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Energy and exergy analysis ofuidized bed dryer based on two-uid modeling

M.R. Assari a,*, H. Basirat Tabrizi b, E. Najafpour c

a University of Jundi Shapor, Dezful, IranbAmirkabir University of Technology, Mech. Eng. Dept., Tehran, Iranc Mech. Eng. Dept., Dezful Branch, Islamic Azad University, Dezful, Iran

a r t i c l e i n f o

Article history:

Received 24 July 2010

Received in revised form

11 November 2011

Accepted 30 August 2012

Available online 12 October 2012

Keywords:

Batchuidized bed dryer

Gasesolid ow

Two-uid model

Exergy

a b s t r a c t

Energy and exergy analysis for batch

uidized bed dryer based on the Eulerian two-

uid model (TFM) isperformed to optimize the input and output and keep the quality of products in good condition. The two-

uid model is used based on a continuum assumption of each phase. Two sets of conservation equations

are applied for gasesolid phases and are considered as interpenetrating continuum. Further this study

considers the two-dimensional, axis-symmetrical cylindrical energy and exergy equations for both

phases and numerical simulation is preformed. The governing equations are discretized using a nite

volume method with local grid renement near the wall and inlet. The effects of parameters such as: the

inlet gas velocity, inlet gas temperature and the particle size diameter on the energy, exergy efciencies

and the availability of gas are sought. Two-uid model prediction indicates good agreement between the

available experimental results and reported non-dimensional correlations and other model predictions

It is illustrated that at the beginning of the drying process, the energy efciency is higher than the exergy

efciency for a very short time. However two efciencies come closer to each other at the nal stage o

the drying. Increasing particle size will decrease both efciencies and the gas availability at the starting

process.

2012 Elsevier Masson SAS. All rights reserved

1. Introduction

Particle drying is an important process in food, pharmaceutical

and chemical industries, which consume signicant amount of

energy. A large number of independent variables such as particle

density, size, shape, permeability, and hygroscopicity can inuence

drying behavior. Fluidized bed drying is one of the most successful

methods. In uidized bed dryer most particles are suspended in

a hot air or stream. Fluidized bed drying, compared with other

drying techniques, offers many advantages such as higher heat and

mass transfer rates due to better contact between particles and gas,

uniform bed temperature due to intensive solid mixing and ease in

a control of the bed temperature and operation. Collectively, their

advantages result in higher drying rates. In uidization phenom-

enon, particles ow in a uid and particle-uid will inuence each

other, and cause a complex phenomenon. Most of this process

occurs in conjunction with other processes such as heat transfer,

mass transfer or heat and mass transfer together.

The uidized bed models can be classied into two broad

groups: engineering models such as two-phase, three-phase

models[1,2]and CFD based models that are based on a continuum

assumption of phases [3,4]. The engineering models comprise

a bubble phase without solids and a dense phase consisting of gas

and solid particles. The dense phase is assumed to be well mixed;

the modeling is applied to each phase separately and incorporates

experimentalndings by others (see Ref. [4]). Two methods have

been typically used for CFD modeling of gasesolid ows, namely

"EulerianeLagrangian" method and "EulerianeEulerian" approach

In the "EulerianeLagrangian" approach, the Lagrangian trajectory

for the study of motion of individual particles is coupled with the

Eulerian formulation for gas. The "EulerianeEulerian" or two-uid

used in the current study provides a eld description of the

dynamics of each phase.

Researchers have conducted several numerical studies to

describe uidized bed drying process. Palancz [5] proposed

a mathematical model for continuous uidized bed drying based on

the two-phase uidization. According to this model, the uidized

bed was divided in two phases involving a bubble and an emulsion

phase. Lai and Chen[6]proposed an improvement for the Palanczs

model. Hajidavalloo and Hamdullahpur [7,8] proposed a mathe-

matical model of simultaneous heat and mass transfer in uidized

bed drying of large particles. They employed a set of coupled

nonlinear partial differential equations based on three-phase

model representing a bubble, interstitial gas and solid phase that* Corresponding author.

E-mail address: [email protected](M.R. Assari).

