## Results  
### Kinetic Evaluation of Caroenoid Extraction  
Applying the Fick's law to the diffusion between two heterogeneous phases (solid matrix and Sc-CO\({}_{2}\)), the carotenoid accumulation rate (d[Ce]\({}_{t=t}\)/dt) in the mobile phase (Sc-CO\({}_{2}\)) was assumed to be described by the following equation:  
\[\frac{\mathrm{d[Ce]}_{t=t}}{\mathrm{dt}}=\mathrm{ke}\times\mathrm{A}\times([ \mathrm{Ce}^{*}]_{t=t}-[\mathrm{Ce}]_{t=t}) \tag{1}\]  
where ke = mass transfer constant (s\({}^{-1}\cdot\)m\({}^{-2}\)); A = area of contact between the two phases: solid matrix and Sc-CO\({}_{2}\) (m\({}^{2}\)); [Ce]\({}_{t=t}\) = concentration of carotenoids already extracted at time t=t (mg/g); [Ce*]\({}_{t=t}\) = H\(\cdot\)[Cs]\({}_{t=t}\) = concentration of carotenoids already extracted if, at that time t=t, the equilibrium between the two phases involved has been reached (mg/g); H = the equilibrium constant related to the partition of carotenoids between the two phases (H = [Ce*]\({}_{t=t}\)/[Cs]\({}_{t=t}\)); [Cs]\({}_{t=t}\) = concentration of unextracted carotenoids at time t=t (mg/g). The differential Equation (1) can be integrated by adopting some simplifying hypothesis [37] and utilizing the equation related to the mass balance of carotenoids between the two phases:  
\[[\mathrm{Cs}]_{t=t}=[\mathrm{Cs}]_{t=0}-[\mathrm{Ce}]_{t=t} \tag{2}\]  
where [Cs]\({}_{t=0}\) = carotenoid concentration initially present in one gram of starting material (mg/g). Then Equations (3) and (4) could be obtained:  
\[[\mathrm{Ce}]_{t=t}=\mathrm{H}\times[\mathrm{Cs}]_{t=0}/(\mathrm{H}+1)\times \left(1-\mathrm{e}^{-(\mathrm{H}+1)\times\mathrm{ke}\times\mathrm{A}\times \mathrm{t}}\right) \tag{3}\]  
\[=\mathrm{H}^{*}\times[\mathrm{Cs}]_{t=0}\times\left(1-\mathrm{e}^{-kt}\right) \tag{4}\]  
where H* = an adimensional constant ranging from zero to one, related to the equilibrium constant H (H* = H/(H + 1)); k = (H + 1) \(\times\) ke \(\times\) A = kinetic constant (s\({}^{-1}\)). The extraction rate (R) calculated as first derivative of the exponential equation:  
\[\mathrm{R}=\frac{\mathrm{d[Ce]}_{t=t}}{\mathrm{dt}}=\mathrm{H}^{*}\times[ \mathrm{Cs}]_{t=0}\times\mathrm{k}\times\mathrm{e}^{-kt} \tag{5}\]reaches its maximum value (Rmax) at the beginning of the extraction, when t is close to zero:  
\[\text{R}_{\text{max}}=\text{k}\times\text{H}^{*}\times\left[\text{Cs}\right]_{ \text{t}=0} \tag{6}\]  
As in [PERSON] et al. [40], the value of Rmax (s\({}^{-1}\)) was assumed as an index to evaluate the efficiency of the SFE system versus the carotenoid fraction of the matrix. In particular, while the constant k provides information on the kinetics of the SFE, the product H\({}^{*}\times\left[\text{Cs}\right]\), which represents the asymptotic value of the extraction curve when t \(\rightarrow\infty\), measures the maximum amount of extractable carotenoids under the working conditions adopted. As reported in a previous paper [41], the identification of the best values to be assigned to the equation parameters H\({}^{*}\times\left[\text{Cs}\right]\) and k was carried out by a commercial statistical program (BURENL\({}^{\text{\textregistered}}\), CMIC, Politecnico Milano, Italy, 1996). Table 1 reports the values assumed by Rmax and by the functional parameters H\({}^{*}\times\left[\text{Cs}\right]\) and k for the two matrices (tomato peels and chili pepper). Figure 1 reports the time evolution of carotenoid extraction from pepper (Figure 1a) and tomato peels (Figure 1b) as a function of the working conditions, together with the data calculated on the basis of the functional parameters reported in Table 1.