Dynamic Vapor Sorption (DVS) Analysis of the Thermo-Hygroscopic Behavior of Arthrospira platensis Under Varying Environmental Conditions

Abstract

This paper presents a new study and analysis of the thermo-hygroscopic behavior of Arthrospira platensis using dynamic vapor sorption (DVS) system. Thermo-hygroscopic characterization is essential for optimizing the drying process and enhancing storage conditions. Therefore, the objective of this work was to investigate the thermo-hygroscopic properties of Arthrospira (Spirulina) platensis using a dynamic vapor sorption (DVS) system. This thermo-hygroscopic analysis focused on three fundamental parameters, namely: the desorption isotherms, the net isosteric heat of water desorption, and the moisture diffusivity. Desorption isotherms were measured at five different temperatures (25 °C, 40 °C, 50 °C, 60 °C and 80 °C) over a relative humidity range of 10–80%. The desorption isotherm data were fitted to five semi-empirical models: GAB, Oswin, Smith, Henderson, and Peleg. The results indicated that the GAB model provided the best fit for the experimental data. The net isosteric heat of desorption was determined using the Clausius–Clapeyron relation. It decreased from 21.3 to 4.29 KJ/mol as the equilibrium moisture content increased from 0.02 to 0.1 Kg/Kg (dry basis). Additionally, the moisture diffusivity of Arthrospira platensis was estimated based on Fick’s second law of diffusion and the desorption kinetics obtained from the DVS equipment. This parameter varied between 1.04 10⁻⁸ m²/s and 1.46 10⁻⁷ m²/s for average moisture contents ranging from 0.003 Kg/Kg to 0.191 Kg/Kg (dry basis). Furthermore, the activation energy for desorption was estimated to be approximately 33.7 KJ/mol.

1. Introduction

Arthrospira (Spirulina) platensis is a microscopic and filamentous cyanobacterium that has existed for over 3 billion years. Its name is derived from its spiral shape, and it belongs to the family of cyanobacteria, commonly referred to as blue-green micro algae. Spirulina platensis is low in calories but packed with nutrients in a very small volume [1]. Indeed, Arthrospira contains a very high protein content, exceeding 60% of its dry mass, making it one of the most significant sources of protein among both protists and plants [2].

At the end of the cultivation phase, the algal biomass exhibits a very high moisture content, posing significant challenges for storage and recovery. Consequently, reducing its water content through dehydration is essential to preserve biomass quality. This dehydration process represents nearly 30% of the overall processing cost for Arthrospira platensis [3]. Various dehydration techniques have been employed to dry the Spirulina platensis biomass, including convective air drying [4,5], solar drying [6], freeze drying [7], spray drying [8], drum drying [9], and hot air drying assisted by capillary draining [10].

A key aspect of our research involves the design of dryers for Arthrospira platensis. This process requires a comprehensive understanding of the desorption isotherms, particularly the relationship between equilibrium moisture content and water activity. According to Leung [11] and Rouquerol et al. [12], water activity measurements techniques can be classified into two main approaches. The manometric methods, where the pressure above the sample is recorded, without direct measurement of mass variation. The other approach is the gravimetric methods, in which the pressure is held constant while the sample’s mass is measured. Among gravimetric techniques, the saturated salts method is commonly used as a standard. However, this method can take several days, or even weeks, to achieve equilibrium, and at high relative humidity, the extended duration can promote bacterial growth, which could invalidate the results. Given that Spirulina platensis is highly hygroscopic and its properties change rapidly over time, it is therefore essential to employ the dynamic vapor sorption (DVS) technique. Desorption experiments conducted with DVS exhibit excellent reproducibility, ensuring unbeatable accuracy of this method [11].

Assessing the thermo-hygroscopic characteristics of Spirulina, including net isosteric heat and moisture diffusivity (in the hygroscopic state) is crucial for optimizing its drying process. Indeed, the net isosteric heat of desorption, typically derived from desorption isotherms using Clausius–Clapeyron relation, is essential for accurately estimating the energy requirements for desorption [13]. Moreover, the diffusion coefficient of Spirulina platensis during desorption is a key parameter for simulating and optimizing the drying process [14].

