A three-equation, one-dimensional thermal model developed by Esence [11] and Fasquelle [12] is used in this study to predict the thermal behavior of our sensitive storage system. It implements 3 balance equations performed at the level of the fluid (1), the solid storage material (2), and the tank walls (3). The pebbles are considered to behave like thin bodies (uniform temperature).

$\begin{array}{l}\epsilon \left(\frac{\partial \left({\rho }_f{C}_{P_f}{T}_f\right)}{\partial t}+\frac{\partial \left({\rho }_f{u}_f{C}_{P_f}{T}_f\right)}{\partial z}\right)\\ =\frac{\partial }{\partial z}\left({\lambda }_{eff,f}\frac{\partial T_f}{\partial z}\right)+h_{eff,f\text{-}s}{a}_s\left(T_s-T_f\right)+h_{eff,f\text{-}p}{a}_{P,int}\left(T_P-T_f\right)\end{array}$ (1)

$\left(1-\epsilon \right){\rho }_s{C}_{P_s}\frac{\partial T_s}{\partial t}=\frac{\partial }{\partial z}\left({\lambda }_{eff,s}\frac{\partial T_s}{\partial z}\right)+h_{eff,f\text{-}s}{a}_s\left(T_f-T_s\right)$ (2)

${\rho }_P{C}_{P_P}\frac{\partial }{\partial t}=\frac{\partial }{\partial z}\left({\lambda }_P\frac{\partial T_P}{\partial z}\right)+h_{eff,f\text{-}p}{a}_{p,int}\left(T_f-T_P\right)+U_P{a}_{p,ex}\left(T_{ex}-T_P\right)$ (3)

The rock bed is treated as a continuous medium. The effective thermal conductivities of the fluid and solid are given by Equations (4) and (5).

${\lambda }_{eff,f}=\epsilon {\lambda }_f$ (4)

${\lambda }_{eff,s}=\left(1-\epsilon \right){\lambda }_s$ (5)

The Reynolds number characteristic of the flows (6) is based on the equivalent diameter of the solids making up the bed.

$Re=\frac{{\rho }_f{u}_f\epsilon d_{eq}}{{\mu }_f}$ (6)

The Wakao and Faguei correlation recommended by Esence [11] is used to calculate the Nusselt number (7).

$N{u}_s=2+1.1R{e}^{0.6}P{r}^{1/3}=\frac{h_s{d}_{eq}}{{\lambda }_f}$ (7)

The effective rock-fluid exchange coefficient is calculated as a function of the rock-fluid exchange coefficient defined in equation (8) [12] .

$\frac{1}{h_{eff,s}}=\frac{1}{h_s}+\frac{d_{eq}}{10{\lambda }_s}$ (8)

The effective exchange coefficient between the fluid and the wall can be obtained through the correlations expressed in Equations (9) and (10) [12] .

$N{u}_p=\left(1-1.5{\left(\frac{d_{eq}}{D}\right)}^{1.5}\right)R{e}^{0.59}P{r}^{1/3}=\frac{h_p{d}_{eq}}{{\lambda }_f}$ (9)

$\frac{1}{h_{eff,p}}=\frac{1}{h_p}+\frac{e_p}{3{\lambda }_p}$ (10)

To establish the heat balance for each collector component, we applied the nodal method, which consists of dividing each solar collector component into a number of nodes to which temperatures are assigned. In this way, the energy balance was applied to the glass (11), the absorber (12), the heat transfer fluid (13), the insulation (14) and the outer face of the solar collector (15).

$\begin{array}{c}{m}_v{C}_{P_v}\frac{\partial T_v}{\partial t}={\alpha }_v{I}_G{S}_v+h_{cv\text{-}ext}{S}_v\left(T_{ex}-T_v\right)+h_{r,v\text{-}ciel}{S}_v\left(T_{ciel}-T_v\right)\\ +h_{r,v\text{-}abs}{S}_{abs}\left(T_{abs}-T_v\right)+h_{c,v\text{-}abs}{S}_v\left(T_{abs}-T_v\right)\end{array}$ (11)

