Jean-Henry Ferrasse

Didier Lecomte
email: lecomte@enstimac.fr

D Ã Epartement

Simultaneous heat-ow di erential calorimetry and thermogravimetry for fast determination of sorption isotherms and heat of sorption in environmental or food engineering

Keywords: Phase equilibria, TGA, Instrumentation, Food processing, DSC, Desorption

HAL is a

Introduction

In various scientiÿc domains, food science and processing, material science, environmental sciences, the combination of water and solid materials is usually described by the water activity a w . Water activity is a major factor in characterizing the quality, the durability and the safety of a product [START_REF] Franks | Hydration phenomena: an update and implications for the food processing industry[END_REF]. Drying operations in the food or environmental processes usually aim at lowering the water activity [START_REF] Bassal | Etude de l'activite de l'eau dans les aliments au-dessus de 100[END_REF]. Sorption equilibrium curves are usually displayed as a number of moles or a mass of water versus the activity of water at equilibrium.

The di erent measurement methods have been classiÿed by [START_REF] Leung | Water activity and other colligative properties of foods[END_REF] and [START_REF] Rahman | Food properties handbook[END_REF]. Two classes of methods are mainly used by researchers; methods where the pressure above a known mass of sample is measured, and methods in which the pressure above the sample is ÿxed and the mass measured. Similar methods are used in adsorption science and are referred to as manometric and gravimetric methods [START_REF] Rouquerol | Adsorption by Powders and Porous Solids[END_REF].

In food science and engineering, [START_REF] Rao | Engineering properties of food[END_REF] report the use of manometric methods but asserts these methods are not suitable for materials containing large amounts of volatile or undergoing respiration process. The freezing point depression technique reported by [START_REF] Rahman | Food properties handbook[END_REF] seems to give reliable values for higher activities (above 0.85), but the range of temperature is narrow.

A very straightforward and widely used gravimetric method is to store a sample in a chamber at controlled temperature and vapour pressure. The mass is recorded at di erent time intervals until a constant value is observed. Constant relative humidity can be obtained by placing saturated salt solutions in the enclosure [START_REF] Rahman | Food properties handbook[END_REF]. Aqueous solutions with di erent concentrations may also be used [START_REF] Collins | The partial pressures of water in equilibrium with aqueous solutions of sulfuric acid[END_REF]. The principal inconvenience of the method is its slowness due to mass transfer both in the material and in the air. Improvements can be made by fractioning the sample and enhancing convective mass transfer in the chamber [START_REF] Ibeka | Moisture isotherms of a processed meat product-Bologna[END_REF]. A group of European research laboratories developed a reference system : reference material, standard equipment and procedure, evaluation of results (Spiess and Wolf, 1987;[START_REF] Jowitt | The certiÿcation of water content of microcrystalline cellulose (MCC) at 10 water activities[END_REF]. The reference material (Microcrystalline Cellulose Avicel PH 101 known as MCC) was chosen for its stability and constant availability.

The use of saturated salt solutions can be avoided by feeding the cabinet with a gas stream at constant vapour pressure, generally moist air. The relative humidity of the air is therefore measured by psychrometric instruments either directly (dew point) or indirectly (wet bulb temperature, capacitive hygrometers). The main di culty of the method is to keep a constant level of moisture in the cabinet and to measure it with a good accuracy. Dew-point sensors are very precise at high activity, but cannot be used at low activities [START_REF] Prior | Measurement of water activity in foods: a review[END_REF]. Capacitive sensors show a good accuracy on a wide range of air moisture content but are sensitive to gas adsorption by organic volatiles and thus subject to ageing. To stabilize the air moisture content, it is possible to achieve a mixture of two air ows of di erent moistures [START_REF] Mã Eot | Rapid determination of sorption isotherms and water apparent di usivity[END_REF] or to saturate air and heat it up to the required temperature [START_REF] Weisser | In uence of temperature on sorption isotherms[END_REF]. [START_REF] Baucour | Development of a new method for fast measurement of water sorption isotherms in the high humidity range. Validation on gelatine gel[END_REF] generates calibrated moist air by successive isothermal pressure drops across diaphragms from an initially saturated moist air. The experimental set-up enables to measure high activity values (0.88-0.98). For temperatures above the boiling point, [START_REF] Loncin | Activity of water and its importance in preconcentration and drying of foods[END_REF] showed that water activity is only a function of temperature at atmospheric pressure. [START_REF] Bassal | Sorption isotherms of food materials above 100 •[END_REF] proposed a method based on the equilibration of food samples in contact with pure vapour. Due to the absence of external di usion and temperature above 100 • C, equilibrium is reached rapidly (48 h).

Other methods (isiopiestic method, suction potential method, osmometric method) are described by [START_REF] Rao | Engineering properties of food[END_REF]. At least, the Landrock and Proctor method (1951), an extrapolation method based on saturated salt solutions, does not require thermodynamic equilibrium.

Except the last cited method, all methods require a long period of stabilization due to internal and external mass transfer, between a few days and a few months depending on the size of the sample, the order of magnitude of mass transfer di usion and the initial moisture di erence between the carrier gas and the surface of the sample. This delay causes measurements problems for biological products at high activities or high temperature where microbial and fungi proliferation and enzymatic oxidation alter the properties of the initial product. For example little work on water activity of sewage sludge is available in the literature [START_REF] Vaxelaire | Desorption isotherms of domestic activated sludge[END_REF], measuring di culties are also reported in this particular scientiÿc area [START_REF] Ferrasse | Dà eveloppement d'outils expà erimentaux pour le dimensionnement de procà edà es de sà echage conductif avec agitation: application  a des boues de stations d'à epurations urbaines[END_REF].

The methods presented in this short review require considerable experimental care. Numerous factors like temperature stability, concentration and temperature gradients, volume ratios (e.g.: sample/cell), frequency of opening, heterogeneity, etc. in uence the results and until now, no method totally meets requirements of accuracy, simplicity and speed [START_REF] Rahman | Food properties handbook[END_REF].

A major disadvantage of the saturated salt methodthough this method is now widely adopted as a standard (Jowitt and Wagsta e 1989)-is the small number of experimental points. Due to the limitation in the use of saturated salts, an average of 8-10 experimental points is used for the determination of sorption isotherms with di culties arising for low and high activities measurements. Sorption isotherm models have been widely described in the literature for food [START_REF] Iglesias | Handbook of Food Osotherms: Water Sorption Parameters for Food and Food Components[END_REF][START_REF] Wolf | Sorption Isotherms and Water Activity of Food Materials: A Bibliography[END_REF], food-based materials [START_REF] Iglesias | Handbook of Food Osotherms: Water Sorption Parameters for Food and Food Components[END_REF], and more recently for waste [START_REF] Vaxelaire | Desorption isotherms of domestic activated sludge[END_REF]. They usually incorporate 2 or 3 parameters, thus conÿdent parameter estimation from very little experimental data is di cult to achieve. In a cooperative study within the framework of the COST 90 program, various European laboratories determined the mean adsorption isotherms of MCC and food materials (apple pectin, potato starch, : : :). The 24 laboratories used the same experimental procedure and the same set-up to perform an important number of experiments. These data were examined in term of mean, standard deviation, repeatability standard deviation and reproducibility standard deviation. It resulted in a mean adsorption isotherm curve for 10 water activities at 25 • C and standard deviations for each point [START_REF] Wolf | Sorption Isotherms and Water Activity of Food Materials: A Bibliography[END_REF].

The ÿtting of sorption isotherms with model equations allows the determination of a function usually expressed as followed:

a w = f(X; T ); ( 1
)
where X is the moisture content:

X = (m -m 0 )=m 0 : (2)
Model equations are very useful to determine the total heat of sorption. Indeed, the total heat of sorption is deÿned by the Clausius-Clapeyron law:

H t = - R M @ ln p v (X; T ) @ 1 T X : (3)
The total heat of sorption is the sum of the latent heat of vaporization and the isosteric heat of sorption:

H t = H st + H v ; (4)
where

H st = - R M @ ln a w (X; T ) @ 1 T X : (5)
The isosteric heat of sorption can be calculated using Othmer's methods [START_REF] Othmer | Correlating vapour pressure and latent heat data. A new plot[END_REF][START_REF] Beristain | Determination of moisture enthalpy of mesquite gum with di erential scanning calorimetry and thermogravimetric analysis[END_REF] using Eq. ( 3) and direct measurements of the vapour pressure p v (X; T )|. It can be alternately computed from the temperature shift of the isotherms, using Eq. ( 5). An average value of H st between two temperatures T 1 and T 2 is obtained by integration of Eq. ( 5) at constant X :