Although the thermo-hygroscopic characterization of Spirulina is an area of growing interest, few studies on this subject have been conducted to date. These studies focus generally on the moisture sorption behavior and the moisture diffusivity under varying temperature and humidity conditions. Among the studies reported in the literature; Desmorieux and Decaen examined Spirulina desorption isotherms using three methods: gravimetric static, dew point temperature measurement, and dynamic vapor sorption (DVS) over a temperature range of 25–40 °C. Their results showed that measurements obtained using DVS resulted in a lower equilibrium moisture content compared to the other methods. Additionally, they concluded that the desorption isotherm was not significantly influenced by air temperature within this range [15]. Oliveira et al. studied the drying behavior of Spirulina in cylindrical pellets (d = 3 mm) at temperatures between 50 °C and 60 °C, with air velocity of 1.5 m/s and relative humidity ranging from 7% to 10%. The diffusivity values were found to range between 0.233 × 10⁻¹⁰ and 0.286 × 10⁻¹⁰ m²/s [16]. Dissa et al. investigated the drying of Spirulina under conditions of T = 50 °C and RH = 10%, using samples of different shapes. For cylindrical samples of 3–6 mm diameter and L = 4 cm, the diffusivity values were found to range from 0.122 × 10⁻¹⁰ to 0.167 × 10⁻¹⁰ m²/s. While slab-shaped Spirulina samples yielded values between 1.79 × 10⁻¹⁰ and 6.73 × 10⁻¹⁰ m²/s at temperatures of 40–60 °C, RH = 10%, and air velocities ranging from 0 to 1.2 m/s [17]. Eloi Salmwandé investigated the moisture diffusivity of Spirulina under controlled drying conditions (T = 50 °C, RH = 6%) using cylindrical samples (d = 20 mm). The diffusivity values were found to range from 1.7 × 10⁻¹⁰ to 6.73 × 10⁻¹⁰ m²/s [18]. Pamella de Carvalho Melo investigated the drying kinetics of Spirulina platensis at temperatures of 30, 40, 50, and 60 °C. This author reported that, firstly, the drying times required for Spirulina to reach constant weight were 7.00, 4.58, 3.83, and 3.25 h, respectively. Secondly, the effective diffusion coefficient increased with temperature, ranging from 3.343 × 10⁻⁸ to 14.881 × 10⁻⁸ m²/s. In addition, the Midilli mathematical model was applied at 30, 40, and 50 °C, while the Diffusion model was used at 60 °C. The activation energy for liquid diffusion was found to be 39.52 kJ/mol [19]. These studies highlight the limited but valuable data available on the thermo-hygroscopic behavior of Arthrospira platensis, particularly in the context of varied environmental conditions. Notably, the dynamic vapor sorption (DVS) technique has proven to be an effective and insightful method for examining moisture sorption and desorption, providing more accurate and lower equilibrium moisture content measurements compared to other traditional methods [20]. Given the promising capabilities of DVS, further research utilizing this technique could significantly enhance our understanding of the moisture diffusion, sorption, and desorption behavior of Arthrospira platensis, ultimately aiding in the optimization of its drying and storage processes.

Therefore, this study aims to (1) determine the desorption isotherms of Arthrospira platensis using the DVS technique over a wide temperature range of 25–80 °C and a relative humidity range of 10–80%, (2) identify and validate the most suitable model for fitting the desorption isotherm data, (3) calculate the net isostatic heat of desorption as a function of equilibrium moisture content using the Clausius–Clapeyron equation, and (4) estimate the moisture diffusion coefficient of Spirulina platensis during desorption process based on Fick’s second law of diffusion, and correlate the moisture diffusion coefficient with both equilibrium moisture content and temperature across the investigated ranges of humidity and temperature.

2. Materials and Methods

2.1. Biomass Cultivation and Preparation

The Arthrospira platensis strain M2 (straight morphology) was provided from Pasteur Institute and cultivated at Bio Gatrana Farm under controlled conditions, in the governorate of Sidi Bouzid, Tunisia. During cultivations, water was supplemented with 20% Modified Zarrouk medium [16]—containing the following components (g/L): NaHCO₃, 16.9; K₂HPO₄, 0.4; NaNO₃, 2.7; MgSO₄ 7H₂O, 0.3; K₂SO₄, 1.5; FeSO₄ 7H₂O, 0.01; NaCl, 1.1; CaCl₂, 0.03; EDTA, 0.08—and micronutrients. At the end of cultivation, the biomass was recovered by filtration and pressed to recover the biomass with a moisture content of 5.6 ± 0.2% (dry basis).

