0716 Articulo Lecho Enpromer 2005 Corregido

7/31/2019 0716 Articulo Lecho Enpromer 2005 Corregido

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2nd

Mercosur Congress on Chemical Engineering

4th

Mercosur Congress on Process Systems Engineering

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grain needs latent and sensible heat to enter the air phase. The model proposes that the phase change takes place

at the air control volume and computational results are compared with experimental data obtained in a cross flow

deep-bed dryer with a local rice variety called Tacuar.

2. Model Development

To model the grain deep-bed drying process, the deep bed was discretized in several thin layers. The heated

air flux (wa) enters the deep bed from the bottom atz = 0, flows across the system and leaves at the top where z

= L. Two phases can be distinguished at each thin layer: the grain phase and the air phase. Uniform temperature

distribution across the grain phase was assumed because the heat Biot number is less than 0.1, but temperature

changes with thin layer position (z) and time. There exist a thermal boundary layer and an interphase condition

to match the continuity of temperatures. The Biot number for mass is greater than 100, so it can be considered

that the concentration boundary layer is too thin to take it into account so we assume the moisture at the air

phase has a uniform profile at each thin layer. In this model, the interphase between the two phases is located on

the grain surface. The latent heat needed to evaporate liquid is computed in the humid air energy conservation

equation.

To predict the average moisture time evolution in the grain phase within each thin layer, a diffusion equation

has been taken with spherical symmetry (Madhiyanon, 2002).

Chen& Pei (1989) have pointed there are three main mechanisms to model the moisture transfer within an

hydroscopic porous material exposed to a convective surface condition: capillary flow of free water, movementof liquid bound water and vapour transfer. When free water content at the surface is greater than a critical value

(approx. 30% of the saturated free water) a constant rate drying appears and mass transfer does not depend on

the inner moisture but on the outer conditions. This stage is followed by a first falling rate where the main drive

mechanism is the capillary flow of free water. The last stage appears when moisture is less than maximum

irreducible water content Xirr and, at this stage (calling the second falling rate) bounded water moves to the

surface and vapour flows through the voids of porous material. For food products bound water refers to cellular

water and no free water diffusion appears (Elbert, 2001).

From this point of view, drying process inside the grain requires a two phase model. Wang and Beckermann

(1993) has pointed isothermal two phase flows (without phase change) can be reduced to liquid flow equation.

In addition, at the second falling rate period, diffusion of bound water take place. Diffusion coefficient is

dependent on temperature with Arrhenius type functionality, while different correlations were proposed to

reflect the moisture content dependence of this coefficient (Zogzas, 1996; Elbert, 2001). Mass transfer equation

at the second falling rate period is given by

( )( )XTXDt

XS =

,. (1)


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Taking account the deep bed itself is a granular porous media we choose the simplest correlation, used at

geothermal system with success (Sondergeld, 1977), Eq. (2), with an Arrhenius function for temperature

dependence, Eq. (3):

=

eqms

eq

bedXX

XXDD 0 (2)

=RT

EDD aexp10 (3)

In Eq. (2), Xms is the maximum sorptive water content and coincides with initial moisture content Xo

(Chen,1989), while the equilibrium moisture, Xeq

, can be calculated by a Chung-Pfost type isotherm (Basunia,

1999):

( ( )[ ]+= lnln 2431 CTCCCX geq (4)

The conservation mass equation states that the moisture loosed by the grain phase enters to air phase:

z

Hw

t

Xa

ss

a

=

(5)

On the other hand, we must apply the energy conservation equation to each thin layer. Inlet moisture air lost

sensible heat to increase rice temperature, to evaporate water coming from the grain and to increase its

temperature from Ts to Tg. This energy balance is given by:

( ) ( ) ( ) ( )[ ]z

HTTcTlw

t

Tcc

z

THccw sgpvsvaa

spwpsss

g

pvpaaa

+

+=

+ (6)

Madhiyanon et al. (2001) have pointed that this equation can be divided in two parts, through a convection

coefficient and an effective area which take account the temperature difference between the grain an the air

phase.

