drying system designs: global balance and...

jfpe_573 1..31

DRYING SYSTEM DESIGNS: GLOBAL BALANCE AND COSTS

KIL JIN BRANDINI PARK1, LUÍS FELIPE TORO ALONSO1,FÉLIX EMILIO PRADO CORNEJO2, INÁCIO MARIA DAL FABBRO3, and

KIL JIN PARK3,4

1Agricultural EngineeringPO Box 6011, CEP: 13083-875

2Research Scientist at Embrapa Agroindústria de Alimentos.

3School of Agricultural EngineeringUniversity of Campinas

Campinas-SP, Brasil

Accepted for Publication October 10, 2008

ABSTRACT

When designing drying systems, heat and mass balance should be studiedas well as constructing and operating costs. These designs, which differ bytype of dryer, depend on the heat-mass relationship. The heat and massbalance on drying systems depend on the type of dryer, the drying process andthe drying rate. One of the biggest problems involving dryers’ selection con-sists in obtaining the required quality of the product with a great variety ofdrying equipments and processes. Many methods are presented in literature,those based on phenomenological concepts and those based on cost aspects.Consequently the criteria to classify and design dryers are also in a largenumber and different for each drying system. Drying system simulation modelscan also be used to design the dryer. Equations for dryer designs and the costsand area calculations are presented to help food engineers. The case studywith specific software using these equations will be presented in future articles.

PRACTICAL APPLICATIONS

In order to increase the aggregate value of food products, research in thearea of dehydration has been widely extended to search products with fewalterations in their sensorial and nutritional characteristics. Drying is one of theoldest and usual unit operations employed in industrial processes. The drying

4 Corresponding author. TEL: +55-19-35211076; FAX: +55-19-35211010; EMAIL: [email protected]

Journal of Food Process Engineering •• (2010) ••–••. All Rights Reserved.© Copyright the AuthorsJournal Compilation © 2010 Wiley Periodicals, Inc.DOI: 10.1111/j.1745-4530.2009.00573.x

1

rate is so small for biological materials compared with other materials that theyrequire a longer drying time. This long drying time induces to high drying cost.Therefore, the understanding of the drying system, process and equipment, isessential for food engineers to design an optimized drying system. Among thecontributions that technological advances did to the understanding of dryingsystems, one is the improvement of computational methods. The progress incomputer science and conveniences created by the personal computer, as wellas the applications of drying process simulation with drying mathematicalmodels are essential.

INTRODUCTION

Any drying system can be designed based on heat and mass transfer.According to Keey 1978, the final design should consider heat and massbalance along with constructing and operating costs.

Lapple et al. 1955, Nonhebel and Moss 1971 and Borde et al. 1997, usedheat and mass transfer relationships to develop the design of the dryer. A modelwas designed for each type of dryer based on simulation models described byJumah and Mujumdar 1993, Mabrouk and Belghith 1994, Zahed et al. 1995,and Fyhr and Kemp 1999.

Designs based on the heat and mass transfer relationship permit:

(1) Determination of the effects of several drying methods.(2) Estimation of the effect of changes in operating conditions on dryer

capacity.(3) Determination of the relationship between laboratory data and theoretical

estimates.(4) Development of new models.

Design Equations

The main equations and considerations according to Lapple et al. 1955,Nonhebel and Moss 1971, Jumah and Mujumdar 1993, Mabrouk and Belghith1994, Kemp and Bahu 1995, Zahed et al. 1995, Borde et al. 1997 and Fyhr andKemp 1999 are presented.

Lapple et al. (1955). Table 1 presents a list of equations that estimate thedrying surface required for the following conditions:

(1) moisture removal;(2) airflow;(3) initial air and temperature condition;

2 K.J. BRANDINI ET AL.

TAB

LE

1.E

QU

AT

ION

SFO

RC

AL

CU

LA

TIN

GT

HE

AR

EA

OF

CO

NV

EC

TIO

ND

RY

ER

S

Dry

ing

cond

ition

sA

bove

tran

sitio

npo

int

Bel

owtr

ansi

tion

poin

t

1.C

onst

ant

batc

hdr

ying

cond

ition

sA

XX

hT

T

Mc

of

cG

BU

b c

=⋅

−(

)⋅

−(

)⋅λ

θ(1

)A

MX

hT

TL

nX X

fb f

c

cG

BU

o f

=⋅

⋅⋅

−(

)θ

λ(2

)

2.V

aria

ble

dryi

ngco

nditi

ons

Adi

abat

icdr

yer

with

coun

ter-

curr

ent

flow

Ac

Gr

c hL

n

M GR

XX

TT

c=

⋅⋅⎛ ⎝⎜

⎞ ⎠⎟⋅−

⋅{}

− −⎧ ⎨ ⎩

⎫ ⎬ ⎭⎧ ⎨ ⎩⎫ ⎬ ⎭

⎡ ⎣⎢p

s c

cf

GB

Us

1λ

⎤⎤ ⎦⎥

−⋅{}

− −⎧ ⎨ ⎩

⎫ ⎬ ⎭⎧ ⎨ ⎩⎫ ⎬ ⎭

⎡ ⎣⎢⎤ ⎦⎥

⎛ ⎝⎜ ⎜ ⎜ ⎜

⎞ ⎠⎟ ⎟ ⎟ ⎟1

M GR

XX

TT

cc

f

GB

Usλ

(3)

Ac

Gr

c h

X X

Gr

M

TT

X

c=

⋅⋅⎛ ⎝⎜

⎞ ⎠⎟⋅+

⋅⎧ ⎨ ⎩

⎫ ⎬ ⎭−

⎧ ⎨ ⎩⎫ ⎬ ⎭{

}⎡ ⎣⎢

ps c

f c

pG

BU

c

s1

λ⎤⎤ ⎦⎥

⎛ ⎝⎜⎞ ⎠⎟

×⎧ ⎨ ⎩

⎫ ⎬ ⎭−

⋅⎧ ⎨ ⎩

⎫ ⎬ ⎭

− −⎧ ⎨ ⎩

⎫ ⎬ ⎭⎧ ⎨L

nX X

M Gr

XX

TT

cc f

p

cf

GB

Us

1λ

⎩⎩⎫ ⎬ ⎭

⎡ ⎣⎢⎤ ⎦⎥

⎛ ⎝⎜⎞ ⎠⎟

(4)