2.2. DVS (Dynamic Vapor Sorption) Apparatus and Experimental Procedure

Dynamic vapor sorption (DVS) experiments were performed using a DVS Advantage apparatus (DVS-ET-1, Surface Measurement Systems Ltd., London, UK). This apparatus is available at LAGEPP (Laboratory of Automation, Process Engineering, and Pharmaceutical Engineering), affiliated with the University of Lyon (France). The system is equipped with a high-precision microbalance (±0.1 μg). Approximately 10 ± 2 mg of Arthrospira platensis biomass was placed on a stainless-steel sample pan. The analysis was conducted at a controlled temperature, under a continuous flow of dry nitrogen (200 mL/min), which was used as the carrier and purge gas.

The relative humidity (RH) was varied in a stepwise fashion from 10% to 80% in increments of 10% RH, using a mixture of dry and saturated nitrogen streams regulated by mass-flow controllers. At each RH step, the sample mass was continuously recorded until equilibrium was reached, defined as a mass change rate lower than 0.002% per minute over 10 min. The corresponding equilibrium moisture content (dry basis) was automatically calculated by the DVS software. The desorption isotherm was obtained by plotting the equilibrium mass (or mass change, Δm/m₀) against relative humidity during the RH decrease. These data were extracted from the DVS output and plotted as the desorption branch of the isotherm.

All experiments were performed in duplicate to ensure repeatability. The data were subsequently used to calculate related thermodynamic parameters, including the heat of sorption and the moisture diffusion coefficient in the hygroscopic state.

2.3. Modeling of the Desorption Isotherms

The desorption curves are influenced by the nature and hygroscopic state of Spirulina platensis as well as the process through which equilibrium is achieved. Numerous theoretical, semi-empirical and empirical models have been proposed in the literature to describe experimental desorption isotherm data [21,22]. The most widely recognized models are listed in

Table 1. Models used to fit desorption isotherm data.

Models	Equation	References
GAB	$X_{eq}={\displaystyle \frac{X_{m}C{Ka}_w}{\left(1-K{a}_w\right)(1+\left(C-1\right)K{a}_w)}}$	[24]
Oswin	$X_{eq}={A\left({\displaystyle \frac{a_w}{1-a_w}}\right)}^B$	[25]
Smith	$X_{eq}=A+Bln(1-a_w)$	[26]
Henderson	$X_{eq}={(-{\displaystyle \frac{\left(\mathrm{l}\mathrm{n}\right(1-a_w)}{A}})}^{{\displaystyle \frac{1}{B}}}$	[22]
Peleg	$X_{eq}=A{a_w}^B+C{a_w}^D$	[27]

. Among these, the GAB model, which has three parameters, is generally considered to be a robust theoretical model for most agri-food products across a broad range of water activity levels [23].

The parameters in the GAB, Oswin, Henderson, Smith, and Peleg models are empirical constants obtained by fitting the experimental equilibrium moisture data (X_eq) to each model. These constants do not have a fixed physical value but rather reflect the sorption characteristics of the material under the studied conditions. Typically, the constants vary with temperature, composition, and structural properties of the material. For instance, in the GAB model, X_m represents the monolayer moisture content, while C and K are related to the energy of sorption in the monolayer and multilayer regions, respectively. In the Oswin, Henderson, Smith, and Peleg models, the constants (A, B, C, D) are determined empirically and describe the shape and curvature of the sorption isotherm. These parameters were estimated using nonlinear regression analysis, and their significance was assessed based on the goodness-of-fit indicators (R² and χ²).

2.4. Statistical Analysis

The chi-square test (R²) and the relative mean deviation χ² were employed to evaluate the fit of the drying models [27,28]. Their mathematical expressions are provided in Equations (1) and (2):

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _1^n{(X{R}_{i,\exp }-X{R}_{i,pre})}^2}}{{\displaystyle \sum _1^n(\overline{XR}-X{R}_{i,pre})2}}}$$

(1)

$${\chi }^2={\displaystyle \frac{{\displaystyle \sum _1^n{(X{R}_{i,\exp }-X{R}_{i,pre})}^2}}{N-z}}$$

(2)

where XR_i.exp is the i-th value of the experimental moisture content. XR_i.pre is the i-th value of the moisture content predicted by the selected model. XR is the average moisture content. N is the number of observations and z is the number of constants.

2.5. Isosteric Heat

The net isosteric heat of sorption, or enthalpy (Q_st), measures the energy changes that occur when water molecules interact with the sorbent during the sorption process [28]. The heat of desorption refers to the energy required to overcome the intermolecular forces between water vapor molecules and the adsorbent surface [29]. Therefore, the heat of sorption serves as an indicator of the intermolecular attractive forces between water vapor and the sorption sites [30].

