Plant Material

The onion varieties on which isotherms were determined were Violet of Galmi, Safari, Gandiol F1, and Orient F1. They came from the RAO cooperative in the Saint-Louis region of Senegal. The onions were collected directly from the field and transported to the laboratory to avoid any source of mixing with other varieties. The initial moisture content (% wet basis) and water activity of the four varieties were as follows:

Drying and Analysis Equipment

Statistical Analysis and Modeling Tools

The various IT tools used for statistical analysis and modeling are:

Study of Desorption Isotherms

The desorption isotherms of onion varieties (Violet of Galmi, Safari, Gandiol F1, and Orient F1, grown in Senegal), taken separately, and of the varieties’ mixture, are determined gravimetrically in an oven with temperature and relative humidity control. Three randomly selected onion bulbs are peeled, washed with 100 ppm chlorinated water, rinsed three times with clear water, wrung out, and chopped with a mincer (thickness of around 1.7 mm). From the prepared bulbs, the test sample consists of ten grams (10 g of the same variety for separate monitoring and 2.5 g of each variety for the mixture). The onion samples taken are placed in dishes above seven saturated salt solutions with known water activities at different temperatures. For temperatures between 50˚C and 70˚C, the salt solutions used have water activity values between 0.05 and 0.96, enabling equilibrium relative moisture to be controlled ( Table 1 ).

This less expensive static gravimetric method is the most widely used [9] [10] [15] - [17] . Experiments are carried out in triplicate without ventilation over a temperature range from 50˚C to 70˚C, with a step of +5˚C. The dishes are weighed three times a day at regular intervals until equilibrium is reached. The constant weight at this point corresponds to the wet mass (X_h) of the samples. After measuring water activity on the dried samples, their dry mass (X_s) is determined by drying at 105˚C for 2 hours. Moisture content and water activity are measured using the reference methods NF ISO 712:2009 and NF EN ISO 17025, respectively. The equilibrium moisture content (X_eq in kg∙H₂O∙kg⁻¹ db) is calculated according to the formula:

$X_{eq}=\frac{X_h-X_s}{X_s}$ (1)

Desorption isotherms (for onion varieties: Violet of Galmi, Safari, Gandiol F1, and Orient F1, taken separately, and their mixture) at temperatures of 50, 55, 60, and 65˚C are obtained by plotting equilibrium content curves (X_eq in kg∙H₂O per kg d.b.) as a function of water activity.

The evaluation of the reproducibility and the repeatability of the measurements is made by the numerical method, which is the LIN coefficient. Lin’s concordance coefficient varies between −1 and 1, where the values −1, 0, and +1, respectively, mean perfect discordance, zero concordance, and perfect match.

The Student’s t-test is used for comparisons between varieties and within the same variety at different temperatures.

Isotherm Modeling of the Onion Varieties (Violet of Galmi, Safari, Gandiol F1, and Orient F1) and Their Mixture at Different Temperatures

Eleven empirical mathematical models are tested to model desorption of isotherms at different temperatures for the four onion varieties, taken separately, and their mixture. The models used are Langmuir [18] , Brunauer-Emmett-Teller (BET) [13] , Henderson [12] [19] , Halsey [12] , Iglesias-Chirife [20] , Oswin [12] [19] , Caurie [12] , Smith [21] , Peleg [22] , and Chung-Pfost [19] . Equations are presented in Table 2 .

The least squares method is used to determine the best model. The statistical parameters are R², which must be close to 1, and χ², which must be close to 0, for the best model with:

$X^2=\frac{{\displaystyle {\sum }_{i=1}^n{\left(X_{eqpred,i}-{\bar{X}}_{eqpred}\right)}^2}}{{\displaystyle {\sum }_{i=1}^n{\left(X_{eqexp,i}-{\bar{X}}_{eqexp}\right)}^2}}$ (2)

${\chi }^2=\frac{{\displaystyle {\sum }_{i=1}^n{\left(X_{eqexp,i}-X_{eqpred,i}\right)}^2}}{n-N}$ (3)

where N is the number of parameters for each model, and n is the number of experimental points.

The evolution of equilibrium moisture content (calculated from Equation (1)) as a function of water activity reflects the fact that the desorption isotherms of the four onion varieties are virtually identical. The equilibrium moisture content and water activity values shown are mean values of the three measurements carried out, with standard deviations ranging from 0.0028 to 0.0047 and from 0.002 to 0.005, respectively.