Adi

abat

icdr

yer

with

para

llel

flow

Ac

Gr

c hL

nM G

R

XX

TT

c=

⋅⋅⎛ ⎝⎜

⎞ ⎠⎟⋅−

⋅{}

− −⎧ ⎨ ⎩

⎫ ⎬ ⎭⎧ ⎨ ⎩⎫ ⎬ ⎭

⎡ ⎣p

s s

of

GB

Us

11

λ⎢⎢

⎤ ⎦⎥⎛ ⎝⎜

⎞ ⎠⎟(5

)A

cG

Rc h

X X

Gr

M

TT

X

c=

⋅⋅⎛ ⎝⎜

⎞ ⎠⎟⋅+

⋅⎧ ⎨ ⎩

⎫ ⎬ ⎭−

⎧ ⎨ ⎩⎫ ⎬ ⎭{

}⎡ ⎣⎢

⎤s s

f c

pG

BU

c

s1

λ⎦⎦⎥

⎛ ⎝⎜⎞ ⎠⎟

×⎧ ⎨ ⎩

⎫ ⎬ ⎭

−⋅

⎧ ⎨ ⎩

⎫ ⎬ ⎭

− −⎧ ⎨ ⎩

⎫ ⎬ ⎭⎧ ⎨ ⎩L

nX X

M Gr

XX

TT

cc f

p

oc

GB

Us

1λ⎫⎫ ⎬ ⎭

⎡ ⎣⎢⎤ ⎦⎥

−⋅

⎧ ⎨ ⎩

⎫ ⎬ ⎭

− −⎧ ⎨ ⎩

⎫ ⎬ ⎭⎧ ⎨ ⎩⎫ ⎬ ⎭

⎡ ⎣⎢⎤ ⎦⎥

⎛ ⎝⎜

1M G

r

XX

TT

cp

of

GB

Usλ

⎜⎜ ⎜ ⎜

⎞ ⎠⎟ ⎟ ⎟ ⎟

(6)

Inte

rmitt

ent

adia

batic

drye

rA

cG

rc h

Ln

M

Gr

XX

TT

c=

⋅⋅⎛ ⎝⎜

⎞ ⎠⎟⋅−

⋅⋅

⎧ ⎨ ⎩

⎫ ⎬ ⎭

− −⎧ ⎨ ⎩

⎫ ⎬ ⎭p

s cp

c

of

GB

Us

11

θλ

⎧⎧ ⎨ ⎩⎫ ⎬ ⎭

⎡ ⎣⎢⎤ ⎦⎥

⎛ ⎝⎜⎞ ⎠⎟

(7)

Non

-adi

abat

icdr

yer

with

coun

ter-

curr

ent

flow

Ac

G hT

TdH

cYY

=⋅

−[

](

)++ ∫

λ

ψξ

1G

BU

G

oo

(8)

Ac

XG h

XG

MH

HT

TdH

Y

Y

=⋅⋅

+[

]−

[]

[]

−[

](

)+ ∫

c

cf

GG

BU

G

o

oλ

ψ

10

(9)

Non

-adi

abat

icdr

yer

with

para

llel

flow

Ac

G hT

TdH

Y

Y

=⋅

−[

](

)+ ∫

λζ

cG

BU

G

o

o

1(1

0)A

cX

G hX

GM

HH

TT

dHY

YY oo

=⋅⋅

+[

]−

[]

[]

−[

](

)++ ∫

c

co

GG

BU

G

λ

ζξ

10

(11)

ξ=(

)−

()

MG

XX

of

(12)

ψ=(

)−

()

MG

XX

cf

(13)

ζ=(

)−

()

MG

XX

oc

(14)

rh

Kc

pc

Hs

=(

)(1

5)

DRYING GLOBAL BALANCE AND COSTS 3

(4) final air and temperature condition.

The following data are required to apply the equations in Table 1.

(1) dry outlet condition;(2) heat transfer coefficients;(3) critical moisture content.

Dryer outlet conditions should coincide with the limitations of dry mate-rial temperature and relative humidity supported by exhaust gases.

Heat transfer coefficients vary according to the dryer design. In the caseof indirect drying, the overall heat transfer coefficient (q = U A DT) is noteasily determined, but some considerations can be applied for normal tem-perature conditions. The coefficients range between 5 and 30 W/m2K withoutagitation of the material and in direct rotational dryers between 30 and 90 W/m2K. Higher values may be obtained by agitation (between 90 and150 W/m2K).

Thermal exchanges are intensified by radiant energy when temperaturesare higher than vapor.

The transition point of constant drying rate to falling rate (referred inliterature as critical point) is essentially a characteristic of the material andcannot be determined without conducting drying tests. It should be under-scored that the critical point is not necessarily constant and may vary accord-ing to the drying rate and thickness of the material.

The equations in Table 1 assume that there is no heat lost by radiation.The product should first be heated to evaporation temperature until a consid-erable drying rate is attained.

The follow procedure is suggested to calculate the drying area:

(1) Select the final condition in the psychrometric chart.(2) Disregard preheating, radiation and super heating.(3) Calculate airflow based on global balance:

GM X X

H H= ⋅ −( )

−f o

f o

(16)

(4) Calculate convection heat transfer coefficient (W/m2K) and velocity (m/s)using one of the three empirical equations:Airflow parallel to a plane surface:

h Vc p= ×0 18627 0 8. . (17)

Airflow perpendicular to a surface:


h Vc p= ×3 2468 0 37. . (18)

Cross airflow (Re > 350):

h V cc p s= × ⋅ −( )⋅3 2468 10 37 0 41. . .ε ϕ (19)

(5) If possible, evaluate the transition point experimentally.(6) Calculate the drying area using the appropriate equation from Table 1

considering;

A A At c d= + (20)

Equations (5) and (6) are restricted to drying under constant conditionsof material temperature, temperature, velocity and airflow. Temperature andmoisture vary, particularly in continuous operations, and in this case Eqs. (7)–(15) are more appropriate. Equation (11) can be applied to most of the avail-able convection commercial batch dryers.

Make the necessary corrections for preheating, radiation and super-heating, multiplying the total area At by a safety factor.

The value “rp” in the equations is the psychrometric ratio and in theair-water vapor mixture, it is considered a unit value and in this case:

h

Kcc

Hs= (20a)

For other mixtures or when the value cannot be experimentally evaluated,“rp” can be deduced from the equation:

rh

K c

D m

c kpc

H s

g g

s g

=⋅

=( ) +( ) +

21

21(21)

In the case of most organic liquids, “rp” is between 1.5 and 2.0.Some types of dryers need specific gas velocity and adequate solid

residence time. The velocity of air in cross-flow dryers ranges from 1.2 to1.8 m/s. If a great deal of dust is formed, particularly during discharge, thevelocity is reduced between 0.3 and 0.6 m/s.