H st = R ln(a w2 =a w1 ) 1 T 1 - 1 T 2 -1 ; (6)
where a w1 and a w2 are the activities of the material at temperatures T 1 and T 2 . Using Eq. ( 6) and the data and conÿdence intervals established by [START_REF] Wolf | Sorption Isotherms and Water Activity of Food Materials: A Bibliography[END_REF] and [START_REF] Iglesias | On the temperature dependence of isosteric heats of water sorption in dehydrated foods[END_REF] and [START_REF] Chirife | Estimation of the precision of isosteric heats of sorption determined from the temperature dependence of food isotherms[END_REF] showed important errors in the calculations of the heats of sorption (typically 39% for microcrystalline cellulose (MCC) at X = 0:08 and 40

• C 6 T 6 60 • C).
An alternative way to calculate the heat of sorption is to make a formal di erentiation of the Clausius-Clapeyron equation (Eq. ( 3)). The method requires that coe cients are expressed as a function of the temperature. It was applied by [START_REF] Bassal | Etude de l'activite de l'eau dans les aliments au-dessus de 100[END_REF] and more recently by [START_REF] Viollaz | Equilibrium sorption isotherms and thermodynamic properties of starch and gluten[END_REF] using the GAB equation. It is shown [START_REF] Balaban-Murat | Estimation of heat of moisture sorption and improved criteria for evaluating moisture sorption isotherm equations for food[END_REF] that the applicability of this method is very much dependent of the type of model used and the range of application of the model. No real sensitivity analysis has been made to validate these methods.

Direct measurement of the heat of sorption is often invoked by authors to cope with these di erentiation problems. Di erential scanning calorimetry (DSC) is used by [START_REF] Beristain | Determination of moisture enthalpy of mesquite gum with di erential scanning calorimetry and thermogravimetric analysis[END_REF] to measure the heat of sorption of mesquite gum in the temperature range (25-195 • C). Thermogravimetric measurements with the same heating rate as the DSC (10 K min -1 ) allow computation of the mass loss of the sample. [START_REF] Mulet | Sorption isosteric heat determination by thermal analysis and sorption isotherms[END_REF] used a very similar procedure with smaller heating rates (2 K min -1 ) for DSC measurements of the heat of sorption of cauli ower and potato starch in the temperature range (25-150 • C). Although the results for both research works are consistent with calculations from GAB's equation, the scanning method involves large temperature ranges-including probably temperatures above the boiling temperature-and one should consider that the heat of sorption obtained from this measurement is an average value over a large range of temperature (from ambient to above boiling temperature) and a large range of water content.

So far simultaneous calorimetric and gravimetric techniques have not been used for food or environmental engineering purposes. In the ÿeld of adsorption science, Berlier and Fr ere (1996) used a Setaram DSC111 for the measurement of heat of sorption and sorption isotherms of CO2 onto activated carbon. The calorimeter is used in isothermal mode and the sample is ÿrst outgased at 513 K with a pressure of 10 -2 mmHg. Then the gas is adsorbed at di erent pressures and the heat and mass gain are recorded. This step-by-step method allowed to record both the adsorption isotherms and the heat of adsorption.

In the present paper, simultaneous gravimetric and calorimetric measurements using Setaram DSC111 apparatus in drying (desorption) conditions are presented. This method has been successfully applied for various products (Ferrasse et al., 2000a,b). Unlike the method described by [START_REF] Berlier | Adsorption of CO2 on activated carbon: simultaneous determination of integral heat and isotherm of adsorption[END_REF], no vacuum is necessary. It will be shown that under some particular assumptions regarding di usional transfer in the sample, the method can provide reliable values of the heat of sorption and sorption isotherms, with a very fast time response. The assumptions and the limits of application of the method will be discussed.

Material and equipment

The experimental set-up is a Setaram DSC111 di erential scanning calorimeter combined with a thermobalance as shown in Fig. 1. The furnace is made of a silver cylindrical block where two refractory tubes are inserted. Thermopiles of Tian-Calvet type [START_REF] Tian | Microcalorim etre  a compensation par e ets Peltier et Joule[END_REF][START_REF] Calvet | Microcalorimà etrie. Applications Physico-chimiques et Biologiques[END_REF] are inserted between the refractory tubes and the furnace (height 20 mm). The temperature of the furnace is controlled by an electric heater and a water cooled coil heat-exchanger. A Pt100 probe inserted into the silver block allows to control the furnace temperature either in isothermal mode or in scanning mode with an error better than 0:01 K. The thermobalance is made of a beam supporting two platinum crucibles (diameter 4:3 mm-height 9:85 mm) which are suspended in the two refractory tubes at the level of the thermopiles. One of the crucibles (S) contains the sample, the other (R) containing a non-reacting substance is called the reference. An electromagnetic device enables to keep the crucibles at a constant position during the measurement of the mass change. The detection limit is 1 g. The crucibles are set at the level of the thermopiles which measure the heat ow exchanged with the furnace. The detection limit of the heat ow is 5 W in isothermal mode and 15 W in scanning mode. The relative error on the heat ow measurement is 1%. The recorded heat ow signal is the di erence between the heat ows measured in the two thermopiles.

Theoretical model

In the following section, two assumptions are made: A.1: Temperature is homogeneous in the crucible (S) and equal to the sample temperature T S . It may vary with time and be di erent from the furnace temperature but the order of magnitude of the di erence should be small and it will be shown furtherhow this di erence may be quantiÿed.

A.2: Moisture gradients are neglected in the material. This can be achieved using very thin samples (≈ 1 mm) and for relatively high moisture di usion coe cients.

These fundamental assumptions will be discussed in Section 5.

Measurement of the heat of sorption

The theory of heat ow di erential calorimeter (Calvet andPrat, 1956, 1963) is based of an energy balance on the reference and the sample crucibles.

The heat ows for the sample and the reference are proportional to the temperature di erences between the crucibles and the furnace:

qS = (UA) S (T F -T S ) (7) and qR = (UA) R (T F -T R ): (8)
The recorded heat-ow is the di erence between the two heat ows qS and qR :

q = (UA) S (T S -T F ) -(UA) R (T R -T F ): (9)
The energy balance equation for the sample and the reference are:

(m(t)c S + C ) dT S dt + UA S (T S -T F ) = P; (10) C dT R dt + UA R (T R -T F ) = 0; ( 11
)
where P is a source term in the sample crucible. The heat capacities of the sample and the reference crucibles can be assumed equal to C . The massic heat capacity of the sample c S is assumed constant.

In the present case, the microcalorimeter is used in isothermal mode. In Experiment S, the furnace temperature is ÿxed and the sample is dried (desorption process). The gas rapidly reaches the furnace temperature both in the sample and the reference cells. Eqs. ( 9)-( 11) yield:

q = P -(m(t)c S + C ) dT S dt + C dT R dt : (12)
It is useful to perform a blank experiment (Experiment B) using an empty crucible. The corresponding calorimetric signal is

qB = (UA) S (T SB -T F ) -(UA) R (T RB -T F ) (13)
and the energy balance for the blank experiment yield:

qB = -C dT SB dt + C dT RB dt : (14)
The corrected calorimetric signal is obtained by subtracting qB from q: qC = q -qB :

Although the correction is very small, it takes into account many e ects such as buoyancy e ects in the cell or di erences of owrates in the uid owing in the two cells.

The assumption is made that the reference crucible undertakes similar ambient conditions during the two experiments; thus the temperatures of the reference can be assumed equal:

T RB = T R : (16)
It is also assumed that rapidly the sample temperature in the blank experiment becomes very close to the furnace temperature, because of the quasi-steady regime in isothermal mode:

T SB ∼ = T F : (17)
Using Eqs. ( 9) and ( 13) and the above approximations, a simple expression of qC is obtained:

qC = UA S (T F -T S ) (18)
and the combination of the energy balance equations ( 12) and ( 14) yield

qC = P -m(t)c S dT S dt + C dT S dt - dT SB dt + dT R dt - dT RB dt : (19)
With assumptions ( 16) and ( 17), this equation reduces to:

qC = P -(m(t)c S + C ) dT S dt : ( 20
)
The application is particularly interesting in a drying or a desorption process where the energy absorbed per unit time by the material is related to the total heat of desorption:

P = -ṁ(t) H t (t): (21)
Thus one obtains:

ṁ(t) H t (t) = -qC -(m(t)c S + C ) dT S dt : (22)
In isothermal mode, the energy equation can be simpliÿed assuming the heat accumulated in the sample crucible is negligible (however it is possible to keep this term). Thus a very straightforward equation can be obtained:

qC = -ṁ(t) H t (t): (23)
The thermogravimetric experiment enables to determine both the TG curve (m(t)) and the DTG curve ( ṁ(t)). It is thus possible to plot at any time the value of the total heat of desorption as a function of the mass:

H t (t) = f(m(t)): (24)
It must be reminded that the heat of sorption is an average heat calculated at the average temperature of the sample and for an average moisture concentration. The more accurate assumptions A1 and A2 are, the more Eq. ( 24) may be considered valid.