Contents lists available atSciVerse ScienceDirect

International Journal of Thermal Sciences

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . co m / l o c a t e / i j t s

1290-0729/$ e see front matter 2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.020

International Journal of Thermal Sciences 64 (2013) 213e219
mailto:[email protected]://www.sciencedirect.com/science/journal/12900729http://www.elsevier.com/locate/ijtshttp://dx.doi.org/10.1016/j.ijthermalsci.2012.08.020http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.020http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.020http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.020http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.020http://dx.doi.org/10.1016/j.ijthermalsci.2012.08.020http://www.elsevier.com/locate/ijtshttp://www.sciencedirect.com/science/journal/12900729mailto:[email protected]

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describe the thermal and hydrodynamic characteristics of bed.

Syahrul et al. [9] carried out a thermodynamic analysis of the

uidized bed drying process of moist particles to optimize the input

and output conditions using energy and exergy models. Dincer and

Sahin [10] developed a model for thermodynamic analysis, in terms

of exergy of a drying process. The two-uid models of Ishii[11]was

used for packed bed dryer by Basirat Tabrizi et al. [12] further

extended for uidized bed dryer Assari et al. [13]. The governing

equations discretized using a nite volume method and compared

with the experimental results. Azizi et al. [14] considered numerical

simulation of particle segregation in bubbling gas uidized beds. Li

and Duncan[15] presented a dynamic model for batch uidized

bed dryers where a simple two-uid model was used to describe

the dynamics of a uidized bed dryer, which includes a bubble

phase and an emulsion phase consisting of an interstitial gas phase

and a solid phase.

Understanding the relation between exergy and energy and

environmental impact are important for energy cost, therefore the

main objective of this study is to conduct an energy and exergy

model to optimize the input and output conditions in uidized bed

drying. A comprehensive model to simulate energy and exergy in

bubbling uidized bed has been described in the previous studies

[e.g. Ref. [18]]. This study implements a two-uid model for the rst

time. The model simulates the drying operation for two-

dimensional cylindrical case that includes the mass and energy

conservations and exergy equation for each phase. Simulation is

carried out with nite volume method. What distinguishes this

paper from others is that the region of gas uidization is in slug

regime and the bubble phase is undistinguishable (Umf 0.9731

and U 4e5 m/s). The model predictions are compared with the

experimental results and other reported predictions. In respect to

our averaging procedure and working under slug regime in uid-

ized bed dryer, the energy and exergy results do not indicate affect

of bubbles directly. However, the bubble effect can be noticed

indirectly in terms of other criteria.

2. Modeling and analysis

In the two-uid models, two sets of equations are used for

gasesolid phases, both of which are considered as inter-

penetrating continuum. The reader should refer to Ishii [11] and

Gidaspow[4] for the fundamental theoretical formulation of two-

uid ow. In this study, those theorems are applied for obtaining

the governing equation with the volume averaging in order to

express the thermal energy and exergy for each phase where the

wet solid particles and the gas stream consider as two separate

uids. The governing equations can be summarized as follows:

Continuity equation for gas and solid phases respectively are:

v

vtrg 3gxg 1

r

v

vrrrg 3gugxg v

vzrg 3gvgxg a _m (1)

v

vtrs 3sxs

1

r

v

vrrrs 3susxs

v

vzrs 3svsxs a _m (2)

here, _m stands for the moisture evaporation from the particle

surface. If the temperature and moisture gradient inside the solid

particles are ignored, then, _mcan be expressed by Palancz[5]:

_m sp

x*pg xg

(3)

The evaporation coefcientspis dened:

sp hvrgDg

kg(4)

Moreover, the transfer coefcient for convective heat transfer

between solid and gas is given:

hv cgugrgJhPr23 (5)

Jh 1:77Re0:44ic if Reic 30

Jh 5:70Re0:78p if Reic < 30

(6)

Reic dpugrg

3gmg(7)

where the value of the moisture content of the saturated drying

medium at the surface of the solid particle, x*pg is indicated as

a function of the temperature and moisture content of the particle

and expressed as:

x*pg F1TsF2xs (8)

where,the functionscan be computed from the tension curve of the

moisture and the absorption character of the solid moisture system.The approximations were given:

F1Ts 0:622 Pw

760 Pw(9)

F2xs

8>:

1 if xs>xscxnsxnsc l

xnsc

xns l

if xs xsc (10)

Pw 10

0:622

7:5Ts238 Ts

(11)

wheren and l are constant (n 3, l 0.01).

