High-intensity infrared drying study

Chemical Engineering and Processing, 32 ( 1993) 3 II- 3 18

High-intensity

311

infrared drying study

Part I. Case of capillary-porous

material

P. Navarri Cetiat-Villeurbanne,

27-29,

Boulevard du II novembre

1918, BP 6084, 69604 Villeurbanne

Cedex (France)

J. Andrieu* UniversitP Claude Bernard Lyon I, Lnboratoire d’Automqtique et de G&es 43, Boulevard du I1 novembre 1918, 69622 ViIleurbanne Cedex (France)

des Pro&d&

LAGEP-URA-CNRS,

Britiment

721,

(Received March 18, 1993; in final form May 7, 1993)

Abstract Experiments involving the drying of a thin horizontal unconsolidated wet sand layer under intensive infrared radiation (TR) have been performed in a laboratory-scale dryer. The drying rates and solid temperature data were much higher than in conventional convective processes. Coupled heat and mass-transfer phenomena have also been analyzed. It was observed that medium-infrared radiation (medium-IR) was more efficient than near-infrared radiation (near-IR) and that the analogy between heat and mass transfer is still valid in the case of high mass fluxes. A simple receding model based on the assumption that IR radiation is absorbed at the surface of the product could predict the drying rates and the temperature profiles up to the zero moisture content. The experimentally determined parameters of the main model were the mass transfer and infrared absorption coefficients.

Introduction The main advantages of infrared heating mode which make it attractive for drying processes are (i) no resistance of gases towards heat flux as in the case of convection and no direct contact required as in the case of conduction, (ii) the radiation can be focused as in the case of light, (iii) the heat-transfer fluxes can be very high (up to several 100 kW m-*) so that heaters

can be very compact and (iv) fast response times are obtained which allow easy and rapid process control. Despite these advantages, however, this heating and drying treatment is only homogeneous in the case of flat and thin materials. Furthermore, although higher thermal levels and drying rates than those obtained in pure convective processes can be achieved, the main limiting factor is the product radiation absorptivity (~1)because the incident radiation is not completely absorbed. This important material characteristic depends on many parameters (radiation wavelength and nature of the product) and should be determined experimentally. In addition, when thermosensitive materials are dried, thermal damages can occur if the IR flux is not con-

*To whom correspondence

025%2701/93/$6.00

should be addressed.

trolled; in these circumstances, it is necessary to know how the operation parameters (IR flux, air velocity) affect the quality factors. Nevertheless, this combined drying mode is very attractive and its applications are more and more numerous (paper, textile, panels, granules, etc.). However, few works have been devoted to the study and modelling of drying processes under the influence of IR radiation [l-7] and very few in the case of high IR fluxes as encountered in industrial applications [5-71. As the first stage of a research programme on the IR drying of thin-layer products, the present work was undertaken with a non-sensitive material-an initially saturated layer of sand-in order to investigate some heat and mass-transfer phenomena occurring during the combined radiative and convective drying of commercial materials.

Experimental All details of: the apparatus and of techniques can be found in ref. 8. is shown schematically in Fig. 1. were conducted under constant drying ambient air: the air temperature and

0 1993 -

the experimental The test section All experiments conditions with air velocity were

Elsevier Sequoia. All rights reserved

312

ii

fs -7 2

Qmd

2

5

IE ’

0 0.05

0

0.1 y

Fig. 1. Schematic representation dow; 2, radiator; 3, sample.

of the apparatus.

1, Glass win-

0.15

02

0.25

(kg.kg1)dry basis

Fig. 2. Drying kinetics as a function of the air velocity.

Qrad = 9.7kW.m-2NIR

controlled

automatically by means culating laboratory-scale tunnel heaters were used as radiative heat ing to different spectral fields, i.e.