The velocity of gas in pneumatic dryers should be greater than twice theminimum velocity of material fluidization, which generally ranges between 15


and 60 m/s, and can be considered in approximate calculations as 24 m/s. Thesolid load is generally between 0.25 and 0.55 kgdry mass/kgdry air, but loads up to1 kgdry mass/kgdry air are tolerated depending on how the dryer manipulates thematerial.

In fluidized bed dryers, the velocity of the gas is approximately 10% ofthe material terminal velocity.

The volume and load of the drying chamber can be evaluated on the basisof the surface area of material exposed for drying:

Exposed surface: S m m=[ ] 2 3 (22)

Chamber volume: c tV A S= (23)

Dryer chamber load: c b tM A S= ⋅ρ (24)

It should be underscored that this calculation should be performed forcontinuous operations, as in a batch the chamber load and volume are syn-onymous of the processing capacity.

In continuous cross-flow circulation equipment, the estimation of “S”should be such that the thickness of the material maintains a transition pointsimilar to that used in the calculation.

In fluidized or cross-circulation beds, the thickness of the bed is not veryimportant, except in the case of ventilator pressure loss effect. However, incross-circulation, the thickness of the bed may interfere in the drying rate,moving the drying rate transition point. Constant drying rate may extend oralter the behavior of the falling rate.

In batch drying, the thickness of the bed is one of the factors involved indrying conditions (the quantity of water to be evaporated from a definedquantity of material during a specific interval of time). However, in continuousdrying, the thickness of the bed is calculated regardless of the surface area.

If the thickness is very small, the transition point will be higher and thefalling rate more concave, consequently the area will have to be larger tocomplete drying.

When estimating transition point, it should be kept in mind that it canchange because of the moisture, thickness and air velocity. Any experimentaldetermination increases calculation reliability.

The engineer should not forget that even though the area calculated for adefined thickness leads to correct performance; it is possible to increase thethickness without significantly increasing the area.

In a cross-flow circulation bed, “S” should correspond to the total surfaceof the particles according to the volume of the material, which is the ExposedSurface. In the case of non-spherical particles, the value can be calculated


using the following equation:

S d= ⋅ −( ) ⋅( )6 1 ε ϕ p (25)

Finally, the residence time is given by:

θ ρ= ⋅⋅

b tA

M S(26)

The gas velocity, the chamber load or the residence time are deter-mined. The volume of the drying chamber should comply with thesedemands. In the case of batch drying, the operation cycle time should bedetermined. This choice may depend not only on the drying process but alsoon the phases before and after drying or on the lowest operation cost.However, it is most important to guarantee performance with minimuminterruptions.

The precise manner of transforming total area regarding residence timeand gas velocity into specific dryer dimensions depends on the shape, arrange-ment and type of dryer. This method can simplify the calculation of drying area(as in a simple drying chamber) or complicate the calculation (as in a rotativedryer involving rotation velocity, inclination angle and other variables thatinterfere in the performance).

In some cases the total area can be disregarded as in the fluidized bedcalculation, where volume of air is calculated according to heat balance and airvelocity needed for fluidization. The drying chamber is calculated to allowsufficient airflow for fluidization.

Even in the case of a fluidized bed, the area has to be calculated in a moresimple manner, as the “S” value and the residence time determine the dryingarea. It is important that the airflow should be sufficient for the removal ofevaporated water. All the water will not immediately be available for evapo-ration. The internal moisture may not diffuse rapidly enough and the materialmay not take advantage of the drying medium. In this case, reduction ofmaterial should be considered and when not possible, the residence timeshould be extended.

Nonhebel and Moss (1971). This is one of the most important worksavailable on dryer designs. It is possible to reasonably estimate the perfor-mance of the dryer during the constant drying period by applying the mass andheat transfer equations and a simple mass balance for each particular type ofair circulation. However, it is almost impossible to estimate the falling dryingperiod without experimental data of the material to be dried. Hence, theauthors recommend drying tests to be conducted, whenever possible, in flex-


ible equipment such as tray dryers, if not, the performance can be deducedfrom an analogy with similar material.

Constant Drying Rate. The mechanism of constant drying rate is one inwhich evaporation from a liquid surface associated to a solid occurs withoutinterference from the latter. The drying rate is then determined by the vapordiffusion rate from the surface of the material. The surface tends to attain thetemperature of the wet bulb, corresponding with the temperature and humidityof the drying gas flow. If the gas conditions remain constant, the temperatureof the surface will also remain constant.

Similarly, the partial pressure and humidity at the surface of the materialwill be the partial saturation pressure and saturation humidity of the wet bulbtemperature.

The instantaneous drying rate is:

dX

dt

h A T TK A p p

K APM

PMP p H H K A H

=−( )

= −( )

= −( ) −( ) = −

c G Sp S G

pG

Aml S G H S

λ

HHG( ) (27)

Integrated to constant drying rate:

RX X M

A

h T TK p p K H Hc

TR S

c

c G BUp BU G H S BU G= −( ) = −( ) = −( ) = −( )0

θ λ , (28)

Falling Drying Rate. During this time, drying is controlled byinternal migration rate of the liquid to the surface of the solid whereevaporation occurs. The heat transferred to the surface falls progressively,compensating the fall in the transference rate of the mass within thematerial. In the same conditions, the temperature at the surface risesup to the temperature of the dry gas when drying is complete. The reductionof heat transfer rate may be expressed as a result of the increase insurface temperature of the material. As previously observed, a combinationof Eqs. (29) and (30) provide a general equation for the falling drying ratetime:

RX X M

A

hd T T

T T

dTR d S

d

cG S

G BU= −( ) = −( )−

∫θ λ 0(29)


RX X M

AdK d p p

dK d H

K p p

K

dTR d S

dp S G

H

p c BU G

H c

=−( )

= −( )

=

( ) −( )

( )

∫ ∫

∫θ 0 0

0 SS G

S BU G −( )−( )

∫ HH H

0

, (30)

Conduction Drying. In relation to conduction, it should be consideredthat the drying rate depends on the quantity of heat supplied for drying:

dX

dt

dq

dt= 1

λ(31)

The rate of heat supplied is given by the relation between temperaturedifferential and the source, material and area of contact:

dq

dtU A T T= ⋅ ⋅ −( )h h s (32)

Integrated to the overall drying time:

RX X M

A

UT Tm

o f b

h m

mh s= −( )⋅

⋅= ⋅ −( )

θ λ (33)

Can be simplified to:

RX X M

A

kT Tm

o f b

h m

ch BU= −( )⋅

⋅= ⋅ −( )

θ λ (34)

Batch Tray Dryer. Convection. Generally, heat is transferred to thematerial by hot air and water is removed in the form of vapor by a current ofair. When heat is transferred by convection, the drying rate can be expressed interms of temperature and humidity of drying gas as:

RX X M

A

h T TK p p K H Hc

TR S

c

c G BUp BU G H S BU G= −( ) = −( ) = −( ) = −( )0

Θ λ , (35)

Radiation. If part of the heat is supplied by radiation, surface tempera-ture of the product at constant rate may surpass the wet bulb temperature of thematerial. In this case, the surface temperature maybe evaluated by trial anderror or by heat and mass balance as simplified in follow relation:


T Th T T

hH H

cg sr r s

cS BU G

s

− + ⋅ − = −( )⋅( * *),

λ (36)

It is important to keep in mind that gas flow affects the heat and masstransfer coefficients. The drying rate increases as gas velocity increases whenother factors are maintained. During constant rate period, heat transfer rela-tionships can be used to empirically assess the impact of gas velocity in anair-water system with airflow parallel to the surface:

h G Sc t0.32367= ⋅( )0 8. (37)

During falling rate, the heat transfer effect is reduced and it is impossibleto predict the effect of air velocity on drying rate.

Cross Airflow. During batch drying with cross airflow, dry gas is insuf-flated through the drying material in a perforated tray. The external conditionshardly change. The effect on the falling rate is also hard to predict, it isgenerally quicker than parallel airflow. The thickness no longer has any influ-ence on water migration. The most important parameter is the mean diameterof the particle. Therefore, the load is not restricted by drying, but by the lossof load and cost of heating the gas. The equation can be reformulated to studythe effect of the mean diameter of the particle on drying rate:

RX X M

A

G S

dH Hc

o c b

c

t

pS BU G= −( ) = ( ) −( )

θ146 0 59.

, (38)

Vacuum. In this process, the solid tends to assume the boiling point of theliquid during evaporation and then remains constant during drying. As thetemperature varies close to the boiling point, it can be assumed thatthe temperature differential remains constant during drying, which means thatresistance to drying does not increase during the process.

The equation used in conduction can be applied to this phenomenon.Transference by radiation is negligible at temperatures below 100C. In order toapply this equation, the thickness of the material should be kept between 2.5and 4 cm with no agitation and no surface hardening:

Dryers with Agitation. Tray drying with agitation may be considered asevaporation below the boiling point of the liquid. Drying can now be evaluatedby the heat flow from the hot wall to the material. Resistance to heat flow isoffered by the material, itself, and the film of solids attached to the wall.


The heat flow at any moment can be obtained by the equation:

dq

dtU A T= ⋅ ⋅h Δ (39)

Overall heat transfer coefficient is obtained by:

Uh h h

= + +⎛⎝⎜

⎞⎠⎟

11 1 1

c vapor c plate c product, , ,

(40)

During drying, the overall coefficient “U” and the difference in tempera-tures decrease until drying ends. If these values can be precisely determined,the total heat flow can be defined graphically or mathematically using thefollow relation:

dq

dtA f X f X= ⋅ ( )⋅ ( )h 1 2 (41)

Unfortunately, heat transfer coefficients and temperature differencescannot be precisely determined. Therefore, in the case of proposals such as theagitated tray dryer project, the following simplified equation is consideredsufficiently precise:

q

tU A T= ⋅ ⋅m h mlΔ (42)

In the case of normal dry material in agitated batches, the value of “Um”is approximately 10 to 85W/m2C, which is usually proportional to the humid-ity of the material and does not depend on the pressure of the operation.

Tray or Belt Continuous Dryers. This group of dryers can be subdividedinto three types:

(1) cross airflow belts;(2) parallel airflow belts;(3) conductive belts.

Cross Airflow Circulation Belt. Drying occurs because of a flow of hotair through the fine layer of material particles that move through the dryer. Thebed is normally not thicker than 5 cm. The material remains static in relationto the tray and the system is similar to batch drying with cross-gas flow.

In the case of convection drying for a constant period, the equationpreviously presented may be used. However, in the case of continuous drying


with cross airflow, the surface area of the particles is not clearly defined,making it more convenient to use the constant dry bed sector “Ac” as thereference area “A” for:

RH O

AK Hc H ml

carried by Gas=( )

=Δ Δ2 (43)

We have:

RH O

A

H O

AK Hc

cH ml*

* *= = =Δ Δ Δ2 2 (44)

Or:

R K HA

AK H

A L

MK Hc H ml

cH ml

cH ml

* *= = ⋅ =Δ Δ Δ (45)

Therefore the mass coefficient is corrected by:

K KA L

MH Hc

* = ⋅(46)

The humidity differential should be expressed as a logarithmic mean:

RX X M

AK H H K Hc

o c b

cH S,BU G ml H ml*

* * *= −( )⋅ = −( ) = ( )θ

Δ (47)

Dryer load is M/A* (kgms/m2) and considering the thickness and bed arehomogeneously distributed, the equation can be rewritten as:

R X X Lc o c s c* = −( )⋅ θ (48)

Therefore when projecting the equipment, the area can be calculated interms of material flow:

θ = A L M* s (49)

A X X M R* c c= −( )0* (50)

Parallel Air-Flow Belt. In this case, drying occurs because of the flow ofhot air over the surface of the moist material. Once again, if the material is


considered stationary in relation to the gas, the system is similar to the batchcross airflow dryer.

The drying is convective and the basic heat as well as mass transferequations are applicable. The exchange area is generally the area of the belt.Hence, the constant time rate can be expressed as a result of dry material flow:

A X X M Rc o c c= −( ) (51)

The effect of gas velocity can be deduced from heat transfer equations forparallel flow and air-water system:

R G S H Hc t S,BU G ml= ( ) −( )0 052 0 8. .

(52)

As drying occurs only at the exposed surface of the material, the thicknessof the bed affects the drying rate. The constant rate is not directly affected asit depends solely on the exposed area, but the effect is on the drying ratetransition point and on the falling drying rate curvature. However, it is notpossible to quantitatively predict the effect of the thickness and hence, the useof experimental data is recommended.

