Epstein N., Grace J.R. Spouted and Spout-Fluid Beds: Fundamentals and Applications

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Drying of solutions, slurries, and pastes 207

Table 12.1. Solutions, slurries, and pastes dried in spouted beds with inerts.

Material

Spouted bed

conﬁguration Inerts

Animal blood

CcSB;

D = 1.04 m

Polypropylene beads,

= 3−4 mm; ϕ = 0.80

Solutions of cobalt carbonate and zinc carbonate

MSB;

D = 0.14 m

Glass spheres,

= 6.00 mm

Animal blood, hydroalcoholic vegetable solutions

CcSB;

D = 0.14 m

Polypropylene beads,

= 3.9 mm

Low-fat milk; corn ﬂour suspensions and starch

factory efﬂuents

CCSB;

D = 0.30 m

Celcon beads,

= 2.5 mm

Solutions of antibiotics; baker’s yeast

JSB;

D = 0.48 m

PFTE cubes;

= 4mm

Sodium chloride solutions and alumina suspensions

CCSB;

D = 0.15 m

Glass and plastic beads,

= 2–5 mm

Ferric hydroxide–zinc hydroxide sludge

HCCSB;

D = 0.154 m

Sand,

= 2.36 mm

Microparticle slurry and salt-water solution

CCSB;

D = 0.104 m

Silica sand; alumina,

= 460–1000 μm

Suspensions of maltodextrins; starch; aqueous

carbohydrate solutions

CCSB;

D = 0.60 m

Polypropylene chips,

= 3.4 mm; ϕ = 0.9

Microparticle slurries

CCSB;

D = 0.104 m

ılica sand,

= 650 μm

Xanthan gum dispersions

2DSB;

0.05 × 0.3 m

Glass spheres,

= 3.8mm

Modiﬁed pulps of tropical fruits

CcSB;

D = 0.18 m

Polyethylene particles,

= 3.0 mm; ϕ = 0.76

Homogenized egg paste; animal blood; xanthan gum

solutions

CcSB;

D = 0.30 m

Glass spheres,

= 2.6mm

Black liquor

CcSB;

D = 0.15 m

and 0.30 m

Glass spheres,

= 3mm

Polypropylene beads,

= 4.22 mm; ϕ = 0.86

Extract of Maytenus ilicifolia leaves

JSB;

D = 0.34 m

Teﬂon beads,

= 5.45 mm; ϕ = 0.96

Homogenized egg paste

CcSB;

D = 0.30 m

Glass spheres,

= 2.8mm

Vegetable starch solutions

JSB; V = 0.035 m

(lab.) and V =

0.660 m

(pilot)

Teﬂon cubes,

L = 5mm

AlO(OH) suspensions, tomato concentrates, bovine

serum albumin

MSB;

D = 0.145 m

PFTE particles,

Size n.i.

Soy residue

JSB;

D = 0.20 m

Silica gel particles,

= 3.33 mm

Carbonate solutions; low fat milk; sewage sludge

CcSB;

D = 0.30 m

Glass spheres,

= 2.2mm

JSB (jet spouted bed); CcSB (conical s.b.); CCSB (conical-cylindrical s.b.); MSB (mechanical s.b.); HCCSB

(half-conical-cylindrical s.b.).

208 Jos

e Teixeira Freire, Maria do Carmo Ferreira, and F

abio Bentes Freire

1112

blower

valves

flow meter

air heater

temperature controller

thermocouples

samplers

spouted bed

paste feeding

cyclone

wet bulb thermocouple

dry bulb thermocouple

air compressor

peristaltic pump

paste tank

pressure transducers

signal conditioner

data acquisition board

PC computer

Figure 12.1. Experimental setup in paste drying experiments.

the coating reaches a critical level that allows it to be removed as a powder by the

successive collisions to which the particles are subjected. The powder is elutriated and

collected by a separation device, such as a cyclone. As the paste is fed continuously, the

stages of spouting, layer formation, drying, ﬁlm fracture, and powder elutriation occur

simultaneously. The time required for a complete cycle depends on the paste rheological

characteristics, adhesion features, attrition rates, and other factors. The drying and layer

removal rates must be high enough to avoid agglomeration from accumulation of paste

in the bed.

12.2.2 Analysis

For pastes of high moisture content, water evaporation is the limiting stage of drying,

which occurs at a constant rate. For the stages involving layer adhesion and subsequent

breaking, the frequency of collisions and the drying rate are the major factors inﬂuencing

the process.