The net isosteric heats of desorption for water at various moisture contents were calculated using the Clausius–Clapeyron equation (Equation (3)), based on data derived from the GAB equation:

$${\displaystyle \frac{\partial ln{a}_w}{\partial \frac{1}{T}}}=-{\displaystyle \frac{Q_{st,n}}{R}}$$

(3)

where a_w is the water activity (dimensionless), T is the absolute temperature (°K), and R is the ideal gas constant (J/mol/K)

Assuming that Q_st,n is independent of temperature at a given specific equilibrium moisture content, integrating Equation (3) results in Equation (4).

$$\ln \left(\mathrm{a}\mathrm{w}\right)=-\left({\displaystyle \frac{{\mathrm{Q}}_{\mathrm{s}\mathrm{t},\mathrm{n}}}{\mathrm{R}}}\right)\left({\displaystyle \frac{1}{\mathrm{T}}}\right)+\mathrm{c}\mathrm{t}\mathrm{e}$$

(4)

To construct the isosteric curves, the following procedure was applied:

-

Selection of fixed X_eq values: Several equilibrium moisture contents were selected within the range common to all measured isotherms.

-

Extraction of corresponding a_w values: For each selected X_eq, the corresponding water activity was obtained at each experimental temperature.

-

Application of the Clausius–Clapeyron equation: For each X_eq, ln(a_w) was plotted against 1/T (K⁻¹). A linear regression of these points provided the slope, which was used to calculate the net isosteric heat of sorption (Q_st,n) according to Equation (4).

This approach assumes that the heat of sorption remains constant over the narrow temperature range studied and that equilibrium conditions were reached before each measurement. All regressions were performed using nonlinear least squares fitting, and only correlations with R² > 0.95 were considered acceptable for calculating Q_st,n values.

2.6. Estimation of Moisture Diffusion Coefficient

The water diffusion coefficient is a crucial property for accurately designing and controlling drying processes, as well as related operations such as storage. The effective diffusion coefficient (D_eff) is calculated from water desorption kinetics using the complete solution of Fick’s second law under the assumptions that humidity is uniformly distributed within the product, the medium is isotropic and homogeneous, the diffusion coefficient (D_eff) is constant and that product contraction is negligible [31]. Second law of Fick is expressed in Equation (5) as follows:

$${\displaystyle \frac{dX}{dt}}=D_{eff}{\displaystyle \frac{{\partial }^{2}X}{{\partial }^{2}r}}$$

(5)

where X is the local dry basis moisture content (Kg/Kg). t is the time (s). r is the spatial variable and D_eff is the diffusion coefficient (m²/s). The general solution to Equation (4) proposed by Crank takes the form of an infinite series [32] as expressed in Equation (6).

$$\mathrm{X}\mathrm{R}={\displaystyle \frac{\mathrm{X}-{\mathrm{X}}_{\mathrm{e}\mathrm{q}}}{{\mathrm{X}}_0-{\mathrm{X}}_{\mathrm{e}\mathrm{q}}}}={\displaystyle \frac{8}{{\mathsf{\pi }}^2}}\sum _{\mathrm{n}=0}^{\mathrm{\infty }}{\displaystyle \frac{1}{{(2\mathrm{n}+1)}^2}}\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{{(2\mathrm{n}+1)}^2{\mathsf{\pi }}^2{\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}.\mathrm{t}}{4{\mathrm{e}}^2}})$$

(6)

where XR is the reduced moisture content, X₀ is the initial moisture content, X is the average moisture content at time t, X_eq is the average equilibrium moisture content, and e is the sample thickness.

When the diffusion time is sufficiently long, all terms in the series become negligible compared to the first term [26], resulting in Equation (7):

$$\mathrm{l}\mathrm{n}\mathrm{X}\mathrm{R}=\ln \left({\displaystyle \frac{8}{{\mathsf{\pi }}^2}}\right)-{\displaystyle \frac{{\mathsf{\pi }}^2{\mathrm{D}}_{\mathrm{e}\mathrm{f}\mathrm{f}}.\mathrm{t}}{4{\mathrm{e}}^2}}$$

(7)

The effect of temperature on the diffusion coefficient generally follows an Arrhenius-type law [26] as expressed in Equation (8):

$$D_{eff}=D_0 \mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{Ea}{RT}})$$

(8)

where D₀ is the pre-exponential factor, Ea is the activation energy (KJ/mol), R is the ideal gas constant (J/mol/°K), and T is the product temperature (°K). Equation (9) can be rearranged as:

$$ln{D}_{eff}=\ln D_0-{\displaystyle \frac{Ea}{R}}.\left({\displaystyle \frac{1}{T}}\right)$$

(9)

Based on these equations, we have studied the evolution of the effective diffusion for different moisture contents, the effect of temperature on the diffusion coefficient and the activation energy.

