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CONJUGATE ANALYSIS OF NATURAL CONVECTIVEDRYING OF BIOLOGICAL MATERIALSLeandro S. Oliveira a , Mauri Fortes b & Kamyar Haghighi aa Dept of Agricultural Engineering , Purdue University , W. Lafayette, IN, 47907, USAb Departamento de Engenharia Mecanlca , Universidade Federal de Minas Gerais , CampusPampulha, Belo Horizonte, MG, 31270-900, BrasilPublished online: 21 May 1994.

To cite this article: Leandro S. Oliveira , Mauri Fortes & Kamyar Haghighi (1994) CONJUGATE ANALYSIS OF NATURALCONVECTIVE DRYING OF BIOLOGICAL MATERIALS, Drying Technology: An International Journal, 12:5, 1167-1190, DOI:10.1080/07373939408960994

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DRYING TECHNOLOGY; 12(5), 1167-1190 (1994)

CONJUGATE ANALYSIS OF NATURAL CONVECTIVE DRYINGOF BIOLOGICAL MATERIALS

Leandro S. Oliveira I J Mauri Fortes2J and Kamyar Haghighi 1

1. Dept of Agricultural Engineering, Purdue University. W. Lafayette, IN.47907, USA

2. Departamento de Engenharla Mecanlca, Universidade Federal de MinasGerais, Campus Pampulha. Belo Horizonte, MG, 31270-900, Brasil

Key Words and Phrases: Numerical Simulation; Finite Volume Technique; Heatand Mass Transfer.

ABSTRACT

This paper presents a numerical analysis of heat and mass transport duringnatural convective drying of an extruded com meal plate. The conjugate problemof drying and natural convection boundary layer Is modeled. The finite volumetechnique was used to discretize and solve the highly nonlinear system ofcoupled differential equations governing the transport inside the plate. Theboundary layer solution was obtained by means of a finite difference softwarepackage that utilizes Runge-Kutta's 5th order method to solve the Inherent,transport equations. A methodology for evaluating the heat and mass transfercoefficients dUring the numerical simulation was developed and successfullyimplemented. The results showed that there is no analogy between heat andmass transfer coefficients for this type of problem.

INTRODUCTION

Drying Is the most widespread heat and mass transport process and one of

the most energy-consuming industrial operations. This unit operation is present In

several industrial sectors which Include. among others, food, grain, wood and

1167

Copyright iC)1994 by Marcel Dekker. Inc.

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1168 OLIVEIRA. FORTES. AND HAGHIGHI

textile Industries. Drying can be defined as the process of moisture removal from

a porous solid, and it is performed for many purposes such as minimization of

transportation and handling costs, and attainment of a desired moisture content

for the preservation or subsequent processing of the product. This paper is mainly

concemed with drying of biological products.

A great deal of theoretical and experimental studies have been carried out in

order to understand the drying mechanisms of hygroscopic porous media, more

specifically cereal grain. when subjected to thermal processing. Comprehensive

reviews on this subject area are available in the literature (Fortes and okos. 1980;

Wannanen et aI., 1993). A full modeling ot dry.ing processes by any of the exist·

ing theories should take Into account the solution at nonlinear diffusion equations

coupled to boundary layer equations as applied to the drying medium. Analytical

(Lobo 8t al.• 1987; Robbins and Ozisik, 1988; Aparecido et et., 1989; Uu and

Cheng, 1991) and numerical (Thomas et aI., 1980; Haghighi and Segerlind, 1988;

Nasrallah and Arnoud, 1989; lrudayaraj et al.• 1990; Ferguson and Lewis. 1993)

solutions are available in the literature for drying models where a conjugate prob­

lem was not considered. These solutions were 'obtained with the assumption that

the heat and mass transfer coefficients were known a priori. The conjugate

approach eliminates the need for convective heat and mass transfer coefficients.

These coefficients can be evaluated during the simulation process.