Equation of motion for gas momentum inr-direction:

v

vt

rg 3gug

1

r

v

vr

rrg 3gu

2g

v

vz

rg 3gugvg

3g

vp

vr br

ug us

a _mug (12)

and in z-direction:

v

vt

rg 3gvg

1

r

v

vr

rrg 3gugvg

v

vz

rg 3gv

2g

3g

vp

vz

bzvg vs rg 3gg a _mvg (13)The rst and second term in the left hand side of above equa-

tions, represent the rate of accumulation and net rate of outow

across the closed surface, respectively. The right hand side terms

are considered the pressure gradient, drag, gravitational force and

the source momentum term due to the evaporation of wet solids,

respectively, and neglected the friction force due to shear stresses.

Solid momentum equation in r-direction:

v

vtrs 3sus

1

r

v

vr

rrs 3su

2s

v

vzrs 3susvs

3svp

vr br

us ug

vsrr

vr a _mus (14)

and in z-direction:

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v

vtrs 3svs

1

r

v

vrrrs 3susvs

v

vz

rs 3sv

2s

3s

vp

vz bz

vs vg

vszz

vz rs 3sg a _mus (15)

The left hand side terms on the above equations are the pressure

gradient, drag, normal solid stresses and the source momentum

term due to the solid evaporation, respectively. In the absence of

the normal components of the solid stresses, which physically

describes the solid phase pressure, the local values of the void

fraction in the uid bed become unrealistically low. Rietma and

Mutsers [16] included such a term in their solid equation of motion.

Kos[17]made measurements of such a term for sedimentation. He

found it to be small compared to the hydrostatic pressure. The

constitutive equation for the normal component of stress is

s s( 3g). Using the chain rule, in thez-direction then

vszz

vz

vszz

v 3g

v 3g

vz G

3g

v 3gvz

(16)

in ther-direction

vsrr

vr

vsrr

v 3g

v 3g

vr G

3g

v 3gvr

(17)

Moreover, G( 3g) proposed by Rietma and Mutsers [16] is as

follows:

G

3g

108:76 3g5:43 (18)

In the equations of motion, br and bz are the drag coefcients

between the gas and the solid particles. The drag coefcients

become

bz 150

32gmg

3g

dpFs

2 1:75rgvg vs 3gFsdp for 3< 0:8 (19)And

bz 3

4CDz

3g 3s

vg vs

rgdpFs

32:65g for 3 0:8 (20)

3g 3s 1 (21)

whereCDzthe drag coefcient inz-direction, is related to Reynolds

number, refer to[4]

CDz 24Resz1 0:15Re0:687sz Resz < 1000 (22a)

CDz 0:44 Resz 1000 (22b)

where

Resz 3grg

vg vs

dp

mg(23)

Furthermore, the expression for the friction coefcient in the

radial direction is the same as that in the axial direction.

Thermal energy equation for the gas phase and solid phase are

described by:

v

vt

3grgIg

1

r

v

vr

r 3grgIgug

v

vz

3grgIgvg

1

r

v

vr

rkg 3g

vTgvr

v

vz

kg 3g

vTgvz

ahv

Ts Tg

a _m

cwgTg g0

(24

vvt

3srsIs 1rvvr

r 3srsIsus vvz

3srsIsvs

1

r

v

vr

rks 3s

vTsvr

v

vz

ks 3s

vTsvz

ahv

Tg Ts

a _m

cwgTg g0

(25

The enthalpy of gas and solid containing moisture can be

expressed, respectively, as:

Ig

cg xgcwv

Tg (26

Is cs xscwTs (27

The term involving work of expansion of void fraction is

neglected here. The two terms on the right hand side of Eqs. (24

and (25) involve the energy terms due to conduction, the third

term is due to evaporation exchange term and the last is the

exchange of energy due to convection.