$\begin{array}{c}{m}_{abs}{C}_{P_{abs}}\frac{\partial T_{abs}}{\partial t}={\alpha }_{abs}{\tau }_v{I}_G{S}_{abs}+h_{r,abs\text{-}v}{S}_{abs}\left(T_v-T_{abs}\right)+h_{cv,abs\text{-}v}{S}_{abs}\left(T_v-T_{abs}\right)\\ +h_{cv,abs\text{-}f}{S}_{abs}\left(T_f-T_{abs}\right)+h_{r,abs\text{-}pq}{S}_{abs}\left(T_{pq}-T_{abs}\right)\end{array}$ (12)

${\rho }_f{C}_{P_f}{V}_{}\left(\frac{\partial T_f}{\partial t}+u_{cap}\frac{\partial T_f}{\partial x}\right)=h_{cv,abs\text{-}f}{S}_{abs}\left(T_{abs}-T_f\right)$ (13)

$m_{pq}{C}_{P_{pq}}\frac{\partial T_{pq}}{\partial t}=h_{cd,pq\text{-}iso}{S}_{pq}\left(T_{iso}-T_{pq}\right)+h_{ray,abs\text{-}pq}{S}_{pq}\left(T_{abs}-T_{pq}\right)$ (14)

$m_{iso}{C}_{P_{iso}}\frac{\partial T_{iso}}{\partial t}=h_{cd,pq\text{-}iso}{S}_{iso}\left(T_{pq}-T_{iso}\right)+h_{cv,iso\text{-}ex}{S}_{iso}\left(T_{ex}-T_{iso}\right)$ (15)

The dryer is divided into a number of fictitious slices in the x direction of flow, defined by the volume delimited by two racks and the walls of the drying chamber ( Figure 3 ). The Equations ((16), (17), (18) and (19)) below describe the heat balances at any given slice of the drying chamber.

${\stackrel{\dot{}}{m}}_{sech}{C}_{P_f}\Delta x\frac{\partial T_f}{\partial x}=h_{cv,f\text{-}pr}{S}_{pr}\left(T_{pr}-T_f\right)+h_{cv,f\text{-}pint}{S}_p\left(T_{P,int}-T_f\right)$ (16)

$m_{pr}{C}_{P_{pr}}\frac{\partial T_{pr}}{\partial t}+m_{sech}{L}_V\frac{\partial X}{\partial t}=h_{cv,pr\text{-}f}{S}_{pr}\left(T_f-T_{pr}\right)$ (17)

$m_p{C}_{P_P}\frac{\partial T_{P,int}}{\partial t}=h_{cd,pint\text{-}pext}{S}_P\left(T_{Pext}-T_{Pint}\right)+h_{cv,f\text{-}pint}{S}_P\left(T_f-T_{pint}\right)$ (18)

$\begin{array}{c}{m}_p{C}_{P_P}\frac{\partial T_{Pext}}{\partial t}=h_{cd,pint\text{-}pext}{S}_P\left(T_{Pint}-T_{Pext}\right)+h_{cv,ex\text{-}pext}{S}_P\left(T_{ex}-T_{pext}\right)\\ +h_{r,ciel\text{-}pext}{S}_P\left(T_{ciel}-T_{pext}\right)\end{array}$ (19)

The expression $m_{sech}{L}_V\frac{\partial X}{\partial t}$ defined in Equation (17) above refers to the mass of water evaporated per unit time with $m_{sech}$ the dry mass of the product and $\frac{\partial X}{\partial t}$ product drying speed expressed in kilograms of water per kilogram of dry

matter in one second (kg_water/(kg_ms/s)).

The instantaneous water content ( $X_t$) of the tomato samples was transformed into reduced water content ( $X_r$), according to equation (21) defined below [13] - [15] :

$X_{red}=\frac{X_t-X_{eq}}{X_0-X_{eq}}$ (21)

where $X_0$ is the initial water content of the product and $X_{eq}$ is the equilibrium water content of the product. The reduced water content $X_{red}$, has been simplified using equation (22), since $X_{eq}$ is negligible compared with $X_t$ and $X_0$ [6] .

$X_{red}=\frac{X_t}{X_0}$ (22)

To simulate tomato drying kinetics, we used empirical models described in the literature. To determine the models, we used the non-linear regression method to establish a correlation giving the evolution of the reduced water content ( $X_{red}$) of tomato samples as a function of time. Fitting was carried out using MATLAB 2022a software. Table 5 presents some empirical models described in the literature.