Determination of the sorption isotherm

To determine the sorption isotherm, three measurements are needed: temperature, mass and water activity of the sample. Of these three parameters, only the mass can be easily determined with a thermobalance. The temperature of the furnace is an approximate value of the actual temperature of the sample. The water activity is not known. A protocol to determine all three parameters from a simultaneous TG-DSC measurement is presented herein. The method used is based on the same experiment (S) as described above.

A new assumption is used in this section: A3: Evaporation of water occurs according to the stagnant ÿlm theory [START_REF] Bird | Transport Phenomena[END_REF]. This assumption will also be discussed in Section 5.

The stagnant ÿlm theory [START_REF] Bird | Transport Phenomena[END_REF]) is used to model the evaporation of water in experiment (S):

ṁ = ln p -p v ∞ p -a w p sat (T S ) : (25)
In Eq. ( 25) the vapour partial pressure p v ∞ in the furnace is ÿxed, and the partial pressure at the surface of the sample is the product of the surface water activity a w and the water vapour pressure p sat (T S ) at the temperature of the sample. The coe cient:

= M v D v (T S )Ap RT S (L -e) (26)
is a function of the temperature in the crucible T S and the length of di usion Le between the sample free surface A and the edge of the crucible (Fig. 2). The vapour-air di usion coe cient D v is calculated at the sample temperature T S by the following equation [START_REF] Berman | Evaporative Cooling of Circulating Water[END_REF]: temperature may be computed from the calorimetric signal qc and the furnace temperature T F using Eq. ( 18):

D v = 2:26 × 10 5 1 p T S 273
T S = T F + qc UA S : (28)
The product thickness e is a function of time. It can be derived from the mass of sample:

e = m(t) A ; (29)
where is an average constant density of sample. Using Eq. ( 25), a direct calculation of a w leads to the following expression:

a w = p p sat (Ts) (exp( ṁ= ) -1) + p v ∞ p sat (T S ) : (30)
The sorption isotherm is thus the plot of a W (deÿned by Eq. ( 30)) as a function of the sample moisture content (Eq. ( 2)), for the corrected temperature T S .

The vapour partial pressure in the furnace can be calculated assuming a perfect mixing of the transfer gas and the water vapour:

p v ∞ = ( ṁ= ṁa )p M v =M a + ṁ= ṁa ; ( 31
)
where ṁa is the mass owrate of the transfer gas (dry air).

Experimental results

In order to avoid concentration gradients during the short heating period, pure water is disposed on the pasty sample. The surface must be as plane as possible and a low thickness is required. Typically 3-7 mg of dry mass samples are used. The calorimetric signal q and the TG signals m(t) are recorded simultaneously (Experiment S). The DTG signal ṁ(t) is calculated by the software associated with the apparatus. The corrected calorimetric signal qC is calculated using qB obtained from the blank experiment (Experiment B).

Experimental determination of the heat transfer coe cient in the crucible

An estimation of the heat transfer coe cient may be obtained from a speciÿc experimental procedure (Experiment F). A fusible medium is used in the same range of temperature as in experiment S and B (Section 3.1) and the microcalorimeter is used in scanning mode with a constant heating rate v. During the phase change, the temperature of the sample remains equal to the phase change temperature T C while the furnace temperature varies linearly with time [START_REF] Gray | A simple generalized theory for the analysis of dynamic thermal measurement[END_REF][START_REF] Speyer | Thermal Analysis of Materials[END_REF].

The overall heat transfer coe cient may be written [START_REF] Gray | A simple generalized theory for the analysis of dynamic thermal measurement[END_REF]:

(UA) S = - 1 v d q dt melting ; (32)
where d q dt melting is the slope of the calorimetric signal at the beginning of the melting process. To determine UA S in the appropriate range of temperature, a sample of pure indium of fusion temperature T C = 156:6 • C is used and the heat transfer coe cient is ÿtted using Eq. ( 32).

The comparison of the computed value using and the experimental corrected heat ow is shown in Fig. 3. The agreement between the two curves is very good, in spite of the simplicity of the model (constant heat transfer coecient, temperature uniformness). When the sample reaches the melting temperature, the sample remains at constant temperature until the phase change is completed.

Determination of sorption isotherms for MCC and Kaolin

Validation experiments were conducted on two products. The ÿrst product is MCC because it has been chosen as a reference material in food and pharmaceutical studies. The second product is kaolin because extensive data on this product were found in the literature [START_REF] Kroes | The in uence of material properties on drying kinetics[END_REF].

MCC: Results for MCC are obtained at two furnace temperatures T F = 50 • C and 100 • C. For T F = 50 • C, the temperature correction (Eq. ( 29)) is small and the average temperature during the experiment is T S = 49:3 • C. Although not originally valid in this range of temperature, these data were compared (Fig. 4) to Oswin model using the original set of parameters obtained by [START_REF] Bassal | Etude de l'activite de l'eau dans les aliments au-dessus de 100[END_REF] reproduced in Annex 1. The results from the COST project [START_REF] Wolf | The water-vapour sorption isotherms of mycrocrystalline cellulose (MCC) and of puriÿed potato starch. Results of a collaborative study[END_REF] are also shown at a lower temperature T S = 25 • C. These data are also consistent with the data obtained with the present method, which seems to prove that the temperature shift of the isotherms is not very signiÿcant. It must be emphasized that the present method could not be used at such a low temperature of T S = 25 • C because of the lack of sensitivity of the thermobalance for small mass owrates. However, there is a good agreement between these two sets of data and the present experimental method, especially for small values of the activity (0 6 a w 6 0:4). At a higher temperature, the drying rate is important and the temperature di erence between the furnace and the sample is more signiÿcant. But the temperature shift of the isotherms is still small and thus temperature changes do not a ect signiÿcantly the quality of the results as long as these changes can be quantiÿed. Results for an average sample temperature T F = 92 • C are presented in Fig. 5. The temperature of the sample has been calculated according to Eq. ( 28) and was estimated at 92 ± 5 • C during the whole experiment. These data are comparable to the original set of data obtained by [START_REF] Bassal | Etude de l'activite de l'eau dans les aliments au-dessus de 100[END_REF] with Oswin's model. (Annex 1). The error was estimated by [START_REF] Bassal | Etude de l'activite de l'eau dans les aliments au-dessus de 100[END_REF] and reproduced on the graph. The agreement between the two sets of experiments is very good in this range of temperature especially for a w 6 0:8. For a w ¿ 0:8, the equilibrium method is very di cult to achieve and no results are available in the literature. The present method allows to determine actual values of a w = f(X; T ) even at a w ∼ = 1, but there is no indication on the error made.

Kaolin: Results for kaolin are shown in Fig. 6. The furnace temperature is set at T F =40 • C. As for MCC, measurements at lower temperatures are di cult to obtain because of the lack of sensitivity of the thermobalance. These data were compared with results obtained by [START_REF] Kroes | The in uence of material properties on drying kinetics[END_REF] at similar temperatures (Annex 1). Although the general shapes of the curves are very similar, a good match is only observed for small values of a w typically a w 6 0:25. The origin of this discrepancy (internal transfer limitations) will be discussed in Section 5.

Determination of the isosteric heat of sorption for MCC

In the case of the isosteric heat of sorption, reliable data for MCC were found in [START_REF] Bassal | Etude de l'activite de l'eau dans les aliments au-dessus de 100[END_REF]. These data were obtained using a formal di erentiation of Eq. ( 6). Results are shown in Fig. 7 at 95 • C.

It has been emphasized in Section 1 that the indirect methods led to important errors in the determination of H st . However, in spite of a 25% error for low water contents, the agreement between the two methods is very good. In particular, it can be seen that for X ¿ 0:4, adsorption effects become negligible and the isosteric heat of sorption H st is equal to zero. The method can thus also be used to quantify the transition moisture content between free and bound water in a biological materials undertaking a desorption process, which cannot be obtained with accuracy from typical S-shaped sorption isotherms for food and biological materials.

Discussions

The good estimation of heat and mass transfer parameters requires the validity of the three mains assumptions A1-A3 given in Section 3. For the external transfer, the relevant parameters are the overall heat transfer coe cient (UA) S and the mass transfer coe cient between the furnace and the crucible. These coe cients are not related by laws of similarity because the heat and mass transfer areas are di erent. Internal heat transfer can be quantiÿed by the thermal conductivity k and the e ective water di usivity D S of the sample.

Heat transfer assumptions

Temperature uniformness in the sample

An important condition for the application of Eqs. ( 7)-( 24) is the uniformness of the temperature in the sample which is achieved for Bi T = (UA) S e=A T k 6 0:1, where Bi T is the thermal Biot number.