The analysis of drying processes as conjugate problems have been recently

considered. Prat (1986) analyzed the conjugate problem ot heat and mass tran­

sport In convective drying of a horizontal layer ot sand. The drying model for the

sand was solved using the finite element method, and the equations for the boun­

dary layer flow were solved by means of the finite volume' method. Whitaker's dry­

Ing model (Whitaker. 1977) was used to desaibe the diffusion of heat and mois­

ture within the sand layer, and Prandtl equations were used for the transport in

the boundary layer. The results showed that the time variation of the heat and

mass transfer coefficients. during the numerical simulation. did not differ

significantly from those obtained analytically for uniform temperature and specific

humidity at the interface. Their variation with distance along the plate surface was

significant and the heat and mass transfer transfer analogy did not apply in this

case. The conjugate drying problem of a material of ,arbitrary thickness pulled

through a heat-transfer agent was analyzed by Dolinskiy et al. (1991). The prob-

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NATURAL CONVECTIVE DRYING 1169

lem was described by a system of coupled differential equations and conditions of

conjugation on the material surface. The finite difference method was used to

numerically solve the system of equations. The heat and mass transfer

coefficients were evaluated iteratively. The results indicated that the conjugate

approach led to decreased rates of drying and moistening compared to results

from conventional approaches. Analogy between the heat and mass transfer

coefficients was not observed in this case.

There is no work reported in the literature regarding conjugate analysis of dry­

ing of biological materials. In this paper, the conjugate problem of natural convec­

tive drying of an extruded corn meal plate is analyzed. The finite volume tech­

nique was applied to solve the highly nonlinear system of coupled partial differen­

tial equations governing the transport inside the plate. The boundary layer solu­

tion was obtained by means of a finite difference method that utilizes a special

shooting technique and Runge-Kutta's 5th order method.

The objectives of this work were to develop and implement a numerical

methodology to evaluate the heat and mass transfer coefficients during the

natural convective drying of a porous slab of biological material.

THEORETICAL ANALYSIS

This section presents the models to be used in the analysis of the simultane­

ous heat and mass transfer within the solid and the transport in the natural con­

vection boundary layer..

Heat and mass transfer within the solid

The drying model developed by Fortes and Okos (1981) was adopted as the

governing equations for the simultaneous heat and mass transfer within the solid

material. This model was developed under the following assumptions: the porous

medium Is Isotropic and continuous; local thermodynamic equilibrium exists; diffu­

sion is the predominant mechanism for moisture migration; gravitational and other

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1170 OLNElRA, FORTES, AND HAGHIGHI

fJeid forces are negligible; and chemical reactions and shrinkage effects are

absent

The flux and conservation equations, as applied to hygroscopic porous bodies

(e.g., biological products), are:

Heat flux rJq ):

:1.=- KrVT- [P/K,R.,nH+ K+~ ~~ + He:;]] R;:' ~~VM (')

Uquld flux <1,):

Mass conservation equation:

aM --t-;Ps¥ =- vM+ oJv)

Energy conservation equation:

sr aM 1. :-tPSCbar -PsLwaf =- V q - LvVJv

(2)

(3)

(4)

(5)

where T is the temperature, M Is the moisture content. p" P" 1'110 are the solid,

the liquid, and the saturated vapor densities, respectively, Cb is the solid heat

capacIty, KT, K" and Kv are the heat conduction coefficient, the liquid. and the

vapor diffusivitles, respectively, H Is the Isothermal equilibrium moisture content,

Rv Is universal constant of gases, and Lv and Lw are the heat of evaporation of

water and the heat of adsorption, respectively.

In the equations above V M. the equilibrium moisture content gradient, is Impli­

citly taken as the isothermal moisture diffusion driving force. Equations (1) to (3)

are substituted In equations (4) and (5) to give the final forms of the mass and

energy transport equations as

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NATURAL CONVECTIVE DRYING 1171

The initial conditions for the above equations are:

M{x,y, 0) =MoT{x,y, O) =To

(8)

where x and yare the spatial coordinates within the porous medium and the zero

indicates lime t = O.

