properties in each i-segment of bed are calculated as the
sum of their individual values. In CSB dryers, perfect mixing of particles is commonly
assumed.
13 ,16–18,57
Zahed and Epstein
19
assumed good, but not perfect, mixing, coupled
with the assumption of isothermal solids to provide a better description of the continuous
drying of wheat in CSB dryers. Madhiyanon et al.
29
showed that the RTD function for
continuously drying rice in a 2DSB with draft plates follows the well-known tanks-
in-series model. Therefore, the specific SB column design determines the best RTD
function to be incorporated in the model formulation.
Input and output streams must be added to the mass and energy balance equations at
the specific segment of the bed. The first relates to the feed of solids, with the known
variables i nlet solids flow rate, Y
p|in
and T
p|in
. The second represents the product discharge
with the unknown variables T
p|out
and/or Y
p|out
. Typically, the feed of solids is at the top of
the vessel column, and solids are discharged near the bottom. However, for modified SB
configurations, these input/output streams may be located at other positions. To extend
this model formulation to an intermittent spouting operation, it is necessary to define
the temporal functions for injecting air and/or for varying the inlet air temperature, to
complete the set of model equations. Jumah
42
and Bon and Kudra
58
give the details
necessary to define these functions.
Finally, the incorporation of the quality of the particulate solids should be in the form
of quantitative inequalities (for severe restrictions) and empirical correlations that can
form an objective function for controlling or optimizing the process.
Drying of particulate solids 201
Table 11.3. Assumptions and main results of CFD models in spouted bed drying of particulate solids.
Assumptions: gas-particle flow
59
Assumptions: drying model Results and applications
• Gas-particle: Eulerian-Eulerian
two-fluid model
• Interphase momentum
exchange: Gidaspow drag
model
60
• Kinetic-frictional solid stress:
granular kinetic theory
(derived by Lun et al.)
59,60
• Turbulence: standard k −ε
turbulence model
• 2D axial symmetry
• Grid: triangular (conical base)
and quadrilateral (cylindrical
region)
• Code: FLUENT
R
(Fluent Inc.,
Lebanon, NH): 6.1
61
and 6.2
62
• Isotropic spherical particles with
volumetric shrinkage and
negligible internal temperature
gradients
• Neglected: heat and mass transfer
between particles; heat losses
• Drying kinetics: constant-rate
period coupled with falling-rate
period (diffusion)
• Transition between these periods
(drying rate in second period
equals that in first period)
• Drying model incorporated into
CFD code by user-defined
function
• Grain drying
61,62
• Validation based on experimental
data of gas–solid flow
• Drying of microspherical porous
particles in a CSB with
draft-tube: computer simulations
predict well the mass transfer
rates, underpredict heat-transfer
rates
61
• Drying simulations for a CSB:
considering decrease in particle
density along drying and its
effects on the gas flow
62
• Same considerations as those
in the preceding row
• Code: COMSOL
Multiphysic
R
(Comsol AB,
Sweden) – 3a with chemical
engineering and heat transfer
modules
8
• Regular cubic particles
(three-dimensional)
• Negligible effect of water
evaporation on air properties
• Linked: heat, mass transfer, and
shrinkage of particle
• Empirical model of changes in
particle volume coupled with
particle moving boundary
model
• Drying of carrot cubes in a SFB
8
• Computer simulations
underpredict heat transfer rates
(need to double the convective
heat transfer coefficient predicted
by SB correlation to achieve
highest accuracy)
8
• Product quality: particle
deformation vs. changes in drying
temperature
8
11.4 Computational fluid dynamic models
Focusing on the 2B-level models, various commercial CFD computer packages have
been used recently to describe and analyze the drying of particulate solids. Wojciech
59
critically reviewed the key aspects of developing a CFD simulation model of a spouted-
bed grain dryer. Table 11.3 summarizes some of the more important assumptions and
results obtained in the SB dryer of particulate solids.
Chapter-specific nomenclature
W
p
solids circulation rate, kg of dry solids/s
Y
g
gas humidity, kg water vapor/kg dry air
Y
p
particle or solids moisture content, kg water/kg dry solids
Subscripts
g gas
in inlet
202 Maria Laura Passos, Esly Ferreira da Costa Jr., and Arun Sadashiv Mujumdar
it spout–annulus interface
lost losses
mf minimum fluidization
out outlet
R reference
w water
0 initial
Superscripts
cr critical
eq equilibrium
sat saturated
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12 Drying of solutions, slurries,
and pastes
Jos
´
e Teixeira Freire, Maria do Carmo Ferreira, and F
´
abio Bentes Freire
12.1 Introduction
In a system consisting of a liquid and solids dispersed within it, such as slurries and
pasty materials, the structure and characteristic properties are defined by the solids
concentration, shape, and size distribution. A variety of solid–liquid mixtures, such as
suspensions, dispersions, sludges, and pulps, is included in these categories.
1
Solutions
with a solute that crystallizes out on evaporation are also included. The drying of
solutions in beds of inert particles of spoutable size was developed in the late 1960s
at the Lenigrad Institute of Technology
2
for applications in which the dried solids are
ultimately required in the form of a fine powder. Spouting with inert particles was
applied successfully by the Leningrad group to dry organic dyes and dye intermediates,
lacquers, salt and sugar solutions, and various chemical reagents.
2
Since then, a large
number of materials have been successfully dried, demonstrating the applicability and
versatility of this method. A partial list of dried materials is given in Table 12.1.
12.2 Drying process
12.2.1 Description
The drying of pastes (or slurries or solutions, all referred to henceforth in this chapter
generically as pastes) is performed in the presence of inert particles, which are both a
support for the paste and a source of heat for drying. The paste may be atomized or
dropped into the bed by a nozzle or dropping device. An example of an experimental
facility for research in paste drying is shown in Figure 12.1. Usually, the paste is
introduced at the bed’s top surface, as there is evidence that feeding at this position leads
to stable operation with a low accumulation of the product in the bed.
3
However, reports
of feeding at the entrance region are also available.
4–6
As the bed becomes wet, the paste coats the inert particle surfaces, forming a thin layer
of slurry material. The sources of heat for drying the layered material are conduction from
the particles and convection from the hot air. The drying process is well understood: The
coating layer gradually dries and becomes fragile and friable. As moisture is reduced,
Spouted and Spout-Fluid Beds: Fundamentals and Applications, eds. Norman Epstein and John R. Grace.
Published by Cambridge University Press.
C
Cambridge University Press 2011.