With the typical values of (UA) S , e and A T , this condition is fulÿlled for k ¿ 3:10 -3 W m -1 K -1 , which is always valid for biological and food materials.

Sample temperature change

Although the calorimeter furnace is kept isothermal, the endothermic desorption e ect creates a temperature di erence between the furnace and the sample. This temperature di erence is larger at the beginning of the desorption process and becomes smaller and smaller as water activity decreases. Using Eq. ( 28), the sample temperature is computed for experiment S. In Fig. 8 kaolin is desorbed at T F = 50 • C and the actual temperature of the sample is plotted versus the moisture content X . An average temperature of T S = 49:3 ± 0:9 • C is observed. For a higher temperature of furnace (T F = 92 • C) the temperature deviation is ±5 • C. Compared to equilibrium sorption measurement techniques for which the error on the temperature is small (±0:5 • C), the present method does not allow a good stabilization of the temperature, but according to [START_REF] Weisser | In uence of temperature on sorption isotherms[END_REF], the sensitivity of sorption isotherms to temperature changes is small. More important is the fact that the sample temperature is di erent from the furnace temperature and can be calculated from Eq. ( 28) at any time.

Temperature uniformness of the gas trapped in the crucible

To be able to apply the stagnant ÿlm theory (Section 5.3), the assumption of isothermal conditions in the gas trapped in the crucible must be valid. A CFD model (FLUENT) has been used to determine the temperature ÿeld in the crucible and the furnace. To account for the endothermic reaction in the sample, a source term was applied in the sample. The furnace temperature was set to 80 • C and a gas laminar ow was applied (0:5 L h -1 ). It is shown in Fig. 9 that the gas temperature in the furnace upstream of the crucible reaches rapidly the furnace temperature, due to the low gas velocities. It is also shown that the gas temperature inside the crucible is close to the sample temperature, thus the assumption of isothermal gas in Eq. ( 25) is valid in the conditions of the experiment.

Mass transfer assumptions

External mass transfer

In order to improve the stagnant ÿlm approximation, a new experimental protocol, very similar to experiment (P) previously described, was performed with pure water.

This experiment allowed to compare the evaporation rate ṁexp (t) from TG experiments and the calculated evaporation rate ṁ(t) obtained using Eqs. ( 26)-( 28) with a w = 1. Corrections were used for temperature (Eq. ( 28)), product thickness (Eq. ( 29)), partial vapour pressure (Eq. ( 31)). As seen in Fig. 10, there is a very good agreement between the computed mass ow rate and the experimental mass owrate which validates the stagnant ÿlm model and the correction made for temperature and thickness change.

Internal mass transfer

An important assumption is the assumption of a homogeneous moisture concentration in the sample. Internal diffusion is responsible for concentration gradients. Thus a di erence should be observed between the average moisture content measured by the thermobalance and the local superÿcial moisture concentration which is in equilibrium with the owing moist air. It is assumed that the sorption isotherms can be locally approximated by a straight line: @a w =@X = C. For poorly adsorptive materials C is large and for highly adsorptive materials C is small.

With these assumptions, convective and adsorptive e ects can be lumped into an overall mass transfer coe cient K deÿned by

K = C M v P V sat (T S )eA ṁa RT S : D v (T s ) L -e : (33)
A mass Biot number can be introduced based on K and the material thickness e:

Bi m = Ke D s :

The mass Biot number compares the internal mass transfer and the external mass transfer. It is generally assumed that internal concentration gradients may be neglected when Bi m ¡ 0:1. Therefore the Biot number was calculated for the experiments with MCC and Kaolin. The value for C for each experimental point was derived from the slope of the experimental sorption isotherm. D v (T S ) and e were calculated using the set of Eqs. ( 28)-(31).

For Kaolin, the internal di usion coe cient is reported in the appendix [START_REF] Kroes | The in uence of material properties on drying kinetics[END_REF]. The following experimental values where used: A = 1:25 × 10 -5 m 2 , L = 10 -2 m, m 0 = 11:11 × 10 -6 kg. For the experiments reported in Fig. 6, the range of the mass Biot number is 2-100 which is well above the critical value of 0.1 although the comparisons are satisfactory especially for small activities, thus low moisture contents. The most satisfying explanation is that the characteristic thickness of the sample is one or two decades smaller than the calculated thickness e due to cracks, bubbles and surface e ects that are observed when the medium shrinks and dries. At larger activities, the internal apparent mass di usion coe cient is larger but the value of e is more relevant to calculate the Biot number and describe the mass transfer process.

For MCC (m 0 = 6:6933 × 10 -6 kg), only one di usion coe cient was reported in the literature (D s = 1:26 × 10 -9 m 2 s -1 from [START_REF] Sun | Temperature mapping in a model food gel using magnetic resonance imaging[END_REF], but no mention was made of the conditions of the experiment. However the results are better than for Kaolin (Figs. 4 and5), but it is di cult to give a straightforward explanation to the excellent results obtained with MCC.

Due to internal di usion, the average water activity is higher than the local water activity at the surface of the sample, thus the average moisture content is higher than the local moisture content at the surface of the sample. The present method produces a plot of the average moisture content versus the local surface activity. Compared to static methods, the sorption isotherm is thus shifted upwards for most experiments (Figs. 456).

However, the method described in the present paper seems well adapted for pasty materials that contain important amount of waster (gels, collo ds, paste, sludge: : :) with a mass transfer limitation when the internal di usion coecients is smaller than 10 -10 m -2 s -1 . The method has been applied with success [START_REF] Ferrasse | Dà eveloppement d'outils expà erimentaux pour le dimensionnement de procà edà es de sà echage conductif avec agitation: application  a des boues de stations d'à epurations urbaines[END_REF], to the determination of sorption isotherms of sludge in the conditions of high temperature drying (100 • C).

Conclusion

Calorimetric and gravimetric signals from a Setaram DSC111 set-up in isothermal mode were used to build sorption isotherms and heat of sorption calculations for water in standard materials (microcrystalline cellulose and kaolin). Unlike most experimental techniques which are based on thermodynamic equilibrium and yield a limited number of experimental points, the developed method uses a dynamic approach and yields continuous functions. Although temperature stability in the sample is di cult to achieve due to the evaporation e ect, temperature shifts have little effect on the sorption isotherms. The sample is small enough to allow temperature homogeneity even with low thermal conductivity materials. With respect to mass transfer, pasty or collo dal materials have smaller di usion coe cients at lower moisture contents thus lower activities but the validity of the method is better at low activities (0 6 a w 6 0:4). This surprising result suggests that the characteristic length at low moisture content is much smaller than the theoretical thickness of the sample due to cracks, bubbles and surface e ects in the tested media. The accuracy is also enhanced at higher temperatures (T ¿ 40 • C), ÿrst because of a better sensitivity of the microbalance and owmeters, but also because of higher internal di usion rates. Unlike most values found in environmental or food engineering, the isosteric heat of desorption is not derived from the shift of sorption isotherms with temperature but from direct measurements. The data for the heat of sorption su er from a lack of comparison but direct measurement instead of calculations would give more precise and e ective information for the design and calculation of dryers at low moisture ranges. The residence time of the sample in the DSC cell is small (typically a few hours for a complete experiment), which makes it suitable for the prediction of thermodynamic properties of heat and moisture sensitive products like food or sludge, whereas more conventional methods would lead to misleading interpretations due to the conjunction of biological activity, temperature and residence time e ects. (1993) and ÿtted by Oswin model. The temperature range is 80-150 • C: K 1 = 6:0585; K 2 = -1:5927 × 10 -2 ; K 3 = 0:3198; K 4 = 2:0875 × 10 -3 ; X = (K 1 + K 2 T )(K 3 + K 4 T ) a w 1 -a w ; with T in • C:

Notation

The relative error for this Oswin law with experimental data was 5.5%.

A.2. Kaolin

(a) Sorption isotherm: Results are reported from Bas [START_REF] Kroes | The in uence of material properties on drying kinetics[END_REF] The temperature e ect is described with the Arrhenius equation. A constant activation energy of 3×10 4 J mol -1 is used.

Fig. 1 .

 1 Fig. 1. Schematic view of the experimental set-up: TG-DSC 111 Setaram.

Fig. 2 .

 2 Fig. 2. Partial pressure proÿle according to the stagnant ÿlm theory in the sample crucible (S).

Fig. 3 .

 3 Fig. 3. Comparison of theoretical and experimental heat ow for a fusible medium. The melting region corresponds to the ÿrst part of the curve.

Fig. 4 .

 4 Fig. 4. Comparison of sorption isotherm for MCC for T F = 50 • C.

Fig. 6 .

 6 Fig. 5. Comparison of sorption isotherm for MCC.

Fig. 7 .

 7 Fig. 7. Comparison of isosteric heat of desorption for MCC for low moisture content.

Fig. 8 .