The boundary conditions are as follows:

- at the adiabatic and impermeable walls:

J/+Jy =o

- at the convective drying interface:

J/ +Jy :::: hm{Pvs - p....)

(9)

(10)

(11)

(12)

where hr and hm are the heat and mass transfer coefficients, respectively, Pvs is

the vapor pressure at the surface, and p.... is the air vapor pressure of the undIs­

turbed air. The subscripts sand 00 stand for surface and undisturbed air, respec­

tively.

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1172

Natural convection boundary layer

OLIVEIRA, FORTES, AND HAGHIGHI

The boundary layer equations governing the natural convection drying process

are (Ferreira sf st., 1991):

Continuity:

Momentum:

au au • ()2uu ax + vay =gtNT- T..) + gf3 (C - C..) + v ay2

Energy Conservation:

st st a2 Tu-+v-=a--ax ay ay2

Mass Conservation:

sc sc a2cu-+v-=D--ax oy oy2

(13)

(14)

(15)

(16)

where u is the air velocity in the x-directlon, v is the air velocity In the )'-direction,

C is the humidity ratio, g is the acceleration due to gravity, v is the kinematic

viscosity, pand p. are the thermal and mass expansion coefficients, respectively,

and a and 0 are the mass and thermal diffusivities. respectively, as applied to the

humid drying air. The inherent boundary conditions are:

- at the undisturbed fluid (away from the solid matrix surtace):

u= 0

T=T...

c=c...

(17)

(18)

(19)

- at the solid-fluid interface, the temperature, the moisture content, and the heat

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NATURAL CONVECTIVE DRYING 1173

and mass fluxes must be continuous. The mathematical expressions tor these

conditions are

(20)

Cf= Cg(21)

(22)

(23)

where the subscripts f and s represent the fluid and the solid surface. respec­

tively. The fluxes Jq • JI. and J.... are given by equations (1). (2) and (3), respec­

tively.

IMPLEMENTATION

Solution methodology

A finite volume code was developed to solve the set of nonlinear coupled partial

differential equations describing the heat and mass transport that occurs during

the drying of a porous medium. The discretization of the differential equations fol­

lowed the methodology developed by Patankar (1980). The coupling terms In

equations (6) and (7) were considered to be a source term in the discretization

process. and. since they both are diffusive terms, they were discretized in the

sarne manner as the other terms in the equations. 80th the temperature and

moisture profiles were assumed to vary linearly inside each control volume. In

order to solve the boundary layer equations, the finite difference code developed

by Ferreira at al. (1991) was used.

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1174 OLIVEIRA, FORTES, AND HAGHIGHI

The Implementation of the proposed methodology works In the following way:

the coupled system of differential equations for the drying of the porous medium

Is solved for a time step at, assuming approximate values for the heat and mass

transfer coefficients. hr and hm• respectively. The values of humidity ratio and

temperature obtained lor the solid surface are then used as boundary conditions

for the boundary layer equations. After the solution of the boundary layer equa­

tions, new values for the heat and mass transfer coefficients are evaluated and

used in the convective boundary conditions for the porous medium drying model.

This procedure is repeated until the convergence of the values for the heat and

mass transfer coefficients is achieved, and only then the next advance in time is

performed. The proposed simulation methodology is carried out until the final time

is reached.

It should be mentioned that in the proposed methodology the boundary layer

problem was assumed to be steady-state. whereas the porous medium drying

process was necessarily transient, This assumption was based on the fact that

the time scales for the natural convection process are much smaller than those

lor the drying process (Prat. 1986).

Finite Volume Discretization

The equations governing the transport inside the porous medium were discre­

t1zed following the lines of Patankar (1980). The calculation domain was divided

Into small control volumes and the differential equations. were integrated over

each control volume (Figure 1). Piecewise linear profiles expressing the variation

of temperature and moisture potential were used to evaluate the required

Integrals.