3. Results and Discussion

3.1. Desorption Isotherms

The sorption isotherm represents the equilibrium relationship between the water activity in a product and its moisture content. At equilibrium and constant temperature, the water activity is considered equivalent to the relative humidity of the surrounding air. Figure 1 illustrates the desorption isotherm by dynamic vapor sorption (DVS) technique at temperatures of 25 °C, 40 °C, 50 °C, and 80 °C, over a water activity range of 10% to 80%, The curves represent the average of two repetitions for each experiment, ensuring consistency and reliability in the results, reflecting the high reproducibility of the tests (p < 5%), despite the lengthy duration of each experiment, which exceeded three days.

The desorption isotherm exhibited a Type III shape, characterized by a slow increase in moisture content at low water activity followed by a steep rise at higher water activity. This behavior indicates weak binding forces between water and the solid matrix, which is typical of materials with low surface polarity or high sugar content [15,16,18,33,34,35].

As the temperature increased, the moisture content decreased at lower water activity values (<0.5), likely due to a reduction in hydrophilic sites caused by physiochemical changes induced by higher temperatures [16]. Figure 1 clearly shows that this behavior was most pronounced for Spirulina between 50 °C and 80 °C, possibly due to physical and/or chemical damage occurring during the drying process. Similar findings were reported by Oliveira, who investigated Spirulina desorption isotherms at 40 °C, 50 °C, and 60 °C using the gravimetric static method. Eloi Salmwendé also reported similar results in his study of Spirulina desorption isotherms at 25 °C and 50 °C using the same method for two different stumps [18]. Furthermore, Desmorieux and Decaen examined Spirulina desorption isotherms using three methods (gravimetric static, dew point temperature measurement, and DVS) in the 25–40 °C temperature range. They found that measurements obtained using DVS resulted in a lower equilibrium moisture content compared to the other methods and concluded that the desorption isotherm was not influenced by air temperature within this range [15]. However, this was not the case in our study, which may be attributed to the pretreatment operations applied to the Spirulina, which likely exerted a significant impact on the isotherm and possibly on its internal structure. According to industrial standards, the moisture content of Spirulina should not exceed 0.075 kg/kg [15]. Based on Figure 1, this corresponds to a maximum relative humidity of 10–15% in the storage atmosphere at nearly all temperatures.

3.2. Fitting of Spirulina Platensis Desorption Isotherms

Five sorption equations were used to fit the experimental desorption data of the microalgae Spirulina, and the derived parameters are presented in

Table 2. Values of the tested model parameters for the different temperatures.

Models	Temperature (°C)	Models Parameters	R2	X2
GAB	25	Xm = 0.1195	0.9997	0.0032
		C = 1.1509
		K = 0.5711
	40	Xm = 0.1125	0.9977	0.0059
		C = 1.0234
		K = 0.6256
	50	Xm = 0.1104	0.9973	0.0053
		C = 0.5313
		K = 0.6844
	60	Xm = 0.0520	0.9984	0.0029
		C = 0.4081
		K = 0.7789
	80	Xm =0.0284	0.9990	0.0014
		C = 0.3897
		K = 0.9079
OSWIN	25	A = 0.1143	0.9997	0.0032
		B = 0.9382
	40	A = 0.0848	0.9965	0.0067
		B = 0.7324
	50	A = 0.0774	0.9950	0.0063
		B = 0.6335
	60	A = 0.0607	0.9921	0.0060
		B = 0.6013
	80	A = 0.0521	0.9946	0.0036
		B = 0.4983
SMITH	25	A = −0.0298	0.9787	0.0299
		B = 0.2491
	40	A = −0.0051	0.9944	0.0085
		B = 0.1406
	50	A = 0.0028	0.9959	0.0057
		B = 0.1112
	60	A = 0.0029	0.9965	0.0040
		B = 0.0843
	80	A = 0.0083	0.9898	0.0050
		B = 0.0608
HENDERSON	25	A = 2.8839	0.9982	0.0086
		B = 0.6572
	40	A = 5.8353	0.9971	0.0061
		B = 0.8727
	50	A = 9.3970	0.9956	0.0059
		B = 1.0333
	60	A = 14.2460	0.9974	0.0034
		B = 1.0940
	80	A = 36.0530	0.9993	0.0012
		B = 1.3561
PELEG	25	A = 1.3354	0.9915	0.0224
		B = 2.7232
		C = 0.5994
		D = 2.7205
	40	A = 1.1723	0.9899	0. 0138
		B = 1.9747
		C = 0.8313
		D = 1.9891
	50	A = 1.1235	0.9890	0.0111
		B = 1.5453
		C = 0.8772
		D = 1.5416
	60	A = 1.0912	0.9958	0.0051
		B = 1.5079
		C = 0.9093
		D = 15.1630
	80	A = 1.0621	0.9975	0.0029
		B = 1.1165
		C = 0.9379
		D = 1.1155