Multiplying the entropy equation by T0 and subtracting the

resulting expression from the energy equation, exergy equation can

be derived. Thus, exergy equation for gas and solid phase

respectively are:

v

vt

3grg

Ig T0Sg

1

r

v

vr

r 3grgug

Ig T0Sg

v

vz 3grgvg

Ig T0Sg

1

T0Tg

1

r

v

vr

rkg 3g

vTgvr

1 T0Tg

v

vz

kg 3g

vTgvz

1 T0Tg

ahv

Ts Tg

1

T0Tg

a _m

cwgTg g0

T0 _S

g

gen

(28

v

vt 3srsIs T0Ss

1

r

v

vrr 3srsusIs T0Ss

v

vz 3srsvsIs T0Ss

1

T0Ts

1

r

v

vr

rks 3s

vTsvr

1

T0Ts

v

vz

ks 3s

vTsvz

1

T0Ts

ahv

Tg Ts

1

T0Tsa _mcwgTg g0 T0

_Ssgen

(29

The entropy of gas and solid containing moisture can be

expressed, respectively, as follows:

Sg cglnTgT0

RglnPgP0

xg

cvwln

TgT0

RvwlnPvwP0

(30

Ss cs xscw lnTsT0

(31

The energy efciency of the dryer can be dened as the ratio o

energy used to evaporate water from the particle to enthalpy

available (incorporated) in the drying air. Thus energy efciency is

given as:

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he a _m

cvwTg g0

_maca

Tg1 T0

Dt

(32)

And _ma is mass ow of inlet air.

Furthermore the product as the rate of exergy evaporation and

the fuel consumption as the rate of exergy, so the exergy efciency

for the particle based on the exergy rate balance can be written as:

hex

1

T0Tg

a _m

cwgTg g0

_Eg1

(33)

Where, in Eq. (33), _Eg1 is the exergy of the inlet gas ow and the

specic exergy can be obtained (see Dincer and Sahin[10]):

eg1

cg xg1cwv

T1 T0 T0

cg xg1cwv

ln

Tg1T0

Rg xg1Rwvln

Pg1P0

T0

"Rg xg1Rwv

ln

1 1

:

6078x

0

g1 1:6078x1g

! 1:6078xg1Rglnx

1

gx0g

#(34)

3. Numerical procedure

The drying process of wheat particles in a 0.15 m i.d., 1.2 m riser

height in a uidized bed is simulated numerically based on the

experimental results of Assari et al. [13]. Two-dimensional, axis-

symmetrical cylindrical equations, supplemented with the consti-

tutive equations and initial and boundary conditions are solved by

nite volume method using a variable mesh size. The momentum

equations of uid bed drying are simulated with the Simpler

Algorithm in order to obtain the velocity and voidage proles. The

exergy, energy and mass transfer equations are solved with anupwind, ADI scheme. It is assumed that the particles will move

downward after collision with the wall. So the higher agglomera-

tion of particles is observed in the walls and inlet of the bed. Non-

uniform grid generation is used that is much ner near the wall and

in the entrance as shown in Fig.1. The meshsize istakento be3 mm

near the wall and 8 mm at the center of the bed. The height of the

mesh is 20 mm at the entrance of the bed and 50 mm at the top of

the bed. The computational domain consists of 10 grids in radial

and 25 grids along the axis of the bed.The convergence criterion for

time zone is specied 104 for relative error between successive

iterations.

Generally, the ow velocities for the gas on the wall surface are

zero in all directions. However, this is not completely true for the

solid particles. Normally a rigid particle strikes it and reboundseither fully or partially. Hence, it is assumed that the particles have

a zero normal velocity at the wall. For the tangential direction along

the wall surface, the particles with the same momentum will move

downward on the walls and no circulation adjacent to the wall is

considered.

4. Results and discussion

In this study the simulation was preformed for wheat. Since

wheat is one of the main commodities of agriculture and has

extensive application in drying systems. Although some other

agricultural materials like corn grains are bigger than the wheat

grains. However this will cause a different pattern in uidized bed

as well as energy and exergy efciency. Thereforewheat with initial

temperature of 25 C is used for drying in uidized bed dryer. It is

assumed to be spherical with an average diameter of 3 mm, density

of 1200 kg/m3

and heat capacity of 1260 kJ/kg

C. The initial andcritical moisture content of solid wheat are at 0.25 and 0.2 (kg/kg),

respectively, with initial moisture content of gas at 0.015 kg/kg. The

operating temperature of bedranges between 70 and 100 C and no

heat transfer through the wall of bed.

Numerical simulation is carried out. Effects of the inlet gas

temperature on temperature of the solid, the energy efciency, the

exergy efciency and the availability of gas are discussed.