Conductive Belt. Drying occurs because of the heat conducted by the beltand generally this dryer operates at vacuum. Drying is the result of directliquid evaporation. The constant and falling periods are not clearly defined.The process can be described by the conduction Eq. (33).

The area can be expressed in terms of dry material flow:

AX X M

R= −( )o f

m

(53)

Drum Dryer. Drying in heated drums is performed by transferring heatfrom the drum wall to the film of material to be dried. The evaporation isexpressed as:

qUA T T

θ tbh h BU= −( )−10 3 (54)

The overall heat transfer coefficient is given by Eq. (40):The falling rate takes 80 to 98% of the drying time. The temperature of

the solid film increases as the heat transfer rate decreases. Individual thermalproperties cannot be calculated during the falling period and Eq. (48) cannot


be applied. In order to apply Eq. (47), the overall heat transfer rate should havea value. The overall coefficient ranges between 60W/m2C, in the case ofmaterials that are difficult to dry, and 400W/m2C, for materials easily dried.

Fluidized Bed Dryer. Firstly, the gas flow is determined to guarantee areasonable relationship between uniform fluidization and material distributionas well heat balance demands. This value should remain constant in all calcu-lations and can be experimentally obtained. Assuming the material to be driedand that the temperature of the gases as well as the bed are the same, it ispossible to determine the relationship between fluidized bed drying and anydrying material.

Moisture balance:

G H H M X Xf o o f−( ) = −( ) (55)

Mass balance:

M X X M K Hf o c m ml−( ) = *Δ (56)

Heat balance:

M c c X T c c X T G c c H T c c H Tfs

ogo f

s w f s w o s G v o G v f g+[ ] − +[ ]( ) + +[ ] − +[ ]( )+ MM X X UA Tf o ml−( ) =λ Δ (57)

Direct Rotative Dryer. The residence time is the main key for designingthe direct rotative dryer. Calculation of residence time is obtained by:

θ = M L M� (58)

The residence time is determined by rotation “w”, drying gas velocity“Vg”, diameter “Ds”, inclination “g” and length “L” of the dryer. According toNonhebel and Moss 1971, the relationship described by Seaman and Mitchellis also largely used:

θω γ

=+( )

L

c D dVs g(59)

Usually “w” is close to 3 rpm in large equipment. A rotation may increasetransference, but greater consumption of energy does not compensate. The


value of “c” depends on the design of the dryer and ranges between 2 and p.The value of “d” depends of the type of material and ranges between 9 ¥ 10-4

and 1.5 ¥ 10-3. The diameter of the dryer can be deduced using:

DVz

Vgs =⋅

⋅ ⋅ −( )4

1π ϕ(60)

Obs.: j is the fraction of the bed being used by the gas flow and theindicated value is 85%.

Jumah and Mujumdar (1993). A computer program for a continuousfluidized bed dryer project is presented below together with the basic calcu-lation used in the development of this program.

In order to design the dryer, the area and height of the bed was calculatedas follows:

Bed area:g g

AG

V� = ⋅ρ (61)

Bed height: c

g

hM

A=

⋅ −( )� ρ ε1 (62)

However, the load of material on the bed is restricted by bed drying capacity:

Drying capacity: v fE M X X= −( )0 (63)

Load: cM M X= +( )1 0 θ (64)

Gas flow: s w f s s w s

g v g g v

G Mc c X T c c X T

c c Y T c c Y

f o

o=

+[ ] − +[ ]( )+[ ] − +

0

0 ff g[ ]( )T f (65)

The gas flow is not restricted by Eq. (55), but by the gas flow required toovercome the loss of load and fluidize the material, which should be experi-mentally estimated.

Mabrouk and Belghith (1994). The model was developed for a batchtray tunnel dryer. The dimensions were based on the the drying curve of aspecific product – grapes. The drying rate and the dryer dimensions werecalculated based on the drying curve.

Initially, the adimensional moisture was defined:


YX t X T

X X T= [ ]− [ ]( )

− [ ]( )e

o e

(66)

The adimensional drying rate was:

R R f Y T R dY dtc = ( ) =, ; (67)

The constant drying rate “Rc” was obtained by:

R l G H Hlc S.BU= ⋅ ⋅ −( )1 0

2 (68)

The following polynomial was suggested for predicting drying ratebehavior:

f Y T Y Y Y m m YZ,( ) = + −( ) −[ ]1 1 2 (69)

Once the drying rate is known, the dryer is designed applying moisturebalance:

ρ ε ρg sGdH

dL

dY

dt= − −( )1 (70)

and the heat balance:

ρg gg

s gGcdT

dLUS T T= −( ) (71)

Combining the balances:

λ ρ ε+ −( )[ ] −( ) = −( )c c TdY

dtUA T Tv w s s g1 (72)

Zahed et al. (1995). The model was developed based on the relationshipbetween the dense and bubbling phase as in Fig. 1:

According to the mass or energy conservation law:

Mass or energy

flow Inlet

Mass or energy

flow Ou

( )⎛⎝⎜

⎞⎠⎟

( )⎛⎝⎜

⎞⎠⎟−

ttlet

Mass or energy

accumulation

rate

=( )⎛

⎝⎜⎜

⎞

⎠⎟⎟

(73)


The mass balance in the dense phase would be:

− = −( ) −( )MdX

dtρ ε ε1 1b mf (74)

And the corresponding energy balance:

q M c c XdT

dtM X X− −( ) −( ) +( ) = −( )ρ ε ε λ1 1b mf s w

sf o (75)

And the humidity balance:

61

K

DH H

G

VH H

dH

dtb g b

bf d

g dd o g mf b

dρ ε ρ

ρ ε ε−( ) − −( ) = −( )�

(76)

Together, the equations result in a general equation that predicts thebehavior of the fluidized bed dryer.

M c c XdT

dtG

Vc c H T T

Ko

ρ ε ε

ρλ

ρ

1 1

6

−( ) −( ) +( )

= +( ) −( ) −

b mf s ws

g dG v o g p

b

�

gg b

bf d

g dd

ε ρD

H HG

VH H−[ ]− −[ ]⎛

⎝⎜⎞⎠⎟�

0

(77)

Kemp and Bahu (1995). Calculations for convective and conductivedryers differ. The main parameter in the first case is the gas flow “G”, whichis calculated by the heat and mass balance. The psychrometric chart is veryhelpful. If an absolute humidity as well as the inlet temperature of the gas are

FIG. 1. FLUIDIZED BED MODEL WITH SOLIDS RESTRICTED TO THE DENSE PHASE


provided, the outlet condition can easily be evaluated. Using the mass balanceequation 55, “G”can be calculated.