 8 Fig. 8. Sample temperature as a function of the water content during sorption measurement.

Fig. 9 .

 9 Fig. 9. Temperature ÿeld in the crucible using FLUENT calculation for T F = 80 • C, and a gas owrate of 0:5 l=h.

Fig. 10 .

 10 Fig. 10. Comparison between experimental mass rate for pure water and calculations using the stagnant ÿlm law.

  = a + b ln X + c ln X + d(ln u) 2 + e (ln u) 2 :

  and ÿtted by the GAB model. The temperature range is 35-55• C: X X mono = C gug ka w (1 -ka w )(1 -ka w + C gug ka w ); Internal di usion coe cient for bound water, i.e. for moisture such as a w ¡ 1 (Bas Kroes, 1999):

	no relative errors are reported :
	Gab-parameters	Temperature
	k	35 • C	0.9
	k	45 • C	0.92
	C gug	35 • C	1 5
	C gug	45 • C	53.4
	X mono	35 • C	0.004
	X mono	45 • C	0.003
	(b)


J Madiouli

J Sghaier

D Lecomte
email: didier.lecomte@mines-albi.fr

H Sammouda

Determination of porosity change from shrinkage curves during drying of food material

Keywords: Porosity, Model, Foodstuffs, Shrinkage, Collapse 1

Based on a very simple model of mass conservation, three experimental properties (solid density, liquid density and initial bulk density) and the simultaneous acquisition of the reduced moisture content and the volume shrinkage during drying, a simple method is proposed to calculate the bulk porosity of a material during drying. This model allows a graphical interpretation to visualize the porosity change by comparing the experimental shrinkage curve with an ideal shrinkage curve. In the present work, several examples were taken from the literature to illustrate the application of this method to foodstuffs (apple, banana, carrot, garlic, pear, potato and sweet potato) with two different processes (convective drying, freeze-drying) and different drying conditions. Porosity calculations including error estimations showed a good agreement with experimental values reported in the literature.

Introduction

Drying is used to store food safely by reducing the moisture content and decreasing the microbiological activity. The drying process usually causes volume and surface change of fruit and vegetables due to moisture evaporation. Changes in the physical properties (density, porosity, etc.) can be observed and influence the product quality and consumer's opinion. Among these properties, porosity, defined as the ratio of the pore volume to the apparent volume of the medium, is strongly dependent on the moisture content and the drying conditions.

Important work on the porosity change during the drying of foodstuffs is reported in the literature [START_REF] Lozano | Total porosity and open-pore porosity in the drying of fruits[END_REF][START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF][START_REF] Madamba | Shrinkage, density and porosity of garlic during drying[END_REF][START_REF] Ratti | Shrinkage during drying of foodstuffs[END_REF][START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF][START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF][START_REF] Azzouz | Moisture diffusivity and drying kinetic equation of convective drying of grapes[END_REF][START_REF] Talla | Shrinkage and density evolution during drying of tropical fruits: application to banana[END_REF][START_REF] Mayor | Modelling shrinkage during convective drying of food material: a review[END_REF]. During drying, the shrinkage is known as ideal or perfect when the volume reduction of the solid matrix is strictly proportional to the mass loss. On the other hand, the pores appear if the volume reduction is smaller than the theoretical reduction due to liquid water removal. The variation of the porosity can result from the change of the apparent volume or the change of the pore volume by two phenomena: shrinkage and/or collapse [START_REF] Rahman | A theoretical model to predict the formation of pores in foods during drying[END_REF][START_REF] Khalloufi | A theoretical model and its experimental validation to predict the porosity as a function of shrinkage and collapse phenomena during drying[END_REF][START_REF] Khalloufi | A fundamental approach and its experimental validation to simulate density as a function of moisture content during drying processes[END_REF]. The collapse causes the decrease of the pore initial volume, although the non ideal shrinkage tends to increase it. The pore volume can be determined by appropriate measurement devices (porosimetry) involving the intrusion of a non-wetting fluid (mercury, nitrogen) in the pore network. It can also be determined indirectly by measuring the bulk density and the solid density. Apparent volume measurement can be achieved without contact (chromatography, laser and stereovision) or with contact from the geometric dimension measurement [START_REF] Madiouli | A method for determination of porosity change from shrinkage curves of deformable materials[END_REF]. The experimental measurement of porosity requires either the interruption of the experiment (this creating a bias in the results), or the use of several samples dried under the same conditions of drying.

In the literature, the shrinkage or the change of apparent volume during drying is determined by fundamental models [START_REF] Vacarezza | Cinetica y mecanismo de transporte de agua durante la deshidrataci on de la remolacha azucarera[END_REF][START_REF] Suzuki | Shrinkage in dehydratation of root vegetables[END_REF][START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF][START_REF] Perez | Modeling the thermal conductivity of cooked meat[END_REF][START_REF] Rahman | Density shrinkage and porosity of calamari mantle meat during air drying in a cabinet dryer as a function of water content[END_REF][START_REF] Katekawa | Study of porosity behavior in convective drying of bananas[END_REF]) and empirical correlations [START_REF] Lozano | Total porosity and open-pore porosity in the drying of fruits[END_REF][START_REF] Ratti | Shrinkage during drying of foodstuffs[END_REF][START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF][START_REF] Azzouz | Drying kinetic equation of single layer of grapes[END_REF][START_REF] Mayor | Modelling shrinkage during convective drying of food material: a review[END_REF]. The latter can only be used for limited prod- ucts and conditions of drying. The fundamental models are based on the mass conservation equations in each phase and can be differentiated by their assumptions and mathematical expressions. [START_REF] Vacarezza | Cinetica y mecanismo de transporte de agua durante la deshidrataci on de la remolacha azucarera[END_REF] presented a model based on two assumptions: the shrinkage is ideal and the material is composed of two phases (solid phase and liquid phase). [START_REF] Suzuki | Shrinkage in dehydratation of root vegetables[END_REF] worked out three models of drying based on the absence of porosity in foodstuffs (model of uniform drying, drying with wet core and drying with semi wet core). [START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF] worked out a linear correlation for fruits and vegetables according to the mass fractions of starch, glucose and cell membranes. [START_REF] Perez | Modeling the thermal conductivity of cooked meat[END_REF] carried out a model of meat shrinkage. They neglected the presence of the gas phase at the beginning of the thermal process and considered its formation during drying. [START_REF] Mulet | Effect of shape on potato and cauliflower shrinkage during drying[END_REF] studied the influence of the sample shape on the shrinkage of potato. [START_REF] Madamba | Shrinkage, density and porosity of garlic during drying[END_REF] and [START_REF] Wang | Changes in structure, density and porosity of potato during dehydratation[END_REF] studied the variation of the physical properties (shrinkage, porosity and density) of fruit and vegetables. [START_REF] Rahman | Density shrinkage and porosity of calamari mantle meat during air drying in a cabinet dryer as a function of water content[END_REF] carried out a shrinkage model which was validated for squid by taking into account the presence of three phases (solid, liquid and gas) during drying. [START_REF] Mayor | Modelling shrinkage during convective drying of food material: a review[END_REF] modified the model proposed by [START_REF] Perez | Modeling the thermal conductivity of cooked meat[END_REF] by introduction of initial porosity parameter.

In the present work, the model of shrinkage by [START_REF] Katekawa | Study of porosity behavior in convective drying of bananas[END_REF] is used using the methodology developed by [START_REF] Madiouli | A method for determination of porosity change from shrinkage curves of deformable materials[END_REF]. The model involves a limited number of material properties (initial bulk density, solid true density and liquid true density), measured at the beginning and at the end of a drying experiment, and the simultaneous acquisition of the reduced moisture content and the volume shrinkage during drying. From a literature review, various drying processes involving different shrinkage behaviours and porosity developments in different foodstuffs are examined. A straightforward expression of the porosity change vs. moisture content is proposed and applied to this set of literature data with a special focus on error estimation.

