The materials used in this study are Spirulina platensis samples from two farm sites located in two different areas:

Ÿ “Loumbila” farm situated at 15 km in the north of Ouagadougou, Burkina Faso. This material sample will be named sample-B,

Ÿ “La Fon del Cardaire” farm situated at Gignac in the south of France. This material will be named sample-F.

Their mode of production is not different from one farm to another, from one country to another. These are the methods that change, adapted to the size of the farm, the workforce and the materials available on site. Indeed, the “sample-B” is grown in open ponds and the “sample-F” in glass ponds. However, if these two products were grown in localities with different climates, their seeding is done by strains of the same type “platensis” in basins whose salinity and alkalinity are provided by sodium bicarbonate (NaHCO₃). Pretreatments were carried out on each biomass harvested to remove a good part of the culture water until reaching a given water content. The methods pretreatment used is different from one farm to another depending on the available materials. Thus, the water content of the sample-B is reduced to about 2.73 kg_w/kg_db by manual pressing while that of the sample-F is reduced to about 3.12 kg_w/kg_db by a spin under the action of vacuum. These water contents obtained after pretreatment are considered as the water contents initial of the samples. The initial characteristic parameters recorded in Table 1 have been determined using standard methods. The volume of a substance corresponds to the space occupied by this substance. For measurements, a container of cylindrical shape with known dimensions is previously weighed empty so that its mass can be measured and subtracted from that of the container loaded mass m. It is then in an oven maintained at 70˚C for 48 hours to determine the anhydrous mass m_s.

First, the filtering of biomass through one or two devices superimposed. In general the first consists of a thin mesh of the order of 300 microns which retains any impurities (lumps, insects, leaves). The second is a filter in nylon is of the

order of 50 μm. After filtering, fresh Spirulina platensis is processed into test-ready samples. A cylindrical shape commonly used in farms for the drying is proposed (Figure 1). To avoid variability and ensure identical initial conditions between samples, the following design process has been adopted: the fresh biomass is moulded in a cylindrical device of 20 mm diameter and 40 mm length composed of a tube and a piston made in Teflon; the cylindrical surface of samples are wrapped with plastic film to prevent adhesion between samples. Both end faces are free to ensure a one-dimensional water transport along sample axis; a sample is kept if its initial mass is m₀ = 0.013 ± 0.0006 kg.

It is assumed that these conditions are met and the samples are uniform in the initial state.

To understand experimentally the distribution of water in the product during drying, we chose to follow the variation of the water content in the product by cutting samples. The water content in a sample slice is determined by considering the mass of this slice at a given instant and that of the solid phase according to the following Equation (1):

w = m ( x , t ) − m s m s (1)

The experimental setup is constituted of a desiccator where the relative humidity is regulated at 6% by a potassium hydroxide KOH solution (Figure 2). The cylindrical samples (Figure 2(a)) are put above the solution (Figure 2(b)) while the desiccator is placed in a thermo-regulated chamber (Memmert UFP 600) where the temperature is maintained at 50˚C. At specific time steps of the drying process, one of the samples is removed from the desiccator and immediately ox introduced in a cutting device to obtain slices of 2 mm thickness. The water content of each slice is determined by differential weighing after 48 hours drying at 70˚C for complete dehydration. In order for the desired water content profiles to be characteristic of a given single sample, all the samples should be in the same state at the process beginning. Thus, they are prepared under the same

conditions as indicated above. The material being symmetrical, the water contents averages of symmetrical points with respect to the median plane are considered. This approach provides to access to the moisture content profiles w ( x , t ) corresponding to the drying times (Figure 2(d)). Since we aim to characterize water transport phenomena in a single sample from measurements obtained on various samples, it is essential to have the same initial conditions in all samples. Although special attention is paid to sample preparation, this methodology may have some differences. For this, the experimental water content profiles are approximated by simple polynomial equations (Equation (2)) introducing three parameters a, b and c at each time step.

w ( x , t ) = a x 4 + b x 2 + c (2)

This expression ensures to have a symmetrical evolution with a no flux condition at the symmetry plane. The Experiments are carried out and lasted at most 120 hours (5 days). As it for a drying period long the material begins to stiffen and the cutting deviate delicate. For low temperature (less than 25˚C), the drying is too slow for reach values with low water content and the range of water content investigated is too narrow. It appears that for longer time period, the material starts to denature and the development of micro-organisms are observed. This could change the process of water transport and should be avoided.