.

It is evident that the GAB model provided the best mathematical fit for describing moisture desorption isotherms, achieving the highest R² (≥0.9973) and the lowest χ² (0.0014) across the tested temperatures and water activity range. This was followed by the Oswin model (R² = 0.9921 to 0.9997, χ² = 0.0032 to 0.0066), with the Smith, Henderson, and Peleg models ranking sequentially lower (see Table 2). The GAB model includes several parameters that effectively describe the experimental data for relative humidity values ranging from 10% to 90%, a range of particular interest for food applications [36]. Indeed, its parameters (X_m, C, and K) characterize the sorption process at the molecular level: X_m represents the monolayer moisture content, C indicates the strength of binding between water and the primary sorption sites, and K describes the multilayer sorption properties.

The monolayer moisture content (X_m) is particularly significant, as it represents the amount of water strongly adsorbed to specific sites and is considered the optimum level at which lipid oxidation, non-enzymatic browning, and enzyme activity rates are minimized [37,38]. As shown in Table 2, X_m values decreased with increasing temperature, which can be attributed to a reduction in hygroscopicity, and to structural and physicochemical changes caused by the elevated temperature [15]. Furthermore, the decrease in X_m with temperature may also reflect changes in the binding energy of the sorption sites, as water molecules become less tightly associated with the surface at higher thermal energies. This trend is often accompanied by a reduction in the Guggenheim constant (C), indicating weaker water–solid interactions, and a slight increase in the multilayer factor (K), suggesting that multilayer water behaves more like bulk water at elevated temperatures. Similar findings were reported by Oliveira, who found that the GAB model provided a better correlation coefficient (above 99%) and a lower percentage error (below 10%) compared to the two-parameter BET model [16].

3.3. Calculation of Net Isosteric Heat of Desorption

The previously identified GAB model (Table 2) was used to calculate the net heat of desorption. The desorption isosteres for Spirulina platensis are shown in Figure 2.

Desorption isosteres are generally described by Clausius–Clapeyron equation [39]. The slope of the isosteres curves allows for the determination of the corresponding net isosteric heat of desorption for each moisture content as illustrated in Figure 3.

Analysis of the curve showing the net isosteric heat of desorption as a function of equilibrium moisture content highlights its significance for low moisture levels in Spirulina. Specifically, the net isosteric heat (Q_st,n) for Spirulina platensis decreased from 21.3 to 4.29 KJ/mol as the equilibrium moisture content increased from 0.02 to 0.1 Kg/Kg (dry basis). This decrease is attributed to the strong binding of water molecules to the solid matrix, which necessitates considerable additional heat beyond the latent heat of vaporization to effectively dehydrate the product. A research study conducted by Moreira et al. (2017) reported that the Q_st,n values for Fucus vesiculosus seaweed decreased from 21.24 to 0.01 KJ/mol as X_eq increased from 0.07 to 0.34 Kg/Kg (dry basis), suggesting that the interaction energy between water and seaweed samples in the high moisture region was comparable to that between pure water molecules [40]. Yu et al. (2019) also observed a decrease in Q_st,n for the probiotic-fermented sea tangle powder, from 15.02 to 0.49 KJ/mol as X_eq increased from 0.02 to 1.29 Kg/Kg (dry basis) at temperatures ranging from 4 °C to 37 °C, which aligns with our findings [41].