Page’s reduced water content model was fitted to the experimental data, and model parameters were determined using non-linear regression analysis. The criteria for assessing the smoothing quality of the experimental results are the coefficient of determination ( $R^2$), the reduced ki-square ( ${\chi }^2$) and the root mean square error (RMSE) [4] [6] . These parameters are calculated according to the relationships:

$R^2=\frac{{{\displaystyle \sum }}_{i=1}^N{\left(X_{red,expi}-X_{red,thi}\right)}^2}{\sqrt{\left[{{\displaystyle \sum }}_{i=1}^N{\left(X_{red,expi}-X_{red,thi}\right)}^2\right]\ast \left[{{\displaystyle \sum }}_{i=1}^N{\left(X_{red,expi}-X_{red,thi}\right)}^2\right]}}$ (29)

${\chi }^2=\frac{{{\displaystyle \sum }}_{i=1}^N{\left(X_{expi}-X_{thi}\right)}^2}{N-n}$ (30)

$\text{RMSE}={\left[\frac{1}{N}\times \underset{i=1}{\overset{N}{{\displaystyle \sum }}}{\left(X_{expi}-X_{thi}\right)}^2\right]}^{1/2}$ (31)

Figure 5 and Figure 6 show the evolution of experimental tomato water content during indirect solar drying with or without a thermal storage unit on 16-17/03/2023; 19-20/03/2023; 25-26/03/2023 and 31/03/2023-01/04/2023. The evolution of experimental water content in the two drying racks follows a decreasing trend as a function of drying time. The difference between the two trays is relatively small, which means that drying is almost uniform across both trays [2] . During the night, or when the storage tank is put into operation, the water content of the product decreases again until the discharge process stops.

Figure 7 and Figure 8 show changes in the average water content of tomatoes during solar drying with or without a thermal storage unit on 16-17/03/2023; 19-20/03/2023; 25-26/03/2023 and 31/03/2023 -01/04/2023. The initial dry-base water contents of the product on these different drying days are 17.6 kg_water/kg_ms (16-17/03/2023); 18.14 kg_water/kg_ms (19-20/03/2023); 19.8 kg_water/kg_ms (25-26/03/2023) and 16.8 kg_water/kg_ms (31/03/2023 -01/04/2023) respectively. And the final moisture content of tomatoes on a dry basis at the end of these different drying days are respectively 0.10 kg_water/kg_ms (16-17/03/2023); 0.13 kg_water/kg_ms (19-20/03/2023); 0.115 kg_water/kg_ms (25-26/03/2023) and 0.15 kg_water/kg_ms (31/03/2023 -01/04/2023).

The calculated values of the statistical parameters used are shown in Table 6 above. The different models are compared on the basis of the values of the coefficient of determination ( $R^2$), the reduced ki-squared ( ${\chi }^2$) and the root mean square error (RMSE). The high value of $R^2$, and the low values of ${\chi }^2$ and RMSE reflect a good fit of the model to the experimental values. The values of these parameters range from 0.9287 to 0.9942 for the $R^2$, from 0.00704 to 0.08641 for the ${\chi }^2$and from 0.02797 to 0.09296 for the RMSE. Among these models, the “Page” model gives the highest value of determination $R^2$ and the lowest ( $R^2=0.9942$) values of ${\chi }^2$ and RMSE (respectively and RMSE = 0.00704). It was therefore chosen to represent the behavior of tomato thin-film solar drying kinetics.

Figure 9 and Figure 10 show a comparison of experimental and simulated results for the average dry-base water content of tomatoes from indirect solar drying ( Figure 9(a) and Figure 9(b) ) and indirect solar drying with thermal energy storage ( Figure 10(a) and Figure 10(b) ). The evolution of simulated and experimental tomato water contents shows good agreement. Moisture content falls rapidly at the start of the day, stabilizing at the end of the day as drying air temperature and product moisture content decrease. Moisture content falls again when the storage tank is switched on during the night, or on the second day of drying. However, a discrepancy is observed for drying from 03/31/2023 to 04/01/2023. This discrepancy may be due to the fact that the model does not take into account the influence of thermal and hygrometric parameters on the continuation of night drying during indirect solar drying. We can therefore conclude that Page’s model provides a satisfactory description of experimental tomato kinetics.