The model states a volumetric pseudo-convective heat transfer coefficient, (ha)eff, that mediated the change

of the rice temperature (solid and liquid phase), defined by:

( ) ( ) ( )t

TXccTTha spwpssssgeff

+= (7)

while we have the following equation to modelling the temperature changes on the air phase:


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( )( )

( ) ( ) ( )[ ]z

HTTcTlTT

w

ha

z

THcc sgpvsvsg

aa

effg

pvpa

+=

+

(8)

This equation states that moisture coming from the grain evaporate at the humid air phase itself. As we show

later, simulation results are more robust in front of this pseudo-convection coefficient (ha)eff so this model solve

the problem of choose a correlation for the heat transfer coefficient. On the other hand, this idea resembles the

evaporative cooling which has been pointed by Thompson et al. (1968), although they supposed grain and air

are at thermal equilibrium. The model takes account that grain must not be in thermal equilibrium with air, at

least, on the earliest steps of the experiment.

3. Parameters Determination

Diffusion parameters (D0, Ea, D1) in Eq. (3) were estimated running several thin layer experiment for

different inlet air conditions (temperature and relative humidity). For the thin layer of grain, we apply the same

mass transfer equation (Eq. (1)) but changing the dependence of diffusion coefficient (D tl) during drying

process. We propose a time-dependent diffusion coefficient for a thin layer (Alvarez and Leguez, 1986), as:

b

tlR

tDDD

+=

2

0

00 1 (9)

In this way, the model takes account that diffusion process is a function of moisture content inside the grain.

In the case of a thin layer of grain, with diffusion given by the Eq. (9), the mass transfer equation admits an

analytical solution for the moisture average in a spherical geometry with fixed boundary condition at the grain

surface: X(r=R0) = Xeq

( )

+

+=

+

= 11

1exp

161

2

22

122

b

0

0

meq0

eqt

R

D

b

m

mXX

XtX(10)

where X0 is the initial moisture of the solid and Xeq is the moisture at equilibrium, calculated using Eq. (4).

Using Eq. (9) for diffusion coefficient variation, the model implicitly takes account that the diffusion

process is a function of moisture content inside the grain, while Eq. (3) shows that diffusion coefficient vary not

only with moisture content but also with temperature in a Arrhenius type dependence. Diffusion parameters (D0,

b) in Eq.(9) were estimated running several thin layer experiment for different inlet air temperatures and D(t)

values were calculated from Eq. (9) for different values of diffusion parameters (D0, b). For a constant inlet air

temperature, Eq. (9) shows that D(t) decreases due to b < 0. This property assures, as time goes on, that moisture

content must decrease slower than in a drying process with constant diffusion coefficient during the falling rate

period. Figure 1 shows the variation of diffusion coefficient D(t) with drying time, represented by Fourier


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number in abscissa (Fo= Dot/R2

o), where the diffusion parameters were determined by fitting Eq. (9) with

experimental data as it was indicated above.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.0 0.5 1.0 1.5 2.0

Fourier number, Fo

Diffusioncoefficient

Dex

1011,

(m2/s)

T=41.0C

T=57.8C

T=81.8C

Fig. 1 Diffusion coefficient dependence

If we choose the same values for D1 and Ea to model the deep bed behavior with a diffusion coefficient given

by Eq. (9), we observe that Ro is no longer the equivalent grain radius but an effective radius for the whole thin

layer and can be applied for each one of the thin layers in which the deep bed was discretized.

4. Computation Performance

At each time the algorithm solves the Eqs. (1) to (4), (7) and (8), using a computational FORTRAN program

for numerical solving of the simultaneous equations. Deep-bed is divided in 128 thin layers to compute Eq.(1)

by Cranck-Nicholson scheme for each one of these layers, taking an initial random condition for the average

moisture at each layer, around its initial average experimental value X0:

( ) [ ]( )1,11.01 +== randX0tz,X 01 (11)

To compute the Cranck-Nicholson scheme, each sphere is divided in 50 nodes with same weight and with

an initial random condition in each node to model inhomogeneous moisture content inside the grain kernel:

( ) [ ]( )1,101.01 +== randX0tz,X 1s0 (12)

and the following boundary conditions:

( )

00,

,0

0,0

0

00

==

==>

=


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Following Chen & Pei (1989) maximum sorptive moisture Xms was taken as the initial moisture content X0.