of a convective recirdryer. Two electric sources correspond-

a near-infrared panel heater (near-IR) made of two standard 2 kW quartz lamps with tungsten wires as. radiative squrces - the wire temperature was greater than 2200 K with the wavelength corresponding to the emitted energy maximum being close to 1.2 pm, and ‘a Iriedium-infrared panel heater (medium-IR) compQed of thret double wires made of Ni-Cr alloy and protected from convection by quartz tubes with a maximum IR intensity around 2.5 pm. The radiative flux transmitted was measured in situ by means of a Schmidt-Boelter type sensor. The experimental conditions were as follows: air velocity u, = 26 m s-‘; air humidity = 20-40%; air temperature T, = 20-22 “C; IR heat flux, Qrad = 9.7- 17.8 kW m-’ (near-IR heater), Qrad = 7.2- 10.3 kW me2 (medium-IR heater). The material used was a 5-mm thick unconsolidated granular bed composed of river sand and water. The drying rates were determined from recordings of the sample weight obtained via an electronic balance. For near-infrared radiation, the sand surface temperature was measured with an infrared pyrometer (accuracy, + 1.5 “C) and a flat thermocouple was-used in the case of medium-infrared radiation. A NiCr-NiAl thermocouple placed horizontally at 4 mm depth in the layer recorded the inner bed temperature.

Results Experimental dutu

Drying experiments were performed as a function of the two main.operating parameters, namely the infrared power and the air velocity, the sand being initially saturated with water. :Under these conditions, the dryingirate curves exhibited the three phases for a conven-

1

I

0 0

0.05

0.1

0.15

0.2

0.25

0.3

r (k&kg- I) dry basis

Fig. 3. Surface temperature as a function of the air velocity. 25

I

I

0.05

at

I

I

0.15

0.2

I

I

0.25

0.3

20 =

4

z gi)

15

10

IE

0 0

y jkg.kg- 1) dry basis

Fig. 4. Drying kinetics as a function of the infrared flux density.

tional drying process as shown in Figs. 2-5. These phases are: ( 1) A transient heating period (phase 0) where the drying rate increases gradually. It is characterized by a temperature gradient within the layer. The length of this period depends on the equilibrium temperature of phase I and represents approximately 30% of the total drying time. (2) A constant drying-rate period (phase I) during which the sample temperature is nearly constant up to the critical moisture content, Xc,,. (3) A period (phase II) during which the rate diminishes and when an evaporation front migrates into the porous body separating the moist part from the dry part. A temperature gradient occurs during this stage due to the low thermal conductivity of the dry zone and the very high IR fluxes used.

313

where &rro= 0.90 and E,,,, = 0.90, and represent the emissivities of the product and the surface, respectively. The convective flux may be calculated either from the general relationship:

QCO””=$h=ln

1+

cp, “(T, - TS) L ”

(3) { or by means of a simplified equation which does not take the effect of mass transfer into account:

20 0

0.05

0.1

0.15

0.2

0.25

0.3

K (k&kg- 1) dry basis

Fig. 5. Surface

temperature

as a function

Q

CO””

of the infrared

density.

The stationary surface temperature (50-85 “C) and the high drying rates (lo-22 kg m-* h-‘) observed during phase I demonstrated the efficiency of the radiative heating employed. Although the incident IR flux density was the, main parameter, it was also observed that the temperature was considerably affected (lo-20 “C less) when air under room-temperature conditions was blown over the sample surface whereas the drying rate was much less affected (10% less). Consequently, thermosensitive products, which would be damaged when IR radiation is used without air removal, can be dried under convective air conditions whilst still maintaining high drying rates. Analogy between heat and mass transfer It is well known that during pure convective drying, the Colburn analogy allows determination of the convective transfer coefficient values to be determined generally k,,, from h, values via the psychrometric ratio q = f(Le) and from values of other coefficients. However, this analogy has been rarely checked in the case of high radiative densities, except recently by Parrouffe [ 91. For this reason, we decided to check this analogy, initially for the case of a pure convective process and then in the presence of high radiative densities. Pure convective drying Drying experiments were undertaken over the temperature range 40-70 “C with wet sand and the heattransfer coefficient values deduced from the heat balance during the constant drying rate period, taking into account the radiative flux emitted by the wall, notably Qwa,,. This balance gives:

Qcmv+ Qwau= Qvvap The flux emitted by the wall was evaluated following relationship.