Fig. 2compares the simulation results of the particles temper-

ature in the uidized bed dryer with the experimental results of

Assari et al. [13]. Some discrepancy between the model and the

Fig. 1. Computational mesh.

Fig. 2. Comparison of model simulation results for temperature of particles with

experimental results[13].


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experimental results exist, which is 4.5%. The deviation between

the experimental and the modeling results are due to the precision

of the measurement tools and the heat loss from the apparatus

walls. This error is more considerable at the beginning of the fallingrate period of drying. The difference between the real heat transfer

coefcient and applied one is also another source of disagreement,

which can be modied based on the material and its thermal

resistance.

Furthermore the present model is compared with two-phase

model of Li and Duncan [15]and is shown inFig. 3. Their model

was based on a bubble phase and emulsion phase. However indi-

cates a remarkably good agreement with our two-uid model.

Moreover to validate the present model, the non-dimensional

parameter including Reynolds number which expresses the non-

dimensional drying air velocity and Fourier number which

expresses the non-dimensional drying time, against energy and

exergy efciency correlations introduced by Inaba et al. [18] are

shown in Figs. 4 and 5 . The maximum difference between our

simulation results and their non-dimensional correlations are

within 24% for energy efciency and 19% for exergy efciency. It

indicates a remarkably good agreement with our proposed model

especially qualitatively. Inabas model is good for bubbling regime

and we are working in slug regime in this study. This difference

illustrates the bubble might increase the energy and exergy ef-

ciency value.

The energy and the exergy efciency simulation of uid bed

dryer are preformed in this study by averaging in time for entire

bed. Due to this procedure the bubbles do not affect our energy and

exergy founding.

The effect of increasing the inlet gas temperature on the solid

temperature on the thermodynamic efciencies and the availability

ofgasin the bed are shown in Figs. 6 and 7. It can be seen from Fig. 6

that the energy efciency is found to be higher than the exergy

efciency. Since exergy is not subject to a conservation law. Exergy

is consumed or destroyed due to irreversibility in drying processBoth the energy and exergy efciencies for the drying of wheat

particles are found to be very low at the nal time of drying process

This can be explained by the fact that the surface moisture evap-

orates very quickly due to high heat and mass transfer coefcients

in uid bed systems. Hence the drying rate is very high at the initia

stage of the drying process, but decreases exponentially when al

the surface moisture evaporates and the drying front diffuses

inside the material. When the inlet gas temperature is increased

from 70 C to 100 C with the inlet gas velocity of 4 m/s, energy

efciency increases up to 25% and the exergy efciency up to 21%

Thus, higher inlet temperatures of drying air can be used which

leads to shorter drying times. Further, the enthalpy and the entropy

of drying air also increase and lead to higher energy and exergy

efciencies. However there is practical limitation due to thedamage of the material. At the nal stage of drying process, the

inlet gas temperature increase does not show any signicant effec

in drying efciencies. If the inlet gas temperature is increased, the

grain temperature also increases and the nal temperature of the

material after long time spans becomes almost equal to the

temperature of inlet drying air. In order to use the energy more

effectively we can reduce the inlet gas temperature regularly unti

the end of drying process.

Fig. 7illustrates the availability analysis of gas inuid bed dryer

It indicates the availability of gas at the start of the drying process is

higher than the nal time. Because differences in the gas and solid

Fig. 3. Comparison of model simulation results for temperature of particles and two-

phase model of Li and Duncan [15].

Fig. 4. Comparison with non-dimensional energy correlation efciency.

Fig. 5. Comparison with non-dimensional exergy correlation efciency.

0

20

40

60

80

100

0 500 1000 1500 2000 2500

Efficiency(%)

Time (s)

Energy efficiency, Tgi=100 C, Vgi=4 m/s


Exergy efficiency, Tgi=100 C, Vgi=4 m/s


Fig. 6. Effect of inlet gas temperature on thermodynamics efciencies.


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temperature at the beginning of the drying process are higher and

tend to decrease over time. This shows that the availability of gas is

decreasing. By increasing the inlet temperature, the availability of

gas at the beginning of the drying process increases and then

decreases sharply. The difference in the availability of gas is much

higher between 70 C and 100 C temperatures but this becomes

smaller as time progresses. So in order to use the energy more

effectively one can reduce the inlet gas temperature at the nal

stage of drying regularly.