Simplifying assumptions are considered to develop the descriptive equations of water transport: the material is assumed to remain diphasic along the experiment, the initial water content in the product is homogeneous, both solid and liquid phases are incompressible, water transport is one-dimensional along the cylindrical sample axis x.

The real mass densities are determined over intrinsic volume of water and solid phases:

ρ w * = m w V w ; ρ s * = m s V s (3)

The material being biphasic, the apparent densities are expressed by [25] :

ρ w = ρ w * w α + w (4)

ρ s = ρ s * 1 α + w (5)

α = ρ w * / ρ s * is the ratio of real densities. Considering u w and u s as the water and solid phase velocities along x, mass balance equations are written [26] [27] :

∂ ρ w ∂ t + ∂ ρ w u w ∂ x = 0 (6)

∂ ρ s ∂ t + ∂ ρ s u s ∂ x = 0 (7)

The mass flow of water with respect to the solid phase is written as follows:

F w = ρ w ( u w − u s ) = ρ w u w − w ρ s u s (8)

For reasons of symmetry, u w ( 0 , t ) = u s ( 0 , t ) = 0 and from balance Equations (6)-(7), we obtain by integration the Equations (9) and (10):

ρ w u w ( x , t ) − ρ w u w ( 0 , t ) = ρ w u w ( x , t ) = − ∫ 0 x ∂ ρ w ∂ t d x (9)

ρ s u s ( x , t ) − ρ s u s ( 0 , t ) = ρ s u s ( x , t ) = − ∫ 0 x ∂ ρ s ∂ t d x (10)

By replacing the expressions 9 and 10 in Equation (8) and considering the Equations (8) and (9), the relation of the water flow becomes:

F w = ρ w * ( w ∫ 0 x ∂ ∂ t 1 α + w d x − ∫ 0 x ∂ ∂ t w α + w d x ) (11)

Deriving the Equation (4), the density gradient of the water is written:

G w = ρ w * ∂ ∂ x ( w α + w ) (12)

Fick’s law (Equation (13)) is a relationship that links the water flow to the mass concentration gradient through a phenomenological coefficient [28] .

F w = − D w ∂ ρ w ∂ x (13)

In the case of geometry and boundary conditions used in experiments, radial water transport can be neglected so that water content is considered uniform in a cross-section at abscissa x. From relations 11 and 13, the water transport coefficient is accessed as a function of the water content.

D w = − F w G w (14)

The experimental determination of the water transport coefficient from the exploitation of the water content profiles is carried out as follows. For a particular time step t j and a given water content w k , the fitting of the associated water content profile (Equation (2)) can be inverted to give the corresponding abscissa x i . Knowing this abscissa, the values of the water flow (Equation (11)) and those of the water density gradient (Equation (12)) can be calculated. In fact, in Figure 2(d), the horizontal line associated with the given water content w k can cut several water content profiles providing a set of values ( x i , t i ) leading to a set of couples { F w ( x i , t j ) , G w ( x i , t j ) } . This set of couples is associated with a single value of the moisture content w k and must correspond to a single value of the coefficient of transport G w . In this way, by scrolling through a range of water content values between 0.7 < w k < 2.5 , this approach makes it possible to determine the evolution of the transport coefficient D w ( w ) as a function of the water content. The precision on the determination of the experimental results is estimated by statistical parameters: the coefficient R 2 (Equation (15)): and the root mean square error (RMSE) (Equation (16)): the RMSE squared error and the coefficient (Equation (16)).

R 2 = 1 − ∑ z = 1 N ( Y e x p , z − Y p r e , z ) 2 ∑ z = 1 N ( Y ¯ e x p , z − Y p r e , z ) 2 (15)

R M S E = ∑ z = 1 N ( Y e x p , z − Y p r e , z ) 2 ∑ z = 1 N ( Y ¯ e x p , z − Y p r e , z ) 2 (16)

The experimental approach presented above made it possible to obtain a representative evolution of the water distribution inside of the two Spirulina platensis samples types studied (Figure 3 and Figure 4). These profiles are accurately estimated by a polynomial expression (Equation (2)) whose parametric coefficients at each time t of measurement are recorded in Table 2 and Table 3. Data for both samples were recorded differently. It appears that the recording time has no influence on the evolution of the parameters. This function constitutes a base of the calculation program for the coefficient of transport. The water content profiles (axial and temporal variation of the water content in the product) show that the drying process is more important at the ends than inside of the samples. A heterogeneous water distribution is observed along the cylindrical axis. Ce qui montre que la perte d’eau par évaporation à la surface d’échange n’est pas complètement compensée par le transport interne. Considerable shrinkage of the solid matrix is observed and the thickness of the samples is decreased by about 0.6 mm. Trends similar to that of axial shrinkage

of the spirulina have been theoretically demonstrated for the alumina gel by Chemkhi (2008) [29] . The profiles make it possible to establish the drying kinetics for various abscissas of x up to a value of the order of 5 mm. These water content profiles will serve as a basis for determining the transport coefficient and its evolution throughout the drying process.