The energy of collisions is affected by a number of variables, including

the solid circulation rate, the mass ratio of paste to inert par ticles, and the drying rate

itself. The inert solids circulation rate governs the time required for a complete cycle of

coating–drying–removal of ﬁlm from a particle. An increase in the drying rate favors an

increase of ﬁlm friability, enhancing the process. Solids circulation rates, mass of inert

particles, and drying rates are parameters related to spouting operating conditions, so it

might be useful to analyze the paste drying process from different perspectives.

The interaction between the ﬂuid and a particle moving through the bed differs with

the distinct regions of a spouted bed – spout, annulus, and fountain – each of which has a

very different distribution of air and particles and a different velocity proﬁle. Typically, a

particle leaves the annular region and enters the spout. Its velocity, initially close to zero,

rapidly increases as it is entrained by the air rising through the spout. The gas inlet region

is characterized by intense momentum transfer, similar to the solids accelerating region

Drying of solutions, slurries, and pastes 209

in a pneumatic riser. As the spout is very dilute, possible interactions among particles

may be neglected and a particle may be analyzed as a single element in contact with

the ﬂuid, subject to weight, buoyancy, and drag forces. Adhesion forces, typically of a

physical–chemical nature and acting on a molecular scale, keep a ﬁlm of paste bound to

the particle surface. The particle and its coating layer experience strong shear from the

airﬂow. The conditions at the interface of a coated particle and ﬂuid are established by the

rheological characteristics of the paste and by the complex interactions at the boundary

layers around the particles. Even in a simpliﬁed analysis, considering a single particle

in the ﬂow, the magnitude of the shear stress depends on many factors, including the

air velocity at the entrance of the spouted bed and the paste rheological properties. The

rheological properties, in turn, are likely to be time-dependent, changing as the drying

proceeds. After short periods, as the material dries, the ﬁlm properties and rheological

characteristics change considerably. Evidence from experimental data suggests that, in

steady-state operation, the moisture content of the ﬁlm coating the particles is only

slightly higher than the moisture content of powder collected at the cyclone exit.

16,19,23

A r igorous analysis of this situation is extremely complex and has not been satisfactorily

addressed.

As the particle moves up the spout, it decelerates toward and within the fountain.

Eventually, as the gravitational force predominates, the particle falls back onto the

surface of the annulus. In this region, the structure is similar to that of a packed bed,

and the particle is in direct contact with others. In the early stages of drying, liquid

bridges may form because of paste surface tension, contrib uting to enhanced particle–

particle interactions. The whole bed of wet particles descends toward the conical base,

from which the particles are again propelled into the spout, and a new cycle starts. As

the drying proceeds and the moisture content on the ﬁlm coating decreases, the liquid

bridges are likely to be replaced by other interaction forces.

An aspect that deserves closer inspection relates to the adhesion forces at the interface

between the coating and inert particles. From theoretical work on adhesion, it is known

that the magnitude of adhesion forces, resulting from chemical, mechanical, or molec-

ular interactions, depends crucially on the substrate properties.

Although these inter-

actions are extremely complex and will not be approached in depth here (see references

elsewhere

24,25

), it is interesting to consider a few points. The substrate for adhesion is

provided by the spouting particles and the interaction at this level is expected to affect the

processes of coating and removal of layered material. As an example, it is known that solid

surfaces – even the glass beads commonly used in spouting and often referred to as

smooth – usually display some degree of surface roughness. The surface roughness con-

tributes to increased contact area, which may enhance signiﬁcantly the strength of the

interface for a speciﬁc substrate–coating pair. Chemical composition, viscosity, surface

tension, and other rheological properties of the coating material also play a role. How-

ever, as the rheological behavior changes with the drying, a given paste may become

sticky or friable, or present other features that affect the interaction forces. As a result, the

ﬁlm removal process might differ for each coating–substrate pair. Detachment of layers

of food pastes dried over glass substrates, for instance, was reported to depend on the

glass transition temperature of the paste.

Even this superﬁcial analysis indicates that

210 Jos

e Teixeira Freire, Maria do Carmo Ferreira, and F

abio Bentes Freire

the term inert may not always be accurate when applied to the substrates. The complex

interactions involved in the interfaces of different substrates and coatings have not yet

been investigated in depth.

For effective drying of a given paste on inert particles, the rate at which the ﬁlm is

coated and removed must be higher, or at least equal to, the rate at which the paste is fed

into the bed.