Knowing the value of “G”and assuming the gas velocity “Vg”, the trans-versal section can be evaluated and the dimensions estimated. An estimate ofresidence time is obtained from the length of the dryer, which can be calcu-lated based on mass and heat transfer:

L V E V X X R A= = −( )S S o f d (78)

In conductive dryers, the chosen parameter is the transfer area A, calcu-lation is based on rate of evaporation, assuming control through heat transfer:

Ah M X X Tc o f= −( )λ Δ (79)

Borde et al. (1997). An algorithm was developed for fluidized bed dryercalculations as well as mass and heat transfer relationships that represented thedrying process established.

Constant Period. The drying rate is given by:

− −( )= ⇒ = −( )

−( )h A T T

L

M

V

dX

dtA

M X X

h T Tc g s

p

o f

c g s

λ λ(80)

The drying time is:

θco c

m

= −( )M X X

AK(81)

Falling Period. The drying rate is given by:

− = −( )⎛⎝⎜⎞⎠⎟

dX

dt

h AT T

X

Xo

n

cs s

d

cλ(82)

The drying time is:

θdm

c fc

f

= − ⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

M

AKX X

X

X

n

(83)

The heat transfer coefficient is deduced from:


hk

dAr V V V

ou

hk

d

cg

pm g o

cg

p

= ⋅ ⋅ ⋅ ⇒ ≤ ≤

= ⋅ ⋅ ⋅

0 15

0 051

0 35 0 25

0 12

. Re

. Re

. .

. AAr V V V0 25. ⇒ ≤ ≤o g a

(84)

With adimensionals:Reynolds number:

Re =V dg p

gν(85)

Archimedes number:

Ard g

=−( )p g

g g

3

2

ρ ρν ρ (86)

Velocity is the restriction for Eq. (84):

Minimum f luidization: m gg

V V= ⋅−( )ρ

ρ ρ3 (87)

Optimum: og

p

V Ard

= ⋅ ⋅0 547.ν

(88)

Drag: ag

p

VAr

Ar d=

+ ⋅⋅

18 0 6.

ν(89)

The mass transfer coefficient for calculating drying time is deduced fromthe Sherwood number:

ShK d

D

Sc

B=

⋅⋅

= + ⋅ ⋅+( )

m p

efρ2 0 6

1

0 5 0 3333

0 7

. Re . .

. (90)

With adimensionals:

Schmidt number: g

g ef

ScD

=⋅μ

ρ (91)


Spaulding number: w g BUBc T T

=⋅ −( )λ

(92)

Fyhr and Kemp (1999). The authors presented another fluidized bedmodel. The design was once again based on mass balance:

nS K S H Hp g m p p G= −( )ρ (93)

And heat balance:

q h S T T nS= −( ) +c p s g pλ (94)

The model that expresses fluidized bed drying is:

dX

dtD X= ∇ − ∇( )ρ eff (95)

In a simplified manner and combining mass and heat balance we have:

X XG

nSH H X

Gc

nST Tf o

pf o o

p

po f= − −[ ]⎛

⎝⎜⎞⎠⎟= − −[ ]⎛

⎝⎜⎞⎠⎟

1 1λ (96)

DRYER DESIGN

When designing and developing dryer models, it is possible to observethat the degree of complexity and know-how involved differs in each type ofpublication.

A factor that reveals this know-how is the analysis of the drying phenom-enon as a global phenomenon that terminates in volume control. In this case,only overall balances and resulting dimensions are needed as observed in Fyhrand Kemp (1999) and Kemp and Bahu (1995). Some authors, like Jumah andMujumdar (1993) and Zahed et al. (1995) specify certain peculiarities regard-ing the equipment, even though they are not related to the drying phenomenon.Lapple et al. 1955, Nonhebel and Moss 1971, Mabrouk and Belghith 1994,extended analysis of the phenomenon by dividing it into two drying phases, theconstant rate and the falling rate. Keey 1978, always used the period divisionand analysis of drying rates to understand the phenomenon. This procedurerequires an experimental determination of the drying curve of the material/dyer.


CONVECTIVE DRYERS

This category includes the spray dryers, continuous batch tray dryers, beltdryers with cross airflow circulation, continuous or batch fluidized bed dryers,pneumatic and direct rotative dryers. Because of their complexity, spray dryersdo not fit into the model formulated below and, hence, will not be included.

Before determining the constant drying rate, the gas flow to be used indrying is determined, Eq. (55).

In dryers with pneumatic and fluidized beds, the need for gas flow thatcan overcome load loss and fluidize the material, invariably guarantees a flowhigher than estimated in Eq. (87).

In continuous drying, the direction of the airflow in relation to productflow should be observed. If it is a concurrent flow, the equation is directlyapplicable. In the case of counter-current flow, the sign of the equation shouldbe inverted.

As the outlet gas condition “Hf” is rarely known before conducting theproject and the gas flow is caused by this condition, it can only be estimated byan interactive process.

As previously observed, the drying process is divided into two periods,the constant drying rate and the falling rate. Therefore, the total drying area isthe sum of the area needed for drying at the constant rate and the area fordrying at the falling rate.

Constant Drying Rate Period Area

During the constant rate period, the heat supplied by the gas to thematerial being dried, considering the balance between gas and material is:

q h A T T= ⋅ ⋅ −( )c c g sat,bu (97)

Similarly, ignoring possible heat loss to the environment, the heat sup-plied removes the liquid in the form of vapor according to the equation:

q X X M= −( )λ o c (98)

Equalizing the two equations we have:

λ X X M h A T To c c c g sat,bu−( ) = −( ) (99)

Therefore, area of the constant drying period is:


AX X M

h T Tco c

c g sat,bu

= −( )−( )

λ(100)

Outlet gas humidity depends on the gas flow. In this condition, thetemperature of the wet bulb, Tbu, depends on the humidity of the gas leavingthe dryer, Hs, at gas temperature, Ts, This factor is calculated according to thepsychrometric relationsip between evaporated liquid and drying gas. The con-vective heat transfer coefficient depends on gas flow. There are several rela-tionships between gas flow and convective heat transfer coefficient. Perry andChilton 1973, provided the following relationship regarding parallel airflowover a plane surface:

hc GSs= × ( )12.2976

0 8.(101)

Treyball 1968 developed the following relationship regarding perpen-dicular airflow over any surface:

hc GSt= × ( )20.7819

0 37.(102)