Presentation of the model

Based on the mass conservation and the volume additivity of the various sample phases, the change of the apparent volume as a function of moisture content is given by the following equation [START_REF] Katekawa | Study of porosity behavior in convective drying of bananas[END_REF][START_REF] Madiouli | A method for determination of porosity change from shrinkage curves of deformable materials[END_REF]:

z = V V 0 exp = 1 -ε 0 1 -ε A(ˇ) X X 0 + B(ˇ) (1)
where

A(ˇ) = X 0 1 + ˇX0
(2)

B(ˇ) = 1 1 + ˇX0 (3) ˇ= S L ( 4
)
where the initial porosity can be derived from a set of parameters:

ε 0 = 1 - 0 S 1 + ˇX0 1 + X 0 (5)
The ideal shrinkage depends on the solid density, liquid density, initial density and moisture content. It can be derived from Eq. ( 1) and assuming ε = 0 and ε 0 = 0 [START_REF] Vacarezza | Cinetica y mecanismo de transporte de agua durante la deshidrataci on de la remolacha azucarera[END_REF]: [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF] formulated a math model which enabled to compute the porosity by assuming the absence of the gas phase at initial state. [START_REF] Rahman | A theoretical model to predict the formation of pores in foods during drying[END_REF] assumed the presence of the initial porosity and function of the shrinkage-expansion coefficient allowing to determine the fraction of the water volume removed during drying and replaced by air. [START_REF] Khalloufi | A theoretical model and its experimental validation to predict the porosity as a function of shrinkage and collapse phenomena during drying[END_REF] presented their porosity model function of moisture content by studying the effect of the air initial volume change when drying in extreme conditions. In this paper, the porosity can be calculated from Eq. (1) and written in the following form:

y = V V 0 ideal = 0 1 + X 0 1 S + X L (6)
ε = 1 - (1 -ε 0 )(A(ˇ)(X/X 0 ) + B(ˇ)) (V/V 0 ) (7)
In another way and according to the work of [START_REF] Madiouli | A method for determination of porosity change from shrinkage curves of deformable materials[END_REF], the porosity can be deduced from the shrinkage curve vs. the reduced moisture content. It can be easily shown that the porosity can be deduced from a combination of the experimental V/V 0 value z and the ideal shrinkage y through the equation:

ε = z -(1 -ε 0 )y z (8)
The previous equation has also a graphical interpretation on the shrinkage diagram. A new baseline (ideal shrinkage biased curve) is deduced from the ideal shrinkage curve by multiplying it by the quantity (1ε 0 ). The porosity is deduced using this baseline.

The calculation of the porosity thus depends on the experimental values of the shrinkage and a set of properties (initial moisture content, solid density, initial bulk density and liquid density). The error of the porosity calculation is then derived from the experimental errors made during the determination of the shrinkage curves and the errors on the properties used in the model.

2.1.

Error calculations for the case of ideal shrinkage

The differentiation of Eq. ( 6) yields:

(V/V 0 ) (V/V 0 ) = 0 0 + X 0 X 0 + 1 + 1 S (1 + ˇX) S + 1 (1/ˇ) + X X + X L ((1/ˇ) + X) L (9)
The calculation of the error of the volume change in the case of ideal shrinkage needs the determination of the errors on moisture content, solid density, liquid density and initial bulk density.

Experimental error of the moisture content

The error of the moisture content follows from:

• the error of the measurement of the mass of the solid phase;

• the error of the measurement of the mass of the sample during drying.

The experimental error of the moisture content can be written as:

X = (1 + X) m m + m s m s (10)

Experimental error of the solid density, the liquid density and the initial bulk density

The experimental error of the solid density, the liquid density and the bulk initial density depend on measurement errors of the masses and volumes. Indeed, the solid density, the liquid density and the initial bulk density are written respectively:

S = m s V s (11) L = m 0 -m s V p0 -V s (12) 0 = m 0 V 0 (13)
The differentiation of Eqs. ( 11)-( 13) respectively gives the errors on the solid density, the liquid density and the initial bulk density in the following form:

S = S m s m s + V s V s (14) L = 2 L m m 0 -m s + V p V p0 -V s (15) 0 = m m 0 + V 0 V 0 (16)

Error of the experimental shrinkage

The error of the determination of the experimental shrinkage results from the error of the initial volume and bulk volume during drying. It is given by the following equation:

V V 0 = V V 0 V 0 V 0 + V V (17)

Error of the calculated porosity

The model of shrinkage elaborated by [START_REF] Katekawa | Study of porosity behavior in convective drying of bananas[END_REF] may be written in the following form:

(

1 -ε) V V 0 = 0 X 0 + 1 1 S + 1 L X (18)
This equation allows to write the expression of the porosity:

ε = 1 - ( 0 /(X 0 + 1))((1/ S ) + (1/ L )X) V/V 0 (19)
The error of the calculated porosity [START_REF] Madiouli | A method for determination of porosity change from shrinkage curves of deformable materials[END_REF], depends on the experimental errors on many parameters. The differentiation of Eq. ( 18) yields:

ε 1 -ε = V V + V 0 V 0 + 0 0 + X 0 X 0 + 1 + S S (1 + ˇX) + ˇ X 1 + ˇX + (ˇX) L L (1 + ˇX) (20)

Application of the model for various foodstuffs under various drying modes

In order to calculate porosities, we applied the shrinkage model for various foodstuffs (apple, banana, carrot, potato, sweet potato, garlic and pear) dried under various modes (convective drying and freeze drying). The data were collected from different authors (Table 1). For these foodstuffs, three properties were listed: the bulk initial density, the liquid density and the solid density -with the experimental methods used to obtain these data (Table 2).

Results and discussion

Using Eqs. ( 10), ( 14)-( 17) and ( 20), the errors on the initial bulk density, solid density, liquid density, initial moisture content, and initial calculated porosity of some foodstuffs were calculated (Table 2).

Convective drying

In the case of convective drying, the seven tested products were classified in various classes according to the shrinkage and porosity changing during drying. The experimental works of [START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF], [START_REF] Lozano | Total porosity and open-pore porosity in the drying of fruits[END_REF], [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF], [START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF] and [START_REF] Katekawa | Study of porosity behavior in convective drying of bananas[END_REF] were used for the application of the model for apple, carrot, potato, sweet potato, garlic and pear in order to calculate porosity changes during drying.

Ideal shrinkage (carrot and potato)

The shrinkage of carrot is reported to be ideal for convective drying [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF][START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF]. Fig. 1 shows the experimental data from [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF].

With the experimental values of S , 0 and L , given by the authors (Table 1), an initial porosity of 0.4% was calculated (Table 2). Eq. ( 19) allowed to determine porosity during the whole drying process. Except for small values of the moisture content (X/X 0 ≤ 0.1), comparison with the experimental porosity clearly shows an excellent agreement and porosity may be neglected, thus confirming ideal shrinkage. The Table 1 -Drying conditions and foodstuffs properties from the reference used in the application of the present porosity estimation model (Eq. ( 19)).

Modes and conditions of drying

Measured properties

Measurement apparatus

Product

Refs.

Convective drying (60 Fig. 1 -Experimental and ideal shrinkage of carrot as a function of the water content: convective drying [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF]; ideal shrinkage with error of measurement (dashes). et al., 1994).

dashed lines on Fig. 1 show the upper and lower positions of the shrinkage curve when the three properties S , 0 and L vary according to the error calculations. The negative values of calculated porosity cannot be accepted for obvious reasons (Fig. 2). For the experimental works by [START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF] shown in Fig. 3, experimental values show an important initial porosity (13%) whereas porosity calculations are in conformity with those performed with Zogzas experimental results (Fig. 4). This discrepancy rather suggests that experimental procedures for porosity measurement should be carefully checked. Another discrepancy is observed: there is little coherence between ideal shrinkage behaviour for low moisture content as seen on Figs. 1 and 3 and the porosity increase observed experimentally. This discrepancy cannot only be attributed to porosimetry measurements. One possible explanation is that constraint's relaxations may introduce porosity after manipulation of samples for stereopycnometry. However, the accuracy of calculations at low moisture content may be questioned. Potato behaves the same way as carrot in the case of the convective drying experiments carried out by [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF] (Fig. 5). The experimental measurements of porosity during drying also show negligible porosity development. However calculations show an important porosity development for X/X 0 ≤ 0.3 (Fig. 6.). For low moisture values, calculations are very sensitive to the y-intercept of the shrinkage curve, whose value is y(X = 0) = 0 /( s (1 + X 0 )). A possible explanation for this discrepancy would be a modification of the amylaceous solid phase density due to heat effects. et al., 1994).

Other experiments by [START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF] and [START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF], showed that potato may be classified with the other foodstuffs described thereafter for the non-ideal case, which seems to show that specific phenomena may occur in the drying of potatoes.

Non ideal shrinkage (sweet potato, garlic, pear and banana)

Sweet potato, garlic and pear are characterized by the same shrinkage behaviour [START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF]. They exhibit a linear shrinkage with a smaller slope than for ideal shrinkage down to a critical value of the reduced moisture content X/X 0 . The experimental porosity of these foodstuffs is small and it remains either constant (sweet potato, potato and garlic) or it decreases slightly (pear). The decrease of porosity in the first stages of drying was interpreted as a collapse of the solid matrix and modelled by [START_REF] Khalloufi | A theoretical model and its experimental validation to predict the porosity as a function of shrinkage and collapse phenomena during drying[END_REF][START_REF] Khalloufi | A fundamental approach and its experimental validation to simulate density as a function of moisture content during drying processes[END_REF]. Although the calculated porosity of pear fits quite well with the experimental results, this decrease was not observed. The calculated porosity remains almost constant around 5% before a sharp increase at low moisture content. For sweet potato (Fig. 7) and garlic (Fig. 8), the fitting is not correct. The experimental porosity is much higher than the calculated one. No straightforward interpretation of this discrepancy can be given.