To reach steady-state operation, the rates of ﬁlm coating and removal must

be equal. The ﬁlm layer over a particle is expected to be thin enough to allow operation

at a constant drying rate, with no signiﬁcant moisture diffusion resistance through it.

As a particle moves along the distinct regions of the spouted bed, the coating dries and

is removed, and the resulting powder is elutriated. The ﬁlm removal depends on the

magnitude of the adhesion force. Powder removal efﬁciency is associated with friability,

which depends on the frequency of collisions and on the properties of boundary layers

around the particles. In addition, it is affected by the paste moisture content, temperature,

and chemical composition. The shear stress acting on coated particle surfaces is affected

by the paste composition and properties, air temperature, and particle slip velocity. The

number of effective collisions is controlled by such variables as the solid circulation

rate, mass of coated material on each particle, and drying rate.

12.2.3 Drying rate

The key response variables of a drying process – air outlet relative humidity and tempera-

ture, and powder moisture content – are closely related to the drying rate, which depends

on previously mentioned characteristics associated with the internal bed structure. The

steady-state evaporation rate (W ) is a function not only of the paste characteristics but

also of spouted bed geometry and conﬁguration, quantity and properties of inert parti-

cles, and all other independent variables of the process. Research efforts have focused

on obtaining the maximum evaporation rate for a speciﬁc set of independent variables

air

, T

, H

, P) and a combination of feed material/dryer/inert particles. Water is com-

monly adopted as feed in preliminary experiments to allow the maximum evaporation

capacity of a system to be estimated. When tested under similar conditions, every other

feed material yields evaporation rates that are lower than, or equal to, those obtained

using water.

6,23

By drying different materials in a conical spouted bed at T

= 100

◦

and U = 1.30U

, Almeida

obtained similar values of W

max

and air outlet relative

humidity for water, sewage sludge suspension, low-fat milk, and 3 percent calcium car-

bonate solution. When drying homogenized whole egg paste at similar conditions, bed

collapse was observed at values of W

max

25 percent lower and air outlet relative humidity

70 percent lower than their respective values for water.

Experimental investigations

6,23

showed that W

max

is increased signiﬁcantly by increas-

ing U and T

and increased only slightly by increasing H

. Pham

proposed a simpliﬁed

model that allows W

max

to be estimated as a function of air outlet temperature and

humidity. In this model, the maximum evaporation rates depend linearly on the air outlet

temperature.

In conical spouted beds operated at constant H

, it was veriﬁed

that the use of a 60

◦

cone angle r esulted in higher W

max

compared with beds with 45

◦

and 30

◦

angles. Results

Drying of solutions, slurries, and pastes 211

are deceptive, however, as the authors observed more intense solid circulation rates in

the beds with smaller cone angles, probably a result of the simultaneous variation of

cone angle and quantity of inert particles required to maintain the same H

for different

beds, so the quantity of inerts varied in the experiments. Laboratory assays with equal

quantities of particles yielded no conclusive results about the effect of cone angle on

solid circulation rates and W

max

There is insufﬁcient information in the current literature to provide a full understanding

of how all such variables affect W

max

12.3 Inﬂuence of paste on the ﬂuid dynamic parameters

Pham

was one of the ﬁrst authors to report clear evidence on changes in dynamic

behavior of spouted beds operating with pastes. In the drying of animal blood, he

observed the formation of dead zones that had not appeared in the operation of dry beds.

Since then, several studies have been carried out to identify the effects of the presence

of a liquid or slurry on spouted bed dynamics. These experiments have been conducted

either in batch mode or continuous operation. Changes in rheological, physical, and

chemical properties commonly observed during dr ying often led to unstable spouting, so

experimental data could be obtained for only a narrow range of experimental conditions.

This problem has been overcome by employing pure liquids, such as water or glycerol,

to keep the beds wet, thus simulating to some degree the presence of a paste. Signiﬁcant

contributions reporting ﬂuid dynamic behavior of wet beds are listed in Table 12.2.

These references provide important evidence that the presence of a liquid or paste does

indeed alter the ﬂuid dynamic parameters, as it engenders interparticle cohesive forces

that may change the solid and ﬂuid circulation patterns. The effects of paste on ﬂuid

dynamic parameters are discussed in the following sections.