The diffusion coefficient (in the hygroscopic state) was estimated, For each desorption temperature, from the desorption kinetics curves, it was estimated based on Equation (7) by plotting the curves of -Ln(aw) as a function of time. Figure 4 illustrates the variation in the diffusion coefficient as a function of moisture content, ranging from 0.003 Kg/Kg to 0.2 Kg/Kg (dry basis), for the different desorption temperatures. The moisture content of Spirulina platensis, at each relative humidity step, is determined by averaging the moisture content during the corresponding desorption kinetic.

The equilibrium moisture content (X_eq) values shown in Figure 4 were selected to cover the range effectively measured in all desorption experiments at different temperatures. These values were chosen to enable a consistent comparison of moisture diffusivity across the relevant moisture content range and reflect the practical range observed in the samples during the desorption process

This figure illustrates the monotonic relationship between the moisture diffusivity coefficient and the water content. Such behavior in liquid diffusion during the drying of foodstuffs has been reported by several authors [17,19,31]. From these results, it can be inferred that the main physical parameters influencing Spirulina diffusivity are the drying temperature and the moisture content. As shown in the figure, the diffusion coefficient values ranged from 2.75 × 10⁻⁸ to 2.88 × 10⁻⁷ m²/s for temperatures ranging from 40 °C to 80 °C during the desorption process. These values are higher than those reported in the literature, as our work focuses on the hygroscopic state of Arthrospira platensis.

The dynamic vapor sorption (DVS) technique involves exposing the sample to controlled variation in relative humidity conditions and observing the changes in weight due to moisture sorption. At lower equilibrium moisture content levels, the moisture in the sample is generally more mobile, allowing them to be readily absorbed or desorbed as the relative humidity changes. This dynamic behavior results in higher measured diffusivity at low moisture content, as the material responds more rapidly to changes in vapor pressure and moisture uptake or release [42]. Furthermore, at low equilibrium moisture contents, the moisture gradient between the material and the surrounding atmosphere is typically higher. This steeper gradient enhances the driving force for water into or out of the material. Since moisture diffusion is governed by Fick’s law, which depends on the concentration gradient, a higher moisture gradient leads to an apparent increase in diffusivity [42].

Previous studies reported lower diffusivity values (which were determined for overall drying kinetics). Indeed, Eloi Salmwandé reported diffusivities ranging from 1.7 × 10⁻¹⁰ to 6.73 × 10⁻¹⁰ under drying conditions of 50 °C, 6% RH, and cylindrical shape (d = 20 mm) [18]. Likewise, Oliveira et al. [16] found diffusivities between 0.233 × 10⁻¹⁰ and 0.286 × 10⁻¹⁰ for drying conditions of 50–60 °C, an air velocity of 1.5 m/s, 7–10% RH, and cylindrical pellets (d = 3 mm).

Dissa et al. [17] observed diffusivity values between 0.122 × 10⁻¹⁰ and 0.167 × 10⁻¹⁰ for drying at 50 °C, 10% RH, and cylindrical shapes (d = 3–6 mm, L = 4 cm). In the same study, this author also investigated Spirulina in a slab shape (L = 6 cm, width = 2 cm), obtaining diffusivities between 1.79 × 10⁻¹⁰ and 6.73 × 10⁻¹⁰ for drying at 40–60 °C, 10% relative humidity, and air velocities ranging from 0 to 1.2 m/s. A sharp increase in the moisture diffusivity coefficient is observed above a water content of 0.1 Kg/Kg (dry basis). This trend has also been reported for Spirulina. Additionally, it is evident that temperatures of 40–50 °C and 60 °C result in very similar diffusivity values.

As the moisture content increases, Spirulina transitions to a state where liquid water becomes more prevalent, thereby reducing diffusivity since liquid water moves differently through the material compared to water vapor. The higher moisture diffusivity observed at low equilibrium moisture content can be attributed to several factors, including a steeper moisture gradient, a more accessible pore structure, increased surface area for sorption, and the presence of less tightly bound water within the material [34,36]. These factors facilitate faster water vapor transport through Arthrospira Platensis, resulting in higher diffusivity values, as observed using the DVS technique.