Fig. 8 shows the effect of gas velocity on efciency of dryer

versus drying time. Following conditions are used; inlet gas

temperature 100 C, inlet gas velocity varies from 4 to 5 m/s. So by

increasing mass ow rate (this is due to the inlet gas velocity

increase) reduces the exergy efciency. This enhances the exergy

into the system, which in turn lowers the exergy efciency, based

on Eq. (33). It is observed that for an increase of about 25% in the airvelocity, the energy efciency decreases 21%, and the exergy ef-

ciency is roughly 20%.Fig. 9shows the effect of gas velocity on the

availability of gas in the bed. It seems there is a large difference

among the availabilities of gas at the initial time of drying, rst 25 s

and then remained the same during the drying process. It would be

wise touse a gas velocity higher at the rst drying stage,and reduce

to the required value for nal stage.

Figs. 10 and 11and illustrate the effect of particle size on the

thermodynamic efciencies andthe availabilityof gas on the drying

process in bed. It is observed with increase of particle diameter, the

energyand exergy efciencies decreaseto 29% and 34% respectively.

Also increase in particle size decreases the availability of gas.

5. Conclusion

The need to understand relation between energy and exergy,

and environmental impact is important in drying industries. Since

lower exergy efciency leads to higher environmental impact and

this affect energy cost. The wheat grain is used in uidized bed

dryer and desirable improve in efciency is a plus sign for energy

consuming. This paper investigates energy and exergy efciency

based on two-uid model for uidized bed dryer. The effect of inlet

gas velocity, inlet gas temperature and particle size is investigated.

It is shown that differences between the thermodynamic efcien-

cies are higher at the start of the process, decrease during the

drying process and all close to zero at the nal stage. This is due to

moisture transfer from the particle at the beginning of the process.

However, the energy efciency is found to be higher than the

0

200

400

600

800

1000

1200

0 50 100 150 200 250

Theava

ilabilityofgas(W)

Time (s)

Tgi= 70 C, Vgi=4 m/s

Tgi= 100 C, Vgi=4 m/s

Fig. 7. Effect of inlet gas temperature on the availability of gas.

0

20

40

60

80

100

0 500 1000 1500 2000 2500

Efficiency(%)

Time (s)





Fig. 8. Effect of inlet gas velocity on thermodynamics efciencies.

0

200

400

600

800

1000

0 50 100 150 200 250

Theavailabilityofgas(W)

Time (s)

Vgi=4 m/s, Tgi=70 C

Vgi=5 m/s, Tgi=70 C

Fig. 9. Effect of inlet gas velocity on the availability of gas.

0

20

40

60

80

0 500 1000 1500 2000 2500

Efficiency(%)

Time (s)

Energy efficiency, Tgi=70 C, Vgi=4 m/s, dp=3 mm

Energy efficiency, Tgi=70 C, Vgi=4 m/s, dp=5 mm

Exergy efficiency, Tgi=70 C, Vgi=4 m/s, dp=3 mm

Exergy efficiency, Tgi=70 C, Vgi=4 m/s, dp=5 mm

Fig. 10. Effect of particle diameter on efciencies.

0

200

400

600

800

0 50 100 150 200 250

Theavailabilityofgas(W)

Time (s)

dp=3 mm, Tgi=70 C, Vgi=4 m/s

dp=5 mm, Tgi=70 C, Vgi=4 m/s

Fig. 11. Effect of particle diameter on the availability of gas.


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exergy efciency all the time. The increase of the inlet gas

temperature, increased thermodynamic efciencies and as a result

the availability of the gas is increased. Higher inlet gas velocity

decreased the thermodynamic efciencies. It would be advanta-

geous to use the higher air velocity rather at the rst drying stage

and then, reduce to the specication value. An increase in particle

diameter size decreased the thermodynamic efciency and the

availability of gas.

Furthermore, this study shows the capability of the two-uid

model to predict accurately the energy and exergy by comparing

the introduced non-dimensional correlations, the model predic-

tions and the experimental results.