This means that the convective heat transfer coefficient is:

hc a GSg

a

= × ( )1

2

(103)

Applying Eq. (101), it changes to:

AX X M

a GSg T T

aco c

g sat,bu

= −( )

( ) −( )λ

1

2 (104)

It was observed that when airflow is perpendicular to solid flow, thesection “Sg” is the drying area, hence:

AX X M

a GA T T

c

aco c

g sat,bu

= −( )

( ) −( )λ

1

2 (105)

Reformulating:


AX X M

a G T Ta

a

co c

g sat,bu

= ⋅ −[ ]⋅⋅ ⋅ −[ ]

⎛⎝⎜

⎞⎠⎟

−⎛⎝⎜

⎞⎠⎟λ

1

1

1

2

2

(106)

Falling Drying Rate Period Area

Just as in the constant rate period, the possible heat loss to the environ-ment is ignored; the heat supplied removes the liquid in the form of vaporaccording to:

q X X M= −( )λ c f (107)

However, during the falling rate period, the temperature of the solidgradually rises above the temperature of the wet bulb with a tendency towardsattaining equilibrium with the gas at the ending of the drying:

q h A d T TT T

= −( )−

∫c d g s

g bu

0

(108)

Equalizing the two equations:

λ X X M h A d T TT T

c f c d g s

g bu

−( ) = ⋅ −( )−

∫0

(109)

Therefore, the drying area during the falling rate period is:

AX X M

hd T T

T Tdc f

cg s

g bu= −( )

⋅ −( )−

∫λ0

(110)

In this condition, the falling drying rate is obtained by:

Rh

d T TT T

dc

g s

g bu

= ⋅ −( )−

∫λ0

(111)

Substituting in Eq. (111):

AX X M

Rdc f

d

= −( )(112)


Therefore, if the mean falling rate value “Rd” is known, it is possible toevaluate the drying area during the falling period:

Besides the equations mentioned earlier, empirical equations like thoseby Page and Fick that have effective diffusion experimentally determined canbe used to calculate the falling drying rate.

Calculation of Drying Rate during the Falling Period

Page’s equation as well as effective diffusion instantly provides the mois-ture value in the solid:

X f t= ( ) (113)

The drying rate is:

dX

dtf t= ′( ) (114)

The value of the mean falling rate “Rd” is the integrated value of rateduring the falling rate period obtained by the interval [Xc, Xf]. Hence, the rateis obtained by:

RX X

dX

dtdX

X

X

dc f

f

c

=−

⎛⎝⎜

⎞⎠⎟∫1

(115)

Similarly, the curve can be integrated with the interval [qc ; qd]:

RdX

dtdtd

c dd

c

=−

⎛⎝

⎞⎠∫1

θ θ θ

θ

(116)

Total drying Area in the Convective Dryer

After determining the drying areas in relation to the falling period andconstant period, the total drying area is obtained by their sum, Eq. (20).

CONDUCTIVE DRYERS

This category consists of drum dryers, batch tray dryers, continuous beltdryers and indirect rotative dryers.


In conductive drying, the rate depends on the quantity of heat supplied fordrying, Eq. (31). The drying heat rate is obtained by Eq. (32). The integratedequation for the entire drying period are Eqs. (33) and (34).

After reformulating Eq. (34), the drying area in a conductive dryer is:

AX X M

k T Tho f

c h bu

= −( )−( )

λ(117)

The “Tbu” condition is easily obtained from the psychrometric relation-ships of the drying fluid. It is even possible to obtain the drying area in termsof the drying rate just like the falling rate period in convective dryers knowing:

Rk

T Tmc

h bu= −( )λ

(118)

Substituting in Eq. (117) we have:

AX X M

Rho f

m

= −( )(119)

It should be emphasized that when a vacuum dryer is being used, thelatent heat of vaporization and the temperature of the wet bulb will depend onthe pressure in the drying chamber.

COSTS

The cost of the dryer is the final phase of the dryer project. Two types ofcosts should be analyzed to finalize equipment selection:

(1) fixed cost;(2) operating costs.

Fixed Cost

As previously cited, the economic factor plays a fundamental role inequipment selection. The area of the equipment is directly related to its costand serves as a base for selecting the best dryer.

Lapple et al. 1955, defined several functions for calculating the cost of adryer in relation to the drying area (Fig. 2).

Based on these correlations, the value of the area obtained in the previousitem is used to perform the calculation:


Cost c Atc= ⋅1

2 (120)

Where “c1” and “c2” are constant values obtain from the correlations.It should be underscored that the cost evaluated is only the cost of

constructing the equipment and does not include possible accessories (e.g., acyclone for a fluidized dryer), which alter the total cost of dryer.

Operating Costs

The operating cost is based on the energy consumed by the dryer. In orderto start the drying process, the material has to be heated. The cost of energyused in heating is obtained in two ways.

Convective Dryer. In a convective dryer, the heat potency applied isgiven by the difference between ambient air enthalpy and enthalpy of airheated by dry gas flow:

q G H H= −( )g a (121)

FIG. 2. COST OF A DRYER IN TERMS OF DRYING AREA AND CORRESPONDINGCORRELATIONS

Obs.: The values in US$ shown in the figure only serve as a quantitative reference.


Conductive Dryer. In a conductive dryer, heating is obtained by thequantity of heat applied during drying:

q M X X= −( )λ o f (122)

The product of potency used during heating, the total drying time and thecost of energy provide the operating cost:

Operating cost C q= eng θ (123)

Finally, in order to use energy consumption as a comparative term fordryers, the operating cost is divided by the total dry material produced.

Specific operating costoperating cost

M=

⋅θ s

(124)

Just like the procedure adopted for designing the dryer, studies involvingcosts also are specific regarding products, processes and drying equipment.According to the calculation of operating costs, studies on the cost of energyfor drying, such as the recent study by Zhang et al. 2007, who comparedstudies on drying wood using/not using dehumidification of dry air, are bib-liographic sources.