(1) via the

=

h;(T, - I”,)

giving the heat-transfer coefficient values h, or hb, respectively. In addition, an apparent value of the heat-transfer coefficient, written as h, ,, could be obtained via a simplified form ofeqn. (1) in which the wall radiation flux was neglected and expressed as: (5)

h, ,(Ta - 7’s) = fi&(T,)

These various values of h,, hb and h, t are listed in Table 1 and show that radiation of the wall cannot be neglected if a true value of h, is to be obtained but that, the effect of mass transfer on h, is small so that the h, and hk values are very close to each other. In addition, these mean h, values can be deduced from literature correlations; in the case where the dynamic and thermal boundary layers do not arise from the same edge, this relationship may be written [8]: Nu, = i[;:;p”‘i’:

for the turbulent

(6)

regime (Re, > 3 x 105),

and (7)

for the laminar regime (Re, < 3 x 105) where z represents the distance between the front edges of the two boundary layers. The values of the mean heat-transfer coefficient, h,, was then obtained via the relationship:

(8) where the local value the turbulent regime regime. The mass-transfer from the relationship

h, * is derived from eqn. (6) for or from eqn. (7) for the laminar coefficient, [ 8, lo]:

k,,

was

calculated

(2) (9)

314 TABLE

1. Experimental

and theoretical

Run No. 2 4 3 S 6 7 8

24.4 24.4 24.2 24.6 31.2 31.5 31.2

heat and mass-transfer

103k, (m s-‘)

h ct (W m-’

12.4 17.2 22.5 25.6 20.2 13.8 13.9

18.2 23.3 28.0 32.6 25.6 18.7 18.8

K-’ )

coefficients

for pure convective

drying

h: (W m-2 K-’ )

hc (W m-z K-r )

f( Wcxp

f( ~)thcor

10.9 16.8 21.5 26.8 18.8 11.4 10.6

10.9 16.9 21.7 27.0 19.1 11.6 10.8

0.77 0.84 0.86 0.92 0.87 0.71 0.71

0.886 0.886 0.886 0.886 0.886 0.886 0.886

where Pt represents the total pressure and T is the mean boundary layer temperature. The deviation in this value was c. 15%. Theoretical values of the psychrometric ratio were then calculated from the classical relationship h &=f(Le)= ’ = k,pc,

SC 2/3 E 0

and compared with the experimental values listed in Table 1. The mean deviation was c. 8%, thus providing firm evidence for the application of the Colburn analogy in the case of a pure convective process [ 131. Infrared and convective drying As in the case of pure convective drying, the mean mass-transfer coefficient, k,,,, was calculated from eqn. (9) with fi values corresponding to the constant infrared drying-rate period. These values are plotted in Fig. 6 as a function of the infrared flux density and in Fig. 7 as a function of the air velocity. It can be seen that the infrared heating density has no significant influence on the k, values and that these values are mainly dependent on the air velocity, as in the case of a pure convective drying. Hence, it appears that the radiative heat-transfer and mass-transfer processes are quite independent, with the mass concentration profiles inside the boundary layer being unaffected by the radiative flux. Finally, the Lewis function f(Le) was evaluated from the values of h, and k, deduced previously using two

. 1

*

3

4

5

6

7

Va(m.s-1) Fig. 7. Mass-transfer

coefficient

as a function

of the air velocity.

different methods: (i) from the h, values obtained in the pure convective experiments, corrected for the mean temperature and for the water vapour pressure at the boundary layer (these values are listed as h, I and f, (Le) in Table 2) and (ii) from the heat-transfer coefficient values, h,, calculated from eqns. (6) or (7) and (8) (these values are listed as h, 2 and f2( Le) in Table 2). From the values listed in Table 2, it is seen that they deviate from the theoretical values by mean values of 16% and 7%, respectively. Because of the uncertainty in the experimental coefficients h, and k,,, (c. 15%), it can be concluded that the Colbum analogy remains valid in the presence of high industrial infrared fluxes (l-20 kW mm2). IR absorption coeficient, u Drying rates and temperature levels depend strongly on the fraction of the incident radiation absorbed by the sample. The mean absorptivity coefficient CL,which is a characteristic of the product and of the radiation wavelength, can be deduced experimentally from a simple heat balance during the constant drying rate period, i.e. from:

30

Qcw + Qabs+ Qwau= Qw,

coefficient

as a function

(10)

in other words, from:

10

Qrad (kY.m-2) Fig. 6. Mass-transfer density.

MIR

5

of the infrared

flux

MT, - r,) +

~Qrar,+ Qwan= mL(Ts)

where Qwal, is given by eqn. (2).

(104

315 TABLE

2. Experimental

and theoretical

No.