12.3.1 Pressure drop for stable spouting

Most researchers listed in Table 12.2 found that the minimum spouting pressure drop

in wet beds is less than that in dry beds (P

ms0

), regardless of whether the feeding is

in the batch or continuous mode. The reduction of P

is attrib uted to the decrease in

airﬂow rate through the annulus

4,28

because the liquid or pasty material contributes to

particle agglomeration and to increased resistance to ﬂow in this region, forcing more

air toward the spout region. The stronger interaction forces in the annulus also contribute

to reducing the number of particles leaving this region toward the s pout, thus reducing

solid circulation rates, a feature that also contributes to lower pressure drop.

Passos and Mujumdar

investigated the effects of cohesive forces on the spouting of

wet particles in a two-dimensional column (see Table 12.2) and evaluated the dynamic

parameters for different liquid contents by changing the amount of glycerol injected into

the bed. They observed that, as the bed was operated at a low liquid content, P

the wet bed was initially greater than for the dry bed. As the liquid content increased,

P

dropped to values as low as 50 percent of the dry bed value (see Figure 12.2).

212 Jos

e Teixeira Freire, Maria do Carmo Ferreira, and F

abio Bentes Freire

Table 12.2. Studies investigating the effect of paste on spouted bed dynamics.

Paste or liquid Bed dimensions Inert materials Variables investigated

Glycerol, distilled water

(BF)

CCSB; θ = 60

◦

D = 0.15 m

= 0.02 m

Glass spheres

Acrylic resin spheres

PVC cylinders

Irregular particles

P

= f (V

)

= f (U

)

= f (V

, T )

Glycerol, water (BF)

CCSB; θ = 65

◦

D = 0.15 m

= 0.02 m

Glass spheres

PVC cylinders

Acrylic resin spheres

P

/P

ms0

= f (V

,μ)

= f (V

, t )

ms0

= f (V

)

= f (t)

Animal blood; homogenized

chicken egg; water (CF)

CcSB; θ = 60

◦

D = 0.3m

= 0.05 m

Glass spheres P

/P

ms0

= f (Q

)

ms0

= f (Q

)

Glycerol (BF)

2DSB

= 0.2m

= 0.015 m

Plastic pellets P

/P

ms0

= f (V

)

ms0

= f (V

)

Glycerol; homogenized egg

paste (CF, BF)

16,19

CcSB; θ = 60

◦

D = 0.3m

= 0.05 m

Glass spheres P

/P

ms0

= f (s)

ms0

= f (s)

Modiﬁed and in-nature

mango pulps (BF)

CCSB; θ = 60

◦

D = 0.13 m

= 0.03 m

Polyethylene particles P

/P

ms0

= f (t)

0 = f (t)

ms0

= f (V

)

Sewage sludge, egg paste,

carbonate solutions

(BF)

CcSB; θ = 60

◦

D = 0.47 m

= 0.05 m

Glass spheres P

/P

ms0

= f (s, t)

f 0

= f (s, t)

ms0

= f (s, t)

(BF) batch feeding of paste; (CF) continuous feeding; 2DSB two-dimensional spouted bed

1.0

0.9

0.8

0.7

0.6

0.5

0.0 0.5

Water

Egg paste

Xanthan gum 1%

1.0 1.5 2.0 2.5 3.0

Feed rate (L/h)

/ DP

mso

Figure 12.2. Variation of P

with feed rate (based on data from Spitzner Neto et al.

This behavior agrees with data from other authors

30,31

and is attributed to differences

in the nature of interaction forces developed by the liquid bridges as the liquid content

increases.

At low liquid content, weak liquid bridges are formed and the liquid acts as a lubricant,

increasing the solid circulation rate and the pressure drop at minimum spouting. As

Drying of solutions, slurries, and pastes 213

the liquid content increases and approaches a critical value (which is a function of

inert material and liquid characteristics), the thickness of the coating layer formed

over the particles grows, and stronger liquid bridges are formed. The mean voidage

of the annulus is reduced, as well as the solids circulation rate, causing a decrease in

pressures drop. Bacelos et al.

measured a small increase in P

at low liquid contents

in batch feeding experiments (with glycerol) and continuous feeding experiments (with

homogenized egg paste). As the variation was within the range of experimental accuracy,

however, the authors assumed that P

was not signiﬁcantly affected by the paste.

Physical and rheological characteristics of the paste also affect P

, as illustrated in

Figure 12.2, which shows experimental data obtained for water, homogenized egg paste,

and xanthan gum solutions.