Nomenclature

a specic surface, 1/m

c specic surface, 1/m

CDz two phase drag coefcient

d particle diameter, m

D molecular diffusion, m2/s_E rate of exergy transfer, kJ/s1

F1,F2 function, as dened in text

g gravity, m/s2

G( 3g) solids stress modulus

hv heat transfer coefcient, kJ/s m2 C

I enthalpy, kJ/kg

Jh heat transfer dimensionless

k thermal conductivity, kJ/m C

l constant

n constant_m moisture evaporation

Pr Prandtl number

P pressure, kPa

R gas constant

Re Reynolds number, as dened in text

r radial distance from the centerline, m

s specic entropy, kJ kg1 K1_Sgen entropy generation, kJ kg

1

T temperature, C

t time, s

u radial velocity, m/s

v axial velocity, m/s

x moisture content, kgw/kgsz elevation, m

Greek symbols

b gasesolid drag coefcient

go heat of vaporization, kJ/kg

3 void fraction

h efciency

m dynamic viscosity, Pa sr mass density, kg/m3

s evaporation coefcient, kg/m2 s

s stress, kPa

Fs spherically of a particle

Subscripts

0 dead state

Dz drag in z-direction

e energy

ex exergy

g gas

ic inlet-cell

p particle

pg gas on the surface of a particle

r radial

rr radial-stress

s solid

sc solid-critical

sz solid-axial

v vapor

w water

wg water evapor

z axial

zz axial-stress

References

[1] J.F. Davidson, R. Clift, D. Harrison, Fluidization, second ed., Academic Press1985.

[2] D. Kunii, O. Levenspiel, Fluidization Engineering, John Wiley, New york, 1969[3] L.S. Fan, C. Zhu, Pinciples of GaseSolid Flows, Cambridge University Press

1998.[4] D. Gidaspow, Multiphase Flow and Fluidization, Academic Press, 1994.[5] B. Palancz, A mathematical model for continuous uidized bed drying

Chemical Engineering Science 7 (1982) 1045e1059.[6] F.S. Lai, Y. Chen, L.T. Fan, Modeling and simulation of a continuousuidized

bed dryer, Chemical Engineering Science 41 (1986) 2419e2430.[7] E. Hajidavalloo, F. Hamdullahpur, Thermal analysis of a uidized bed drying

process for crops. Part I: mathematical modeling, International Journal oEnergy Research 24 (2000) 791e807.

[8] E. Hajidavalloo, F. Hamdullahpur, Thermal analysis of a uidized bed dryingprocess for crops. Part II: experimental results and model verication, International Journal of Energy Research 24 (2000) 809e820.

[9] S. Syahrul, I. Diner, F. Hamdullahpur, Thermodynamic modeling ofuidized

bed drying of moist particles, International Journal of Thermal Sciences 42(2003) 691e701.

[10] I. Dincer, A.Z. Sahin, A new model for thermodynamic analysis of a dryingprocess, Heat and Mass Transfer 47 (2004) 645e652.

[11] M. Ishii, Thermo-uid Dynamic Theory of Two Phase Flow, Eyrelles, Paris2006.

[12] H. Basirat Tabrizi, M. Saffar-Avval, M.R. Assarie, Two-dimensional mathematical model of a packed bed dryer and experimentation. in: Proceeding othe IMECHE Part A, Journal of Power and Energy 216 (2) (2002) 161e168.

[13] M.R. Assari, H. Basirat Tabrizi, M. Saffar-Avval, Numerical simulation ofuidbed drying based on two-uid model and experimental validation, AppliedThermal Engineering 27 (2007) 422e429.

[14] S. Azizi, S.H. Hosseini, G. Ahmadi, M. Moraveji, Numerical simulation oparticle segregation in bubbling gas uidized beds, Chemical Engineering andTechnology 33 (2010) 421e432.

[15] M. Li, S. Duncan, Dynamic model analysis of batch uidized bed dryersParticle & Particle Systems Characterization 25 (2008) 328e344.

[16] K. Rietma, S.M.P. Mutsers, The effect of interparticle force on expansion oa homogeneous gas uidized bed, in: Pro. Int. Symp. On Fluidization, vol. 321973. Toulouse, France.

[17] P. Kos, Gravity thickening of water treatment plant sludges, Journal AWWA69 (5) (1997) 272e282.

[18] H. Inaba, S. Husain, A. Horibe, N. Haruki, Heat and Mass Transfer Analysis oFluidized Bed Grain Drying, Memoirs of the Faculty of Engineering, vol. 41Okayama University, 2007, 52e62.


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