Numerous empirical parametric equations describing the moisture diffusivity of biological materials as a function of moisture content have been reported in the literature, with a comprehensive compilation provided by Zogzas et al. 1996 [43]. The influence of temperature on diffusivity can be effectively described by an Arrhenius-type relationship [25]. However, the impact of moisture content has not yet been framed within a widely accepted general model. The diffusion coefficient can be expressed by the following Equation (10):

$$D_{eff}=A\times exp\left({\displaystyle \frac{-B}{T}}\right)$$

(10)

Here, A generally follows the Arrhenius law and can be expressed as

$$A=a\times exp\left({\displaystyle \frac{-Ea}{RT}}\right)$$

(11)

By plotting Ln(A) against (1/T), the activation energy can be determined. Similarly, the variation in parameter B with temperature can be analyzed. The values of A and B are obtained by fitting the experimental data describing D_eff = f(X_eq) using an exponential function of the form a × exp(b × x). Therefore, for moisture contents ranging from 0.003 to 0.2 Kg/Kg (dry basis) and temperatures from 40 °C to 80 °C, the liquid water diffusivity of Spirulina platensis can be expressed by the following empirical Equation (12):

$$D_{eff}\left(X.T\right)=13.709\times {10}^{-08}\exp \left({\displaystyle \frac{-\mathrm{33,704}}{RT}}\right)\times exp\left[\left(-0.1566T+59.284\right)X\right]$$

(12)

The activation energy represents the energy barrier that must be overcome for the diffusion process to occur effectively [41,42]. For liquid diffusion in S. platensis, the activation energy (Ea) was found to be 33.71 KJ/mol, which aligns with the findings of Zogzas et al. [43], who reported activation energies for agri-food products ranging from 12.7 to 110 KJ/mol [43]. Similar findings were reported by Pamella de Carvalho melo [19], who found an activation energy of 39.52 KJ/mol in [19]. It is important to note that, in drying processes, the lower the activation energy, the greater the water diffusion within the product [44,45]. In other words, a smaller amount of energy is required to initiate the physical transformation, specifically the conversion of liquid water into vapor [46].

Table 3. The effect of temperature and air relative humidity on the moisture diffusion coefficient.

Effect of Temperature at HR = 60%		Effect of Relative Humidity at T = 50 °C
T (°C)	Deff (m2/s)	HR(%), X0 (Kg/Kg Dry Basis) (%)	Deff (m2/s)
40	2.75 × 10−8	10, X0 = 2.98	1.60 × 10−8
50	4.05 × 10−8	30, X0 = 5.53	3.14 × 10−8
60	7.16 × 10−8	50, X0 = 8.71	4.01 × 10−8
80	15.7 × 10−8	70, X0 = 15.24	5.76 × 10−8

shows the effect of temperature and air relative humidity on the diffusion coefficient. The moisture content indicated corresponds to that of the product at the beginning of each humidity stage:

The effect of temperature and relative humidity on the moisture diffusion coefficient are often interdependent. Generally, increasing the temperature leads to an increase in moisture diffusivity due to the increased kinetic energy of the water molecules and a greater moisture gradient. Conversely, a lower relative humidity increases the moisture gradient between the material and the surrounding air, leading to higher diffusivity. Specifically, temperature increases the moisture diffusion coefficient by enhancing molecular movement, reducing viscosity, and increasing vapor pressure, whereas air relative humidity affects the diffusion process by modifying the moisture gradient and the extent of water binding within the material [34,38,44].

4. Conclusions

The purpose of this work was to determine the desorption isotherms of Spirulina platensis using the dynamic vapor sorption technique and to establish predictive correlations for its thermo-hygroscopic properties based on experimental data and physical principles.

Based on our results, the following conclusions can be drawn:

The desorption isotherms of Spirulina platensis, obtained over a temperature range of 25 °C to 80 °C exhibited type III behavior according to BET classification, commonly observed in many agri-food products. Among the empirical and semi-empirical sorption models tested, the GAB model proved to be the most suitable for predicting desorption isotherms of Spirulina platensis within the studied temperatures and water activities ranges. The selection of this model is based on the highest values of the coefficient of determination (R²) and the lowest values of the standard error (χ²). In addition, the net isosteric heat of desorption decreased from 21.3 to 4.29 KJ/mol as the equilibrium moisture content increased from 0.02 to 0.1 Kg/Kg (dry basis). The diffusion coefficient (in the hygroscopic state) varied between 2.75 10⁻⁸ m²/s and 15.710⁻⁸ m²/s over the temperature range of 40 °C to 80 °C, corresponding to an activation energy of 33.7 KJ/mol.

The thermo-hygroscopic investigation conducted in this study provides a fundamental basis for optimizing the drying process of Spirulina biomass and effectively reducing its energy costs. Based on this study, the next step of this work is to develop a physical drying model in order to predict the drying rate of a thin biomass layer undergoing convective drying, assisted by capillary drainage. This model can identify the optimal drying parameters by integrating the drying conditions and the capillary properties of the drying support.