Almeida

observed that the P

behavior may change

as a result of varying the concentration of a given paste. By operating a spouted bed

with carbonate solutions of different concentrations, the author showed that the pressure

drop decreased for solutions with concentration close to 3 percent, but increased for

9 percent concentrated solutions. At intermediate carbonate concentrations of 5 percent,

both types of behavior were observed, depending on the feed rate. Higher feed rates

usually led to a decrease in pressure drop, behavior attributed to spouting instabilities

generated by liquid accumulation in the bed.

12.3.2 Minimum spouting velocity

Results obtained for minimum spouting velocity of wet beds are contradictory, par-

ticularly for beds containing glycerol. Schneider and Bridgwater

identiﬁed an initial

increase in U

at low glycerol contents and a decrease after spouting with a maximum

glycerol content. Some researchers

16,19,28

measured a reduction in U

as the glyc-

erol content increased, as a result of the lower solids circulation rates observed in wet

beds, as discussed previously. With lower solids circulation rates, a smaller air velocity

is required to maintain spouting. However, in a “two-dimensional spouted bed” (see

Chapter 17) operating with glass beads and polypropylene particles, Passos and

Mujumdar

reported an increase of U

with glycerol content. A similar result was

reported by Santana et al.,

who operated a half-column conical spouted bed. The

different bed geometries employed by those authors may be partly responsible for

these apparently contradictory results; additional research is needed for a conclusive

analysis.

For liquids and pastes other than glycerol, several authors reported a decrease in

as the paste content increased.

16,19

These authors mentioned that the dynamic

behavior of conventional pastes is very different from that in beds wetted by water or

even glycerol (see Figure 12.3). The distinct behavior may be attributed to differences

in interaction forces on the particles: whereas only liquid bridges are formed with

glycerol, conventional pastes generate different kinds of bridges, associated with powder

formation as the coating material dries.

As already mentioned, experimental results

indicate that the moisture content in the ﬁlm of paste coating the spouted particles is

very similar to the moisture content of the collected powder,

16,19,23

thus corroborating

that the particles at that stage are not linked by liquid bridges.

214 Jos

e Teixeira Freire, Maria do Carmo Ferreira, and F

abio Bentes Freire

1.14

1.12

1.10

1.08

1.06

1.04

1.02

1.00

0.98

0.96

0.94

0.92

0.90

0.00 0.02 0.04

ms0

s (-)

0.06

Glycerol

Egg paste

0.08 0.10 0.12 0.14 0.16

Figure 12.3. Variation of U

with saturation degree (based on data from Spitzner Neto et al.

1.2

1.1

1.0

0.9

0.8

Pulp composition (kg/kg water)

n* Sugar

2 2.73

1.05

water

1.51

1.51 4.10 1.06 1.06

1.06

1.05 1.51 1.04 1.06

1.04 1.06

2.89

Fat Fiber Starch Pectin

46 810

Time (min)

Figure 12.4. Effect of paste composition on fountain height (based on data reported by

Medeiros

12.3.3 Fountain height

In dry beds, the fountain height increases as the airﬂow rate is raised. In wet beds, stable

spouting is not achieved at short fountain heights owing to the stronger interparticle

forces, which require greater airﬂow rates to maintain spouting.

4,28

In tests with glycerol,

Spitzner Neto

veriﬁed experimentally that the fountain height increased as the amount

of liquid fed to the bed increased, maintaining a spheroidal contour and linear dependence

on the airﬂow rate. Medeiros et al.

carried out drying experiments for pulps of different

tropical fruits, in batch operation, a condition for which both the paste rheology and the

bed saturation changed with time. They compared fountain heights measured with time

for dry and wet beds and observed a marked dependence of fountain height on pulp

composition (see Figure 12.4).

Drying of solutions, slurries, and pastes 215

From the previous discussion, it is clear that the presence of paste inﬂuences such

hydrodynamic parameters as P

, U

, and H

. Paste also affects bed voidage, perme-

ability, solid circulation rate, and other ﬂuid dynamic parameters. Despite the evidence

from many experimental studies, however, it is still not possible t o predict reliably the

effect of the liquid on the ﬂuid dynamics for a given paste. A severe shortcoming is that

the modeling of the ﬂuid dynamics of wet spouted beds has so far not been successful

in effectively incorporating the paste characteristics into the description of the spouting

dynamics. Advances in modeling of the drying process on inert particles are considered

next.

12.4 Modeling of drying

Modeling of ﬂow in a spouted bed commonly considers the three different regions

(spout, annulus, and fountain) separately, to take into account their distinct structures.