Rewriting equation (7) in a condensed form:

(24)

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NATURAL CONVECTIVE DRYING

N

n OYnr------ -------] TEW w P Ie,,"

I

1I

IIII

OYs------_______ .J

S

IAI 'v ill

5~x ~xow -

Figure 1: Two-dlmensicnal control volume.

where

1175

and

v r aH dpvo]D22 =Kr+ LV"l'lPlio aT + H dT

The discretized form of equation (24) is the following:

apTp =aETE + awTw+ aNTN + asTs + b

where

(26)

(27)

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1176 OLIVEIRA. FORTES. AND HAGHIGHI

and

where

and

D22•as=--!u

(Sy)s

ap = aE + aW + aN + as + a~

s .ILeS,dxdyc>: ~y

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

where S, represents the source term In the d1scretlzed equation.

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NATURAL CONVECTIVE DRYING 1177

The source term, Sr. is discretized in the same manner the differential equa­

tion was:

(37)

where

and

where

D21ass = -_·-tu~y)s

app = aEE + aww + aNN + ass

(38)

(39)

(40)

(41)

(42)

(43)

fJMp

T=(Mp- Mp)

M(44)

where Mp is the moisture potential at the previous time step. The discretization of

equation (6) is similar to the one presented for equation (7).

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1178

Examples

OLIVEIRA, FORTES, AND HAGHIGHI

To illustrate the performance of the proposed methodology, the example prob­

lem of a rectangular.slab of extruded com meal under natural convective drying

was selected and successfully implemented. The schematic diagram of the prob­

lem is presented in Figure 2. The lett side of the slabts assumed to be adiabatic

and Impermeable. The heat and mass transfer through the upper and lower sur­

faces are assumed to be negligible compared to the transfer on the vertical sur­

face at the right. The thermophysical properties are listed in Table 1.

In order to validate the proposed methodology, the finite volume code was

verified by reproducing the experimental results and numerical data obtained by

Fortes and Okos (1981) for the convective drying process of cylindrical extruded

corn meal. The numerical analysis was performed for two different temperatures

for the undisturbed air: T_ =5f1JC and T. = 8rPC. The initial conditions were

To=3rP C and Mo =0.3295 (dry basis) for both analyses. For the drying model

solution, a finite volume mesh of 7x19 control volumes and a time step size

At = 1 sec were used. A qualitative description of the natural convection flow is

shown in Figure 3.

The contour plots for temperature and moisture content distributions within the

slab, for the final drying times of If =600 sec and tf = 1800 sec are presented in

Figures 4-7. It can be seen that the upper right comer of the slab dries faster than

the remainder. The reason for that is a nonuniform drying along the solid surface,

caused by a decrease In the moisture gradient between the solid and the air as it

flows down, t.e., the moisture content in the air increases as it flows along the sur­

face, due to the removal of moisture from the solid. As a further consequence,

the local convective mass transfer coefficient decreases in the flow direction.

Simultaneously, the air cools down, releasing heat to the solid surface and lead­

Ing to a smaller temperature gradient between the surface and the air stream.

This cooling of the air causes the heat transfer coefflclent to decrease along the

solid surface. It Is clear from Figures 6 and 7 that a drying front moves from the

dryIng surface towards the Impermeable surface at the left of the slab. It Is also

clear that the temperature and moisture distributions are not uniform throughout

the slab. Based on this fact, we can conclude thai: biological products with large

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NATURAL CONVECTIVE DRYING 1179

12cm

j_---2cm--.

Figure 2. SChematic diagram of the problem.

TABLE 1

Thermophysical Properties.