I 03k, (m s-‘)

IRCI lRC2 IRC3 1RC4 lRC5 lRC6 lRC7 lRC8 lRC9 IRClO IRCll lRC12 lRC13 lRC14 IRMl lRM2 lRM3 lRM4 IRMS

11.0 15.0 18.7 19.8 22.8 12.8 14.4 20.1 21.8 24.2 12.0 11.9 18.6 18.3 13.1 15.7 18.5 22.6 25.0

Sample

heat and mass-transfer

coefficients

for combined radiative drying

:iiJ m-2 K-! )

:iG m-2 K-1 )

10.8 16.1 20.8 26

9.9 13.8 17.4 20.8 24.1 10.0 13.9 17.6 21.1 24.4 9.9 9.9 17.1 16.8 9.9 13.8 17.6 21.1 24.4

10.7 16.3 21.1 26.4 10.5 10.5 20.4 20.0 10.7 16.3 21.1 26.4

mean deviation with respect to

fi ( wcxp

fAW,,,

0.87 0.95 0.99 1.16

0.80 0.82 0.83 0.93 0.94 0.69 0.85 0.17 0.85 0.88 0.73 0.74 0.84 0.86 0.67 0.17 0.84 0.82 0.87

0.14 1.ooo 0.92 1.06 0.78 0.79 1.oo 1.02 0.74 0.92 1.00 1.03 16%

7%

f( LGeor

Fig. 8. Mean IR absorption operating parameters.

coefficient

values as a function

of the

Figure 8 shows that mean values of CIfor both the near- and medium-infrared radiation are independent of the radiative intensities; in addition, it should be noted that the absorptivity was better with medium IR (c. 0.94) than with near IR (c. 0.82). Furthermore, these data have been confirmed by an independent optical method (for which the uncertainty was higher), so that mediumIR heaters are best suited for drying processes [8].

Drying curve modelling

Our main objective was to develop a simple and sufficiently realistic model - with reasonable calculation times - using a reduced number of equations and

system parameters which would be applicable to different materials for the purpose of dryer modelling and control. Two regimes may be distinguished during the drying process: (1) A surface regime, including phases 0 and I, during which the front surface of the material receiving the radiation stays saturated. (2) An internal regime, during which the vaporization front migrates into the material body, separating a dry zone and a wet zone containing all the remaining water.

Surface regime model In this model, the process is initially entirely governed by the rate of radiative heat transfer and by conduction through the wet solid during the heating period (phase 0). Then, during the isoenthalpic period, the surface temperature and the drying rates are constant, and the process is governed by the external mass-transfer rate across the boundary layer. During these first two periods, simulations have shown that the internal moisture content profile is flat up to the critical moisture content XC,,= 0.045 kg kg-’ and that the sand layer can be considered to be opaque to radiation because we never observed any temperature inversion between the core temperature and its surface (even for near IR fluxes) as should be observed for a transparent or semi-transparent material. In fact,

316

the experimental data showed that the surface temperature was always higher than the sand layer core temperature [2, 81. Hence the heat-transfer process may be described by Fourier’s law as extended to a heterogeneous wet material, whose thermophysical properties (P ,,cr,J vary with the mean mositure content and temperature:

(11) The initial and boundary

conditions

were:

resistance in the dry layer which is equal where 5 = l/s0 is a form resistance factor:

to txr/D,

(17) where P,,, s and P,+ p are the water vapor partial pressure at the vaporization front and in the air core, respectively. The heat transfer is still described by the classical heat equation: In the dry zone (0 < x < xf)

Initial conditions t = 0;

T(x) = To

Boundary conditions x = E (rear face adiabatic); x = 0;

h,(T, - r,) + Qabs +

In the wet zone (xf < x < E)

8T - = 0

ax

Qwa,,= fiL(T,)

(13)

- 4, g (14)

This last relation expresses the thermal balance at the front of the sample receiving radiation; the quantity Qaabsdepicts the absorbed flux at the sample surface, Q,,, depicts the radiative flux exchanged with the tunnel wall evaluated via eqn. (2) and 2, aT/C+x is the conductive flux towards the material body. The drying rate is still given by eqn. (9), where the value of k, may be calculated from the value of h, given by eqns. (6) or (7) and (8) and the psychrometric ratio [8, 10, 111. It should be noted that in this regime the term Qabs= aQr, in eqn. (14) is the main term, evaluated by introducing the wet sand absorptivity CI, which is assumed constant as long as the surface stays saturated, i.e. until the critical moisture content is attained. In ternal regime model In this model, the vaporization front is located at a distance x = x, from the front face, delimiting a dry zone (&, cP,,J where heat and mass transfer become more and more limiting as drying proceeds. Amongst all possible approaches, the front position xf is best determined for a plane slab via the relationship proposed by Schadler and Kast [ 121: (15) which leads to the following 1

m =-&XC,, ,2

J

E dx, -xr dt

(18)