Both within and between varieties, tests comparing the average equilibrium moisture content at different temperatures show no significant differences (p-values > 5%).

Desorption isotherms for the onion varieties: Violet of Galmi, Safari, Gandiol F1 and Orient F1, established at 50˚C, 55˚C, 60˚C and 65˚C ( Figure 1 ), show that, at constant temperature, the lower the equilibrium water content, the lower the water activity. This is due to the fact that the more desorption proceeds at constant temperature, the more water is bound to onion constituents. Increasing temperature leads to a decrease in equilibrium moisture content for fixed water activity values. This effect of temperature is explained by the increased agitation of water molecules, facilitating desorption. This has also been demonstrated by previous work on numerous food products, such as lemon citron [23] , spearmint [16] , pumpkin [24] , garlic [25] and different varieties of onion [10] [11] [13] [26] .

Furthermore, at identical temperatures, the decrease in equilibrium moisture content is more pronounced for Orient F1, followed by Gandiol F1, Safari, and Violet of Galmi. The impact of variety on isotherms is linked to differences in dry matter content (14.44%, 11.89%, 13.01%, and 10.87%, respectively, for Violet of Galmi, Safari, Gandiol F1, and Orient F1). Indeed, the higher the dry matter content, the more hydrogen bonds there are, slowing down desorption. However, an exception was noted between Gandiol F1 and Safari, due to the presence of soluble compounds such as sugars, which are probably more important in Safari, slowing down desorption compared with Gandiol F1 [14] . As shown by [27] , the influence of the variety on desorption isotherms is more marked for a_w above 0.6.

Of the eleven mathematical models tested, only four ( Table 3 ) are presented for the selection of the best model, as the other seven (BET, GAB, Halsey, Iglesias-Chirfie, Caurie, Smith, and Chung-Pfost) have mean R² values below 0.6 for all temperatures combined.

The mean R² and mean χ² values ( Table 3 ) obtained with the different models show that, whatever the temperature, the best models are Henderson for the Safari and Gandiol F1 varieties, Oswin for the Violet of Galmi, and Peleg for the Orient F1 variety. The comparison between the R² of the Henderson and Oswin models tested on the Violet of Galmi variety shows a non-significant difference, with a p-value (0.67) > 5%, while that between the R² of the Henderson and Peleg models on the Orient F1 variety reveals a significant difference, with a p-value (4.94 × 10⁻⁵) < 5%. Despite this significant difference, the Henderson model (R² of 0.9799, close to 1) can also be retained as the best model for the Orient F1 variety. Thus, only the Henderson, Oswin, and Peleg models achieve less dispersion and a better fit, with R² values between 0.97 and 0.99 and χ² values between 0.0016 and 0.029, respectively ( Table 3 ). The Henderson model describes the isotherms of all four varieties well, while the other two do not fit all of them (Oswin model for the Violet of Galmi and Gandiol F1 varieties, and the Peleg model for the Safari and Orient F1 varieties).

This model differs from those found in the literature on modeling onion isotherms [9] - [13] [26] . This difference corroborates the importance of cultivation practices, variety, shape, and size on isotherms [14] [15] [27] . However, the sigmoidal shape of the four onion varieties’ isotherms ( Figure 2 ) studied in our experiments is identical to that of the isotherms of several food products. The different zones observed are [17] :

The equation of the Henderson model, which provides less dispersion and a better fit with experimental data, is:

$X_{eq}={\left[-\ln \left(1-a_w\right)/B\right]}^{1/A}$ (4)

The modeling of desorption isotherms at temperatures of 50˚C, 55˚C, 60˚C, and 65˚C with the best Henderson model is shown in Figure 2 .

Whatever the desorption temperature, the curves in Figure 2 show an almost perfect match between experimental equilibrium moisture contents and those predicted by the Henderson model.

For all combined temperatures and varieties, the average equilibrium time is between 9 and 13 days.

The desorption isotherms predicted with the Henderson model reflect that, for fixed equilibrium moisture contents, each 5˚C increase in temperature leads to an increase in water activity. However, an exception is observed with equilibrium moisture contents < 1 kg water/kg db (for the varieties: Violet of Galmi and Gandiol F1) and < 0.7 kg water/kg db (for the varieties Safari and Orient F1). For these low equilibrium moisture contents, water activity is identical regardless of the desorption temperature. The Henderson model parameters ( Table 4 ) show that B parameter values increase with temperature, whereas A parameter values decrease for all four varieties.