Property Expression Unit

H 1 - 9Xp[ - 2. 736(T - 273.15)7.242MJ·3,9J -Kr 0.1133- 2.936fvf + 25.44MJ - 38.71M' W/mK

pg 1445.0 Kg/m.:l

Cb 4180(0.343+ M)/(1 + M) JIKgK

A 1.59x1(T.:I mhw 1569. 928Jf! (T - 273. 15F242MJ·379 (1 - H)/H J/Kg

hv 3.11x 1at> - 2.3&<.1cflT+ hw J/Kg

K, ~149x1(T'~8-2747n sx, 9.085>< 1(T8(T - 273. 15)0.3016 (J-fJ.25 _ H,·25) m21s

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1180 OLIVEIRA, FORTES, AND HAGHIGHI

airflow

Figure 3. Schematic diagram of the natural convection flow.

flat surfaces under convective drying will be subjected to localized high tempera­

ture and moisture gradients. leading to nonuniform product quality.

Figure 8 shows the mass average moisture content variation with time for

T_ =SCPC and T_ =BCPC. It is interesting to note that. even for natural con­

vective drying, the drying rate increases rapidly with an increase in the air tem­

perature.

The time variation of the heat transfer coefficient is represented in Figure 9.

Curves 1 to 7 correspond to the following locations at the surface: Xt =0.17 em.

x2 = 0.50 em, X3 =0.78 em, X4 =1.00 em. Xs =1.20 em, x6= 1.50 em. and

x7 = 1.80 em. The heat transfer coefficient decreases monotonously with time as

the drying front recedes to the Interior of the slab.

Figure 10 shows the variation of the mass transfer coefficient with time. In the

initial stages of drying, the energy brought about by the air stream is not enough

to promote evaporation of the liquid on the solid surface and the mass transfer

rate Is very slow, i.9., the mass transfer coefficient is very small. As the evapora­

tion rate increases, hm Increases. reaching a maximum when all the liquid on the

surface has been evaporated. When the surface Is dried out. an evaporation front

recedes to the interior of the slab. The mass transfer from within the slab towards

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NATURAL CONVECTIVE DRYING

..,.

..,.

..,.

~ ,'"....-....-

J. TI~

•

1181

• .~ ••,£ ,I" .•• ..11 .•12 .'U .•" .'t' .&:111I

(a)

....

....

..,.

...,~ ,II'

.-.-.....• 16.:===!:.t:::=:::!o::=:!:::c:::!.:::=!:::.::!=6c!:::t::!::,d::z:!:::.!::::::l,

• ..a." ... ." .'11 .•12 .•;... .11' .•11 ...I

(b)

Figure 4. Temperature distributions for (a) t,=600 sec and (b) t,= 1800 sse

(T_ = Sd'Cl

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1182 OLIVEIRA, FORTES, AND HAGHIGHI

t ~J

~t \

l \~1\

•• .M2 0.' .ft6 .In .•t. ..12 .• u .". ..'. .GII

I

.-

....

,...

..,.

,'"

....

.-... ,Itt

..-

(a)

• -1 n•

~\· .... ,0'"•

tI. fa

•• .., ••' ... .... .It. .112 .11.4 .au .111' .III:M

I

."

....

...

.-

...

..,

.-

.-.-

(b)

. Figure 5. Temperature distributions for (a) t,= 600 sec and (b) t,= 1800

sec (T.. =8f1'C).

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NATURAL CONVECTIVE DRYING

~ .It'

.-.-...

....M;! .•• .••• .nl ."'.. . '12 ..'1. ..,. .". .IIQI

•(a)

.-...

....• 1b==l::=:::I:::===l::=:::I:::===l::=:::I:::===!;c!;,1!;;!;!;

• .IG ..... .... .11' .'1' .IU .•14 .•16 .•'. .l12li

•(b)

1183

Figure 6. Moisture distributions for (a) t,= 600 sse and (b) t, = 1800 sse(T_ = 5CfC).

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1184 OLIVEIRA, FORTES,AND HAGHIGHI

.-...

...• .~ ."4 .••~ .••• .111 .•12 .814 .•" .•'.. .13

•(a)

.... fI"="'========"Fi'

.•t •

.• u

.....•11,1

........-.-......