In Figure 5 and Figure 6 are carried the variations water content at various abscissa as a function of time for both types of samples. The results are classic: the surfaces dry faster than the heart of the product. But, the differences between the curves are quite considerable. In fact, the water discharged from the zones close to the surface is not compensated by the water flowing from the heart towards the surface under the effect of the large gradients of the liquid phase, the motor terms of the flow. The study is carried out for abscissa between 5 mm and 16 mm. Because, out of this interval, the variations of water content and concentration gradient are too small and the experiments could not be exploited.

From these profiles above, we have represented on the one hand the evolution of the water transport flow and that of density gradient at different thicknesses in the initial configuration of the sample ( x ≤ 16 mm) as function of time (Figure 7(a) and Figure 7(b)) and secondly their variation as function of the water content (Figure 7(c) and Figure 7(d)). On the sample-B and sample-F, we note that the transport flow and the gradient have a similar appearance at each abscissa. They increase that the further one moves away from the median plane (x = 0). The gradients grow from the drying beginning and decrease towards the process end. For a given abscissa, the water transport flow becomes constant and has some inflection points that reflect the influence of an instantaneous contraction of the material during drying which contributes to migrate the water.

Decreasing curves indicate that the flow of water carried by filtration from the interior to the exchange surface is not compensated for in the pores. The main interest of the representation of the gradient as a function of time and as a function of the water content lies in the identification for each slice, a critical period and a critical water content (Figures 8(a)-(d)) to from which the contribution of the solid contraction to the transport of water becomes very negligible. We have just chosen to draw the curves of Figure 7(a), Figure 7(b), Figure 8(a) and Figure 8(b) on abscissa less than 13 mm. Because beyond these values, the rigidity of the products deviates more important and the exploitation of the data becomes delicate.

In Figure 9 and Figure 10, a relation between the mass flux and the density gradient for each material studied is shown. This function is quasi-linear for each slice considered. What approaches us to Fick’s law which states that in a

homogeneous medium, the flow of the liquid phase can be described by a linear relation (Equation (13)).

From the water content profiles, density gradients and fluxes presented above, the range of water content that can be studied is 0.7 < w_w < 2.5 kg_w/kg_dm. That is, from a horizontal line intersecting at least two water content profiles. In this range, the water flow and the apparent density gradient of water are determined by Equations (11) and (12). For each water content value, a relatively linear correlation is observed between the water flow and the gradient. Therefore, the water transport coefficient is determined from the slope of each linear approximation. For both strains (sample-B and sample-F), the dependencies of the water transport coefficients with respect to the water content are presented in Figure 11. It is noted that the experimental results are practically concordant. We can

therefore consider that the two results are representative of a single evolution of the water transport coefficient of Spirulina type platensis. Experimental data are accurately described by an exponential equation (Equation (17)).

D w ( w ) = 2.78 × 10 − 10 + 3.33 × 10 − 12 exp ( 3.12 w ) (17)

An optimization algorithm based on a nonlinear regression procedure made it possible to obtain a good correlation between the experimental data and those of the theory with a correlation coefficient R² > 97% and a standard deviation RMSE < 1.16 × 10⁻¹⁰ m²∙s⁻¹.

There is a sharp decrease in the transport coefficient for water contents above 2 kg_w/kg_dm and a low decrease for low water contents. This trend has already been observed with rubber [30] , agar gel [31] [32] , clay [33] , coffee [34] and wood [35] . W = 2 kg_w/kg_dm represents a critical mean value from which the contribution of solid shrinkage to water transport is negligible. In the case of Spiruline platenis, previous work [36] [37] has shown that the solid acquires a mechanical strength and becomes progressively rigid during the last stages of drying. It should be remembered that the proposed destructive method satisfactorily assesses the water transport coefficient of Spirulina platensis over a wide range of water content.