(12)

drying rate:

(16)

The drying rate expression is also given by eqn. (9) but this must be modified to take into account the

(1% The initial condition of this regime (phase II) is given by the temperature profile calculated at the time tcr corresponding to the critical moisture content X = XC,. Under these conditions, the boundary conditions are: x = E (rear face); g

(20)

= 0

(21)

with Qabs= cldQrad,where tld is the mean absorptivity

of

the dry sand x =xr(vapourization = mL(Txf)

front);

-I,

(E)+

+$).:

(22)

This last relationship expresses the heat flux continuity at the vapourization front. The system defined by eqns. (15) -(22) may be solved using a finite difference method, the position of the vapourization front being attached to a discretization node [8]. The validity of this model was checked by comparing the calculated and the experimental drying rate, and the surface temperature and inner curves for the main operating parameters, i.e. for the infrared power and air velocity; the core temperature was calculated at a position located 4 mm below the front surface. The critical moisture content XC’,,was determined graphically from the experimental drying curves m(R) = f(X) using the tangent method. The temperature profile curves [Figs. 9(a) and 10(a)] show that the thermal gradients between the surface and the core of the sand layer are predicted perfectly, particularly for the phase 0; further-

317

in the temperature curve. The precision of the model diminishes (5- 10 “C maximum deviation) during the

20 0

0

400

200

6W

1000

800

1200

Time (s)

(a)

0

0.1

0.05

(b)

0.15

0.25

0.2

Ti (kg.kg-I) dry basis

Fig. 9. Comparison between the model and experiment with respect to (a) the temperature profiles and (b) the drying rates [Q& = 9.7 kW m-* (near IR)].

Qrad = 145klm-2 NIR

0

100

200

300

0 (b)

400

500

600

700

800

Time(5)

Cal

0.05

0.1

last period (phase II) for which the thermal gradients inside the material are very important. This deviation may either be due to a small imprecision in the location of the thermocouple or to an imprecision in the thermal conductivity due to porosity variations, as confirmed via a parameter sensitivity study [8]. Figures 9(b) and 10(b) show that the agreement between the calculated and experimental drying rates is satisfactory only in the surface regime (phases 0 and I) with some discrepancy arising during the last period (phase II) [8]. The simulations shown in Figs. 9 and 10 were undertaken by assuming a sharp variation in the mean absorptivity of the product between the dry state (c(J and the wet state (a,), when the surface reached the critical moisture content Xc,. In fact, the introduction of a linear variation in this parameter improves the precision of the model [8]. A complete parameter sensitivity study has also shown the considerable importance of the fraction of energy absorbed (MQ~,,~values) and of the critical moisture content, X,,. Other comparisons between theory and modeling can be found in ref. 8. Nevertheless, despite the numerous parameters involved - some of which being known with quite low precision - this simplified model is capable of predicting accurately the main drying curves, e.g. the surface temperature curve which is the main parameter for process control. Vinally, during extended simulation, this modelling was coupled with heat and mass balance -equations for a co-current dryer in order to emphasize the advantages of infrared heating in terms of dryer length or drying time relative to a purely convective process [8].

Conclusions

0.15

02

0.25

R (kg.kg- I) dry basis

Fig. 10. Comparison between the model and experiment with respect to (a) the temperature profiles and (b) the drying rates [Q,, = 14.5 kW me2 (near IR)].

more, we observe that the model is capable of describing the movement of the vapourization front at the thermocouple level, which corresponds to a breakdown

High-intensity IR drying combined .with convective air flow has been studied experimentally as a function of the radiative heating flux and the air velocity. The curves for the drying kinetics show three periods: a heating period, a constant rate period and a period over which the rate declined. Constant rate period analysis gave absorptivity values which were in agreement. with those obtained by independent optical methods. We have observed that medium-infrared conditions were best suited for drying processes due to their better absorptivity. For given aerothermic conditions, the mean masstransfer coefficient was independent of the IR flux density and was mainly related to the air velocity. Furthermore, for our configuration, the analogy between heat and mass transfer is still applicable in the presence of high IR radiation densities.