I ..... .... .... .... ,.U .112 .1" .•,. .•,. .1:3

•(b)

Figure 7. Moisture distributions for (a) t,= 600 secand (b) t,= 1800 sec(T_ =8CfC).

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0.33

(al

M 0.265

(bl

0.2I I I0 5500 11000

Time

Figure 8. Mass average moisture content for (a) T. == 5CfC and (b) T. = 8Cf

C (If= 10800 sec).

3

0 5500 11000Time

(a)

13

hr 8

~ -.-I-=::

3

0 5500 11000Time

(b)

Figure 9. Heat transfer coefficient variation for (a) T. = 5Cf C and (b) T. =8(fJ C (If = 10800 sec).

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1186

4

hm 2.8

1.6

OLIVEIRA, FORTES, AND HAGHIGHI

o 5500Time

(a)

11000

6.5

hm 4.5

2.5

o 5500Time

(b)

11000

Figure. 10. Mass transfer coefficient variation for (a) T. =S(fJ C and (b) T. =8(fJC(tf = 10800 sec).

Its surface is all in the vapor phase and it is driven by a moisture potential gra­

dient At this stage. the mass transfer coefficient decreases as the moisture gra­

dients decrease. As drying proceeds. vapor diffusion lowers. and total mass

transfer becomes relatively constant, leading to a constant mass transfer

coefficient at the final stages of the drying process. It Is interesting to note that

the maximum values of hm shifts to the right along the time axis from one curve

to another as the distance along the surface increases. Locations closer to the

leading edge dry out faster than the other locations on the surface.

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NATURAL CONVECTIVE DRYING 1187

There are no data available in the literature regarding heat and mass transfer

coeffICients obtained from conjugate analysis of drying of biological materials. A

great number of the heat and mass transfer coefficients utilized In drying simula­

tion stJdles or experiments have been obtained by utilizing analogies between

heat and mass transport phenomena The results obtained in this study indicated

that the use of such analogies may be misleading. The associated errors are car­ried over the theoretical and experimental data obtained for the heat and mass

transfer coefficients.

CONCLUSIONS

A numerical analysis of the conjugate problem of natural convective drying of

a slab of biological material was presented. A methodology for the evaluation of

the heat and mass transfer coefficients during the numerical simulation was pro­

posed and sucx:essfully implemented. This methodology can be applied to the

analysis of any drying process, as long as it is treated as a conjugate problem.

The results Indicated that the use of transfer analogies for the type of problem

analyzed may be misleading because no analogy was observed for the transfer

coefficients evaluated.

NOMENCLATURE

C Humidity ratio

D Thermal ditfusivity

H Isothermal equilibrium moisture content

J Flux

Kr = Heat conductlon coefficient

K, =Uquid dittusivity

K; = Vapor diffusivity

Lv ... Heat of evaporation

LtIII = Heat of adsorption

M '" MoistJre content

Pv =Vapor pressure

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1188 OLIVEIRA. FORTES. AND HAGHIGHI

R = Universal constant of gases

T =Temperature

9 =Acceleration due to gravity

u, v .. Velocities

x, y .. Cartesian coordinates

Greek Letters

ex = Mass diffusivity

~, ~. .. Thermal and mass expansion coefficients

v =Kinematic viscosity

p = Density

Subscripts and Superscripts

o .. Initial

b .. Body

f '" Fluid, final

I '" Liquid

m co Moisrure

q = Heat

5 '" Solid. surface

T· '" Heat

v. Vo = Vapor, saturated vapor

- = Undisturbed fluid

ACKNOWLEDGEMENT

Leandro Oliveira wishes to thank the Brazilian govemment agency CAPES for

his scholarship.

REFERENCES

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NATURAL CONVECTIVE DRYING 1189

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1190 OLIVEIRA,FORTES, AND HAGHIGHI

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