It is difﬁcult to describe the complex circulation dynamics and particle–paste–air inter-

actions. Modeling attempts have relied on three main approaches: (1) totally empirical,

(2) based on differential or integral balances applied over each bed region, and (3) based

on analysis of interparticle forces.

Models based on dimensionless g roups arranged empirically, aided by physical or

statistical procedures, may be useful for preliminary investigations and to enhance

physical understanding of the phenomenon under study. They have not been consistently

applied to model the drying of pastes in s pouted beds and are not treated here. An

example of such an application may be found elsewhere.

12.4.1 Models derived from heat and mass transfer balances

Models based on differential or integral balances are listed in Table 12.3. The assump-

tions adopted are not always physically consistent, but some authors have shown that by

arbitrarily ﬁtting thermal and mass parameters, it is possible to obtain good predictions

for drying rates, outlet air humidity, and outlet air temperature.

The models originally proposed by Kmiec

and Pham

were implemented by

Oliveira

to describe water evaporation and drying of animal blood in a conical bed of

D = 300 mm. Modiﬁed versions of the original models were also proposed, including an

additional term to take into account thermal losses through the column walls. In two of

these models it was assumed that evaporation occurs mainly in the annulus,

a hypoth-

esis contradicted by experimental data, as, according to Oliveira, only about 40 percent

of evaporation occurred in this region.

Nonetheless, good predictions of evaporation

rates and humidity at the bed exit were obtained with both Pham’s original model and the

modiﬁed version, with deviations within ±10 percent. The model of Kmiec

yielded

larger deviations. Only t he modiﬁed version of Pham’s model provided good estimates

of gas outlet temperature, with deviations in the range of ±10 percent.

The mass and

thermal parameters in the model were ﬁtted from experimental data obtained in the

same equipment. Thermal parameters were ﬁtted from experiments on heat transfer in

216 Jos

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abio Bentes Freire

Table 12.3. Models describing paste drying in spouting with inert particles.

Model Model characteristics Assumptions and shortcomings

Kmiec

Global model

(based on drying of

particulate solids)

– Every par ticle is coated with an identical mass of paste

– Heat exchange only between the gas and moist coating

– Negligible thermal losses

Pham

Based on application of

enthalpy balances for

gas and par ticles

– Drying occurs only in the annulus

– Paste does not affect bed dynamic patterns

– Negligible thermal losses

Barrett and Fane

Qualitative model – Paste does not affect bed dynamics

– Requires a very complex mathematical analysis

Oliveira and Passos

Bed is divided into n

segments, and mass

andenergybalances

are applied to each

– Evaporation and convective heat transfer occur only in

the annulus

– Bed dynamics are not affected by the paste

– Inert particles have no ﬁlm coating as they enter the

spout

– Spout air has constant humidity, which is equal

to that at the entrance

Almeida

Model applies global

mass and energy

balances with a lumped

approach

– Bed is modeled as a well-mixed vessel

– Each paste requires a ﬁtted equation to describe drying

kinetics

the dry bed. Mass transfer parameters were obtained from experiments involving water

evaporation and drying of animal blood and 10 percent concentrated Al(OH)

suspen-

sions. The Lewis analogy for heat and mass transfer was veriﬁed for the experimental

conditions investigated. Because the model has not been conﬁrmed using parameters

estimated from literature correlations, there is insufﬁcient evidence that Pham’s model

can be generalized to other experimental conditions.

Almeida

applied lumped heat and mass transfer balances to predict gas outlet

temperatures and air humidities with time in transient drying of pastes in a spouted

bed of inert particles. The ﬁtted kinetic equations varied according to both the type of

paste and its feed rate. This lumped approach provided very good predictions for these

variables, but it also depended on a ﬁtting procedure for the drying rate, which must be

performed individually for each paste.

12.4.2 Models derived from analysis of interparticle forces

Interparticle forces are extremely important in the analysis of dynamic behavior of inert

particles in spouted beds containing pastes. In a review of these forces in wet ﬂuidized

beds, Seville et al.

have mentioned the presence of Van der Waals forces, as well as

forces created by liquid bridges and sintering. The size of spouted bed particles is usually

in the range of 1 t o 6 mm. For such large particles, Van der Waals forces are not expected

to be signiﬁcant, as compared with the other two. Liquid bridges exhibit both dynamic

and static forces and are dissipative of energy. The static liquid bridge force results from

the sum of the surface tension force and the force arising from the pressure deﬁcit in