For banana, the shrinkage curve has the same form as for sweet potato, garlic and pear (Fig. 10). Fig. 11 et al., 1983).

using the data from [START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF]. For the major part of the drying process, porosity remains practically constant, but, when the value of X/X 0 is smaller than 0.3, porosity increases. As for potato, this increase is hardly observed using the porosity model, which may be interpreted as a specific behaviour of amylaceous foodstuffs. The porosity of samples during drying seems to depend on the drying temperature, 0,0 0,2 0,4 0,6 0,8 1,0 0,0

Non-ideal shrinkage (apple)

The experimental results for apple were obtained by [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF] and [START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF]. The shrinkage of apple is linear with a lower slope than the ideal shrinkage 0,0 0,2 0,4 0,6 0,8 1,0 0,0 0,2 0,4 0,6 0,8 1,0 Shrinkage V/V0 Reduced moisture content X/X0 experimental shrinkage ideal shrinkage Fig. 13 -Experimental and ideal shrinkage of apple as a function of the water content: convective drying [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF]; ideal shrinkage with error of measurement (dashes). Fig. 14 -Experimental and calculated porosity of apple as a function of the water content: convective drying [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF]. (Fig. 13). This linear behaviour of the experimental shrinkage results in the increase of porosity, starting from its relatively high initial value (18%) to reach an end value of 60% (Figs. 12 (plot 1) and 14). In comparison with the other foodstuffs, apple is characterized by a high porosity during drying.

The results present a small deviation compared to experimental porosity. This deviation can be due to the errors of experimental measurement of the initial bulk density and the non-homogeneity of the samples. Fig. 13 clearly shows that the ratio between the volume of air and the initial volume remains almost constant above X/X 0 = 0.4.

Freeze-drying

In the case of freeze-drying, the foodstuffs have rather a high porosity at the end of drying. This phenomenon follows 0,0 0,2 0,4 0,6 0,8 1,0 0,0 from the absence of capillary forces during sublimation of the frozen solvent. The final porosity of the foodstuff reaches 90% for apple and it remains between 80 and 85% in case of carrot, potato and banana. By applying the shrinkage model for apple, potato, carrot and banana in the case of freeze-drying (the temperature is -35 • C and the pressure is 0.04 mbar), the experimental shrinkage of these products is linear with a smaller slope than that of the curves of the ideal shrinkage (Figs. 15,17 and 19). The porosity calculated by the model is close to experimental porosity and it varies linearly, which suggests the conservation of a porous skeleton of various foodstuffs during drying (Figs. 12,16,18 and 20). The error of the calculated porosity does not exceed 5% for all the experiments.

Conclusion

With the mere knowledge of the following parameters: solid density, liquid density and initial bulk density and the follow up of the quantities X/X 0 (reduced moisture content) and V/V 0 (volume shrinkage), the proposed method allowed to calculate the bulk porosity (including closed and opened pores) whereas the direct experimental methods may be unable to account for crack formations or closed porosities. We have validated this model by applying it to experimental data from the literature, carried out under various drying modes (convective drying and freeze drying). For two of the studied materials (carrot, potato), the initial porosity was negligible. For the others (apple, banana, sweet potato, garlic and pear), the initial porosity was relevant. Except for garlic and partly for sweet potato, a good agreement between the calculated and the experimental porosity. When the experimental porosity was obtained by stereopycnometry, the agreement with calculated porosities was good, which suggests that the samples treated by these measurement techniques are representative of the bulk behaviour of the product. This was particularly the case for the experiments in convective drying by [START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF] for carrot (Fig. 3) and apples (Fig. 14) and by [START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF] for pears (Fig. 9) as well as for all the experiments conducted in freeze drying for carrot, potato and banana [START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF]. In many cases, differences between experimental and theoretical porosities are observed at the initial stage of drying. These differences can be attributed to the sampling protocol and experimental device rather than to the heterogeneity in the solid. Differences observed at low moisture contents may be attributed to crack formation or chemical reactions occurring in the solid phase at the drying temperature, although errors also become very meaningful and may affect the accuracy of the calculations.

Because the present method is based on simple and accurate methods (mass and length measurements) and relies on complementary density measurements, because it does not require to stop experiments and cut samples in the foodstuffs, it is a powerful tool to obtain information on the porosity change of a drying foodstuff under various external conditions. It should be completed by more local measurement techniques in order to attribute porosity change to physical (collapse, expansion, cracks formation) or chemical phenomena (crystallization, glass transition, reactions in solid or liquid phase) and observe the scale of porosity formation.

Fig. 2 -

 2 Fig.2-Experimental and calculated porosity of carrot as a function of the water content: convective drying[START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF].

Fig. 3 -

 3 Fig.3-Experimental and ideal shrinkage of carrot as a function of the water content: convective drying[START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF]; ideal shrinkage with error of measurement (dashes).

Fig. 4 -

 4 Fig.4-Experimental and calculated porosity of carrot as a function of the water content: convective drying[START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF].

Fig. 5 -

 5 Fig.5-Experimental and ideal shrinkage of potato as a function of the water content: convective drying[START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF]; ideal shrinkage with error of measurement (dashes).

Fig. 6 -

 6 Fig.6-Experimental and calculated porosity of potato as a function of the water content: convective drying[START_REF] Zogzas | Densities, shrinkage and porosity of some vegetables during air drying[END_REF].

Fig. 7 -

 7 Fig.7-Experimental and calculated porosity of sweet potato as a function of the water content: convective drying[START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF].

Fig. 8 -Fig. 9 -

 89 Fig.8-Experimental and calculated porosity of garlic as a function of the water content: convective drying[START_REF] Lozano | Shrinkage, porosity and bulk density of foodstuffs at changing moisture contents[END_REF].

Fig. 10 -

 10 Fig. 10 -Experimental and ideal shrinkage of banana as a function of the water content: convective drying (Krokida and Maroulis, 1997); ideal shrinkage with error of measurement (dashes).

Fig. 11 -Fig. 12 -

 1112 Fig.11-Experimental and calculated porosity of banana as a function of the water content: convective drying[START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF].

Fig. 15 -

 15 Fig.15-Experimental and ideal shrinkage of carrot as a function of the water content: freeze drying[START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF].

Fig. 16 -

 16 Fig.16-Experimental and calculated porosity of carrot as a function of the water content: freeze drying[START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF].

Fig. 17 -Fig. 18 -

 1718 Fig.17-Experimental and ideal shrinkage of potato as a function of the water content: freeze drying[START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF]; ideal shrinkage with error of measurement (dashes).

Fig. 19 -

 19 Fig.19-Experimental and ideal shrinkage of banana as a function of the water content: freeze drying[START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF]; ideal shrinkage with error of measurement (dashes).

Fig. 20 -

 20 Fig.20-Experimental and calculated porosity of banana as a function of the water content: freeze drying[START_REF] Krokida | Effect of drying method on shrinkage and porosity[END_REF].

Table 2 -

 2 Properties of the studied foodstuffs with errors and experimental methods to obtain these properties for the references used in the application of the present model.