318

A simple opaque model, using the in situ measured value of the IR flux absorbed by the sample and the thermophysical properties of the material, is capable of simulating the surface temperature profiles and the evolution of drying rates. This simple analysis can be introduced in a continuous dryer model to simulate the efficiency of the combined infrared and convective drying.

abs a cr d V

CETIAT and in this work.

-

Subscripts

S

Acknowledgments

The authors wish to thank ADEME, EDF for their financial aid and interest

kinematic viscosity, m2 s-i product or wall emissivity,

f h wall pro

radiation absorbed air critical content dry solid surface saturation or solid saturation water vapour front position wet or humid wall product

Nomenclature

Tl CP D E k, b,

Le =a’/D L” m M” p, P”

Pr = v/a‘ R

Qrad &li; SC = v/D T 2 R x t

thermal diffusivity, m2 s-’ surface area, m2 heat capacity, J kg-’ K-i water vapour diffusion coefficient, m2 ss’ sand layer thickness, m mass-transfer coefficient, m 5’ heat-transfer coefficient, convective W m-* K-i Lewis number, vapourization latent heat, J kg-’ drying rate, kg mm2 s-’ mol of water, kg kmol-’ total pressure, Pa water vapour partial pressure, Pa Prandtl number, universal gas constant, J kmol-’ K-’ infrared flux density, kW m-’ convection flux density, kW m-* radiative flux density emitted by the wall, kW me2 Schmidt number, temperature, K or “C air velocity, m s-l local moisture content (dry basis), kg kg-’ mean moisture content (dry basis), kg kg-’ spatial coordinate, m time, s

References

Hasatani, Y. Itaya and K. Miura, Hybrid drying of granular materials by combined radiative and convective heating,

1 M.

Drying

3

4

5

6

7

8

9

10 11

P f

; -%I’

density, kg me3 mean solid absorptivity, thermal conductivity, W m - ’ K- ’ Boltzmann constant, W mm2 K-4 form resistance coefficient, material porosity, -

Technol., 6 (1988) 43-68. N. Arai, Y. Itaya and N. Onada, Drying of optically semitransparent materials by combined radiativeconvective heating,.Drying Technoi., 1 (1983) 193-214. M. Nishimura. M. Kuraishi and Y. Bando. Effect of internal heating on infrared drying of coated films, ueat transfer Jpn. Res., 12 (1983) 59-71. S. Yagi, D. Kunii, S. Okada and R. Toyabe, Studies on the infrared drying of powdery and granular solids in falling rate period, Kagaku Kogaku, 21 (1957) 486-491. M. Dostie, J. N. Seguin, D. Maure, Q. A. Tonthat and R. Chatigny, Preliminary measurements on the drying of thick porous materials by combination of intermittent infrared and continuous convection heating, Proc. 6th Znt. Drying Symp., Versailles, Vol. 2, (1988), pp. 71-77. M. Dostie, Optimization of a drying process using infrared, radiofrequency and convection heating, Proc. 8th Int. Drying Symp., Montreal, Vol. A, (1992), pp. 679-684. J. P. Nadeau, J. R. Puiggali, W. Aregba and M. Quintard, Infrared tunnel dryer: a kinetic study, Proc. 6th Znt. Drying Symp., Versailles, Vol. 1, (1988), pp. 217-223. P. Navarri, Etude du sechage par rayonnement infraiouge. Application a un produit capillaro-poreux et a une enduction, Thbe de doctorat, Universite Caude Bernard Lyon I, 1992. J. M. Parrouffe, Combined convective and infrared drying of a capillary body, Ph.D. Thesis, McGill University, Montreal, Canada, 1992. R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, John Wiley and Son, New York, 1960. W. M. Kays and M. E. Crawford, Convective Heat and Mass Transfer, McGraw Hill, New York, 1980. N. Schadler and W. Kast, A complete model of the drying curve for porous bodies; experimental and theoretical studies Znt. J. Heat Mass Transfer, 30 (1987) 2031-2044. M. Prat, Modblisation des transferts en milieux poreux. Changement d’tchelle et conditions aux limites, Tht%e de doctoraf, INP Toulouse, 1980.

2 M. Hasatani,

12

13