NOMENCLATURE

e Bed porosity adimensionaleb Bubbly phase porosity adimensionalemf Minimum fluidization porosity adimensionalf Material sphericity adimensionalg Dryer inclination m/mj Fraction of bed adimensionall Latent heat of vaporization kJ/kgmg Dynamic viscosity of gas kg/msq Drying time or residence time hqc Constant drying period time hqd Falling drying period time hqm Conductive drying time hqtb Drying time in drum dryer hr s Density of solid kg/m3

rb Apparent density of material kg/m3

rg Density of gas kg/m3


w Dryer rotation rpmx Implicit functiony Implicit functionz Implicit functionA, A* Area m2

A� Drying bed area m2

a1,2,3, . . . Equation constantsAc Drying area for constant periods m2

Ad Drying area for falling periods m2

Ah Contact area with heat source (conduction) m2

Ar Archimedes Number adimensionalAt Total area m2

B Spaulding number adimensionalCeng Energy cost $/kWhC Equation Constantc1,2,3, . . . Constant cost equationcg Specific heat of gas kJ/kg Ccs Specific heat of dry material kJ/kg Ccv Specific heat of vapor kJ/kg Ccw Specific heat of water kJ/kg CDb Bubble diameter mDeff Effective diffusivity m2/sDg Gas mass diffusivity kgmol/h mdp Mean diameter of particle mDs Diameter of dryer mE Residence time hEv Drying capacity (evaporation rate) kgw/hF Equation constantf1,2,3, . . . Any generic function –G Gas flow kgar seco/sGd Dense phase gas flow kg/sG Gravitational acceleration m2/sDHml Logarithmic mean of humidity differece CH Absolute humidityHS Absolute humidity of saturated gas kgw/kgar seco

HS,BU Absolute humidity of saturated gas attemperature of moist bulb Tbu

kgw/kgar seco

Hd Absolute humidity of intersticial gas duringdense phase

kgw/kggar seco

Hf Absolute final humidity of gas kgw/kgar seco

HG Absolute humidity of gas kgw/kgar seco

H0 Absolute initial humidity of gas kgw/kgar seco

Hp Absolute humidity at surface of particle kgw/kgar seco


hc Convective coefficient of heat transfer W/m2 °Chr Convective coefficient of heat radiation W/m2 °CKH Convective coefficient of mass transfer in

terms of relative humidityg/sm2

KH*

Convective coefficient of mass transfer interms of relative humidity defined byEq. (46)

g/sm2

Kp Convective coefficient of mass transfer interms of pressure

g/sm2bar

Kb Convective coefficient of mass transfer at theboundary of the bubble.

m/s

kc Thermal conductivity of material W/m Kkg Thermal conductivity of gas W/m KKm Convective coefficient of mass transfer kgw/m2 sL Characteristic length mLs Length of dryer mM Material flow kgms/sm1,2 Curve adjustment constantsMb Material load for batch process kgms

Mc Material load in drying chamber kgMl Hold-up per unit length of rotative dryer kgMs Dry mass of material kgms

mg Molecular mass of gas kgmol/kgN Massa flow kgw/sPMA Molecular weight – water g/gmolPMG Molecular weight – dry gas g/gmolPG Gas pressure PaPBU Humid bulb saturation pressure PaPs Saturation pressure PaQ Heat flow WR Universal gas constant J/(gmol)K or

m3bar/(gmol)KRe Reynolds number adimensionalRc Constant drying rate kgw/m2 sRd Falling drying rate kgw/m2 sR* Drying rate corrected for reference area A* kgw/m2 sRm Conductive drying rate kgw/m2 srp Psychrometric ratio adimensionalS Exposed surface of material m2/m3

Sh Sherwood number adimensionalSc Schmidt number adimensionalSg Generic drying section m2


Sp Particle surface m2

Ss Section parallel to gas flow m2

St Section perpendicular to gas flow m2

T Temperature CDT Temperature difference CDTml Logarithmic mean of temperature differece CT Time sTBU Temperature of wet bulb in the ambient CTG Temperature of gas CTh Contact temperature with heat source CTr

* Absolute temperature of heat radiating source KTs Surface temperature of material CTsat,bu Saturation temperature of wet bulb CTs

* Absolute surface temperature of material KT f

g Final temperature of gas CT o

g Initial temperature of gas CT f

s Final temperature of material CT o

s Initial temperature of material CU Overall heat transfer coefficient W/m kUm Overall conductive heat transfer coefficient W/m kV� Volume of drying bed m3

Va Drag velocity m/sVc Volume of drying chamber m3

Vg Velocity of gas m/sVp Velocity of particle m/sVs Velocity of solid m/sVz Velocity of gas outlet m3/sX Moisture of material on a dry base kgw/kgms

X Mean moisture of material on a dry base kgw/kgms

XTR Transition moisture content kgw/kgms

Xd Moisture content of material during fallingperiod

kgw/kgms

Xe Equilibrium moisture content of material kgw/kgms

Xf Final moisture content of material kgw/kgms

X0 Initial moisture content of material kgw/kgms

Y Adimensional moisture adimensional

REFERENCES

BORDE, I., DUKHOVNY, M. and ELPERIN, T. 1997. Heat and mass transferin a moving vibrofluidized granular bed. Powder Handling Process.Clausthal Zellerfeld 9(4), 311–314.


FYHR, C. and KEMP, I.C. 1999. Mathematical modelling of batch and con-tinuos well-mixed fluidised bed dryerrs. Chem. Eng. Process. 38, 11–18.

JUMAH, R.Y. and MUJUMDAR, A.S. 1993. A PC program for preliminarydesign of a continuous well mixed fluid bed drier. Drying Technol. 11,831–846.

KEEY, R.B. 1978. Introduction to Industrial Drying Operations, p. 367,Pergamon Press, Oxford, UK.

KEMP, I.C. and BAHU, R.E. 1995. A new algorithm for dryer selection.Drying Technol. 13, 1563–1578.

LAPPLE, W.C., CLARK, W.E. and DYBDAL, E. 1955. Drying: Design &costs. Chem. Eng.

MABROUK, S.B. and BELGHITH, A. 1994. Simulation and design of atunnel drier. Renewable Energy l5, 469–473.

NONHEBEL, M.A. and MOSS, A.A.H. 1971. Drying of Solids in the Chemi-cal Industry, p. 301, Butterworth & Co, London, UK.

PERRY, R.H. and CHILTON, C.H. 1973. Chemical Engineer’s Handbook,5th Ed. McGraw-Hill, New York.

TREYBALL, R.E. 1968. Mass Transfer Operations, 2nd Ed. McGraw-Hill,New York.

ZAHED, A.H., ZHU, J.X. and GRACE, J.R. 1995. Modeling and simulationof batch continuous fluidized bed dryers. Drying Technol. 13, 1–28.

ZHANG, B.G., ZHOU, Y.D., NING, W. and XIE, D.B. 2007. Experimentalstudy on energy consumption of combined conventional and dehumidi-fication drying. Drying. Technol. 25, 471–474.


drying system designs: global balance and...

Documents