Equations (2), (3) and (7) were computed by 4th order Runge-Kutta scheme, with the following boundary

conditions:

ing

ing

ss

HHzt

TTzt

TTTTLzt

==

==

==


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the initial average moisture (Xo) and Figure 3 shows the outlet temperature as function of time for three inlet

temperatures.

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

90.0

100.0

0 10000 20000 30000 40000

time (s)

Outlettemperature(C)

Numerical DataT=38.3C

Experimental DataT=38.3C


Experimental DataT=57.8C


Experimental Data

T= 86.3C

Fig. 3 Comparison between predicted (solid lines) and experimental data for outlet temperature of air

Deep-bed characteristics were summarized at Table 1, including the volumetric pseudo-convective

coefficient (ha)eff chosen for these plots. The model can not handle temperature variation on the axis deep-bed

perpendicular to the flow direction, so the later mean outlet temperature as Tin(eff) was chosen to perform the

simulation.

Table 1: Model Parameters

Table 2 shows the accumulative error in experimental drying for the three temperatures assayed. This error is

defined as:

Tin(oC) 39.7 60.4 89.3

Tin(eff)(oC) 38.3 57.8 86.3

Xo (db) 0.2082 0.2294 0.2244

Hin(kg/kgD) 0.0074612 0.0093423 0.0097324

T(

o

C)21.2 25.2 26.6

ss (kg/m3) 534.1 534.1 534.1

w (m/s) 0.3 0.3 0.3

Ro(m) 0.00165 0.00165 0.00165

L (m) 0.105 0.105 0.105

S(m2) 0.0176 0.0176 0.0176

(ha)eff 8000 8000 8000


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cpv Specific heat of vapor (J kg-1 K-1)

cpw Specific heat of water (J kg-1 K-1)

D Diffusion coefficient (m2 s-1)

Do Diffusion coefficient at initial time (m2 s-1)

Dtl Diffusion coefficient for the thin layer (Eq. 9) (m2 s-1)

Ea Activation energy per unit mass (J kg-1)

(ha)eff Volumetric pseudo-convective heat transfer coefficient (W m-3 K-1)

H Absolute air humidity (kg moisture / kg dry air)

Hin Inlet absolute air humidity (kg moisture / kg dry air)

Hsat Saturation absolute air humidity (kg moisture / kg dry air)

lv Vaporization heat of water (J kg-1)

D1 Constant of Arrhenius-type equation (Eq.3) (m2 s-1)

L Deep bed depth (m)

NX Number of moisture average data (-)

NT Number of outlet temperature data (-)

Q Heat flux per unit volume (J m-3)

R Ideal gas constant (J kg-1 K-1)

Ro Equivalent particule radius (m)

R Radial direction of spherical coordinate (m)

T Time (s)

Tg Air Temperature (K)

Tin Inlet air Temperature (K)Tout Oulet air Temperature (K)

Ts Grain Temperature (K)

Ttr Triple point Temperature (K)

T Environment Temperature (K)

wa Axial drying air flux per unit of deep-bed cross sectional area (m s-1)

X Grain moisture content (kg moisture / kg dry grain)

__

X Average grain moisture content (kg moisture / kg dry grain)

Xeq Equilibrium moisture content (kg moisture / kg dry grain)

Xms Maximum sorptive moisture content (kg moisture / kg dry grain)Xo Initial average moisture content (kg moisture / kg dry grain)

Z Axial direction for cylindrical coordinate (m)

Greek Symbols

a Density of dry air (kg m-3)

ss Apparent density of dry grain (kg m-3)

Relative humidity

Accumulative error

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0716 Articulo Lecho Enpromer 2005 Corregido