	Product	Initial bulk density (kg/m 3 )	Solid density (kg/m 3 )	Liquid density (kg/m 3 )	Initial moisture	Initial calculated
					content (kg/kg)	porosity (%)
	Lozano et al. (1980): convective drying
	Apple	853.5 (1.1)	1540 (1.2)	1000	7.45	18.2
	Lozano et al. (1983): convective drying
	Carrot	1026 (1.1)	1530 (2)	1000	7.55	1.6
	Garlic	980 (1.1)	1383 (2)	1000	2.28	10.3
	Pear	993 (1.1)	1473 (2)	1000	7.03	4.7
	Potato Sweet potato	1050 (1.1) 1038 (1.1)	1316 (2) 1559 (2)	1000 1000	4.5 2.55	-0.4 6.7
	Zogzas et al. (1994): convective drying
	Apple Carrot Potato	888 ± 7.6 (3.1) 1056 ± 9.0 (3.1) 1041 ± 8.9 (3.1)	1510 ± 37 (3.2) 1610 ± 45.82 (3.2) 1500 ± 39.95 (3.2)	1040 ± 4.78 1020 ± 3.85 1020 ± 3.73	7.4 ± 0.001 9.82 ± 0.002 9.71 ± 0.002	18.2 ± 3.69 0.4 ± 5.35 1 ± 5.31
	Krokida and Maroulis (1997): convective drying
	Apple Banana Potato Carrot	927.2 ± 14.7 (3.1) 1053.9 ± 16.8 (3.1) 1046.2 ± 16.8 (3.1) 1059 ± 16.8 (3.1)	1650 ± 10.6 (4) 1920 ± 7.3 (4) 1610 ± 6.6 (4) 1770 ± 10.4 (4)	1000 ± 0.82 1000 ± 0.82 1000 ± 0.77 1000 ± 0.75	9.27 ± 0.04 4.93 ± 0.001 7.01 ± 0.002 9.69 ± 0.003	9.4 ± 8.17 3.4 ± 5.46 3.1 ± 6.47 0 ± 8.94
	Krokida and Maroulis (1997): freeze drying
	Apple Banana Potato Carrot	927.2 ± 18.4 (3.1) 1053.9 ± 21.0 (3.1) 1046.2 ± 21 (3.1) 1059 ± 21.0 (3.1)	1650 ± 12.7 (4) 1920 ± 9.1 (4) 1610 ± 9.1 (4) 1770 ± 9.1 (4)	1000 ± 1.02 1000 ± 1.02 1000 ± 0.97 1000 ± 0.98	9.27 ± 0.001 4.93 ± 0.001 7.01 ± 0.001 9.69 ± 0.001	11.31 ± 10 3.4 ± 6.82 0.5 ± 7 1.6 ± 7.15
	Katekawa and Silva (2004): convective drying
	Banana at 30 • C and RH = 45%	1025 (5.1)	1497 (5.2)	1000	3.15	5.7
	Banana at 40 • C and RH = 27%	1025 (5.1)	1497 (5.2)	1000	2.85	6.3
	Banana at 50 • C and RH = 15%	1025 (5.1)	1497 (5.2)	1000	2.5	7.2
	1.1, buoyancy properties; 1.2, volume meter; 2, pneumatic psychrometer; 3.1, n-heptane; 3.2, Stereopycnometer Beckmann 930; 4, Stereopyc-
	nometer Quantachrome; 5.1, n-hexane; 5.2, helium pycnometer.


1st International Conference on Engineering for Waste Treatment Beneficial Use of Waste and by-Products

The International Conference on Engineering for Waste Treatment: Beneficial Use of Waste and by-Products, 'WastEng2005' (17 -19 May 2005, Albi, France) was a first attempt to bring together engineers and researchers in different fields of expertise to cover scientific, technological and regulation issues concerning waste treatment. This event, organized by the Ecole des Mines d'Albi-Carmaux (France), Paul Sabatier University of Toulouse (France), the CNRS PROMES Center in Odeillo (France), Tsinghua University (China) and the New Jersey Institute of Technology (USA), gathered 275 participants coming from 40 countries. Topics covered by the conference ranged from issues related to basic research to industrial development of new technologies and processes for the treatment and beneficial reuse of wastes: agricultural, industrial and municipal solid wastes, polluted soils and sediments as well as gas and liquid emissions.

During the three day meeting, presentations and discussions highlighted significant topics:

. New cutting-edge technologies and research on the pretreatment methods, processes and environmental and sanitary evaluation strategies for better reuse of waste materials. . Detailed descriptions of relevant concepts including their risk, environmental and economical assessments. . Examples of new processes and pilot scale or industrial demonstrations. . Legislative updates and tendencies. . Barriers to conquer from the technical, legislative and society acceptance points of view to progress in the beneficial reuse of wastes.

During the post-conference workshop, held on Friday May 20th at the Ecole des Mines d'Albi, a working group of experts summarized the advances revealed during the various sessions. A consensus was reached to define the most promising results and needs for future research:

. Multidisciplinary networks need to be reinforced on the European and World scale in order to generate fruitful exchanges between specialists in process engineering, civil engineering, environmental and other sciences devoted to the treatment and beneficial reuse of wastes.

. More co-operation is needed at different levels of government, academia and industry. Support is required to build demonstration pilots for emerging technologies. . Social stakes have to be taken into account to promote recycled materials and accept products and energy made with wastes each time this is possible. This policy will advantageously reduce waste stockpiling and land filling, help to preserve natural resources and reduce the fossil fuel consumption.

From the 230 papers in the proceedings, a selection of short papers was made for publication in this special issue with a special emphasis on waste-to-energy issues, waste beneficial use as agricultural matter or industrial material, and new process developments for the degradation of chemicals.

The production of sludge by an increasing number of urban or industrial wastewater plants and the difficulties in finding a suitable disposal route was an important discussion topic at the conference. Several papers related to wastewater and sludge treatment are included in this special issue as sludge can be viewed as an excellent illustration of the major trends in waste processing: degradation or destruction, energy conversion, agricultural reuse or transformation into high grade 'industrial like' materials.

The reclaiming of wastewater in dry areas such as in Perth (Australia) can be associated with the injection of the treated wastewater into coastal aquifers (Li et al.) allowing for further water purification, restoration of groundwater levels and prevention of saline water intrusion.

Reduction of sludge quantity can be performed using a combination of various treatments and anaerobic digestion. Paul et al. compared disintegration techniques, such as thermal (958C), ozonation or hydrogen peroxide treatment.

A collective paper under the auspices of the EU framework program, the BIOWASTE project (Schmidt et al.) assessed the sustainability of sludge recycling on agricultural land. The benefits of sludge land spreading were assessed using a combination of scientific approaches in various fields (biotechnology, ecotoxicology, plant toxicology, microbiology, chemistry, LCA. . .) with important implications in current EU regulatory work.

The production of carbonaceous sorbents for the treatment of industrial wastewater is an alternative option for sludge beneficial use. In their paper Rio et al. optimized the different stages of carbon activation with sludge as raw material and compared this valorization technique with standard disposal routes. Similarly, Orescanin et al. used bauxite waste (red mud) for the purification of industrial wastewater containing heavy metals.

As far as the energy yield is concerned, sludge may be considered as a potential solid fuel, but the main obstacle for sludge derived fuel production is the cost of energy for drying. A thermal drying process is described by Peregrina et al. Based on the concept of frying sludge with recycled cooking oil for the synthesis of a high grade solid fuel, this process is assessed by LCA (life cycle assessment) at the earliest stage of its design. Sludge degradation rates through anaerobic digestion can be enhanced by a thermal pre-treatment (130 -1708C) with a benefit in biogas yields (Bougrier et al.). Gas production from anaerobic digestion can also be applied to agricultural by-products such as olive pulp with a special focus on hydrogen production (Koutrouli et al.). The incineration of municipal solid waste cannot merely be considered as a pure combustion problem, due to the heterogeneity of the solid load and the presence of heavy metals. Me ´nard et al. developed a 3D model of MSW incineration on a moving grate including thermodynamics calculations of heavy metals speciation in the solid and gas phases at different locations of the bed, post-combustion zone and boiler.

Incineration of low calorific value gases like VOCs both saves energy and allows for reduction in hazardous emissions. Porous media burners are studied both numerically and experimentally. These techniques allow a better flame stability and a good heat recovery.

The degradation of highly contaminated waste streams also necessitates the development of new processes or the optimization of existing ones. Kano et al. developed a non-thermal process for dechlorinating PVC, using mechanochemistry to transform organic chlorine into water-soluble chloride. The degradations kinetics of pharmaceutical effluents (typically halogenated pyridines) in wastewaters were studied by Papadaki et al. using UV radiation. Kawasaki et al. designed a supercritical water reactor for the destruction of polychlorinated biphenyls (PCB) and dioxins. These examples illustrate the variety and the high potential of mechanical, thermal and chemical processes for waste treatment.

Since the WasteEng2005 conference, one year ago, economic changes have occurred worldwide and the constant increase of the price of raw materials and energy have brought the enormous amount of energy and material hidden in solid and liquid wastes under the spotlight. The application of the Kyoto protocol in 163 countries impacts the substitution by heavy industries (chemicals, cement, pulp and paper) of large tonnage of raw materials (including gaseous, liquid or solid fuels) by equivalent quantities of processed waste. There is a resulting demand for research in efficient technological solutions for gas separation and treatments in waste, biomass and by-products combustion, and water removal from waste and pollution control. Lastly, the development of inexpensive wastewater and solid waste processes will allow increased sustainability in developing and newly industrialized countries. This is a formidable challenge for researchers and chemical engineers. The next international conference, WasteEng2008, will focus on the Valorization of Waste as Energy and Useful Materials. Note: The proceedings of WasteEng2005 can be obtained by contacting Jean-Louis Dirion (dirion@enstimac.fr) or Ange Nzihou (nzihou@enstimac.fr). The conference web site is still available at http://www.enstimac.fr/heberges/ wasteeng2005.

  It will be held in Patras, Greece, 3-5 June 2008. Professor Didier Lecomte, Ecole des Mines d'Albi-Carmaux, France Editor, Energy and Alternative Energy Sources and Professor Ange Nzihou, Ecole des Mines d'Albi-Carmaux, France Chair of the organizing committee, WasteEng2005