The parameters of the Henderson model evolve linearly with temperature. With each 5˚C step increase in temperature, the B parameter of the Violet of Galmi variety, which is the highest, rises faster (on average +0.097), followed by that of the Gandiol F1 variety (on average +0.067), while those of the lower Safari and Orient F1 varieties are virtually identical (on average +0.052). As for parameter A, those of the Violet of Galmi, Safari, and Orient F1 varieties, which are almost identical, are higher than those of the Gandiol F1 variety, with decreases of −0.177, −0.148, −0.120, and −0.122, respectively. However, from one variety to another, there was no significant difference between the evolution of the B and A parameters as a function of temperature (p-values > 5%).

According to the work on drying kinetics modeling [29] , the variety Violet of Galmi, with the slowest reduced drying speed across all temperatures, is the limiting variety for drying the four varieties’ mixture. To this end, the two best models (Henderson, followed by the Oswin model) of the Violet of Galmi variety were tested on desorption isotherm data from the four varieties’ mixture. The statistical parameters are shown in Table 5 .

Table 5 shows that the Henderson model, with mean R² and mean χ² values of 0.9930 and 0.0363, respectively, is the best model for the mixture of the four varieties.

Experimental desorption isotherms for the four-variety mixture at temperatures of 50˚C, 55˚C, 60˚C and 65˚C and those predicted by modeling with the best Henderson model are shown in Figure 3 . These desorption isotherms for the mixture of the four varieties, compared with those for the varieties taken separately ( Figure 2 ), reflect a fairly similar trend. Furthermore, tests comparing mean equilibrium moisture contents at different temperatures between the mixture and the separate varieties reveal no significant difference (p-value > 5%).

As observed for individual varieties ( Figure 1 ), as temperature increases, equilibrium moisture content and water activity decrease for the four-variety mixture ( Figure 3 ).

It can be seen from the predicted isotherms that, for fixed equilibrium moisture contents, each 5˚C step increase in temperature leads to an increase in water activity in the mixture of the four onion varieties, except for equilibrium moisture contents < 0.4 kg water/kg db. The Henderson model parameters for the mixture of the four onion varieties are shown in Table 6 .

The evolution of the A and B parameters of the mixture of four onion varieties as a function of temperature is similar to that of the individual varieties. In fact, the B parameter increases with temperature, while the opposite is observed for A. With each 5˚C step increase in temperature, the B parameter increases on average by +0.057, and the A parameter decreases by −0.121.

According to Figure 3 and Table 6 , it appears that the desorption isotherms of the four onion varieties mixture between 50˚C and 60˚C are closer to those of the Safari variety (p < 5%). This can be attributed to the compositional dominance of the Safari variety within this temperature range. The Safari variety exhibits a water retention capacity and desorption dynamics that are shared, or largely dominated, by the mixture at these temperatures. Specifically, Safari has unique cell wall structures and hygroscopic properties that facilitate relatively higher water retention at moderate temperatures. This explains why, as the temperature increases within this range, the mixture tends to follow a similar moisture content and water activity pattern as Safari. Additionally, a water balance between the Violet of Galmi and Orient F1 varieties could also explain why the moisture content and dry matter values of the four onion varieties mixture tend towards those of the Safari variety between 50˚C and 60˚C.

At 65˚C, the elimination of water leads to an increase in dry matter for the four onion varieties mixture, which becomes similar to that of Gandiol F1. In fact, a significant difference (p > 5%) was observed between the predicted moisture contents of the four onion varieties mixture and those of Orient F1 and Violet of Galmi at different temperatures. This situation can be attributed to a more pronounced water elimination at higher temperatures. The Gandiol F1 variety, known for its resistance to dehydration and its ability to maintain dry matter at elevated temperatures, likely influences the behavior of the mixture at 65˚C. At this higher temperature, the water retention properties of the other varieties become less significant, and the mixture’s behavior aligns more closely with that of Gandiol F1, which shows greater stability in dry matter and moisture content at 65˚C. The observed increase in dry matter and decrease in moisture content at this temperature are therefore a direct result of this transition.