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\caption{2° tratto $[117.5\,\mathrm{s}\text{--fine}]$: dati raw (blu) e curva di fit (rosso tratteggiato).} \end{figure} +\clearpage + +% ───────────────────────────────────────────── +\subsection{Raffreddamento doppio esponenziale} + +\subsubsection*{Motivazione} + +Osservando i fit singoli sovrapposti ai dati raw, si nota che nessuno dei due esponenziali riesce a descrivere l'intera curva: il fit del 1° tratto ($\tau_1 \approx 13\,\mathrm{s}$) decade troppo rapidamente nella fase finale, mentre il fit del 2° tratto ($\tau_2 \approx 17\,\mathrm{s}$) non coglie la dinamica iniziale più ripida. Questo suggerisce la presenza di due contributi termici sovrapposti con costanti di tempo diverse. + +\begin{figure}[h!] + \centering + \includegraphics[width=\textwidth]{plot_confronto_fit.png} + \caption{Confronto tra il fit del 1° tratto (rosso) e del 2° tratto (verde) sovrapposti ai dati raw: nessuno dei due descrive correttamente l'intera curva.} +\end{figure} + +Il raffreddamento viene quindi modellato come \textbf{combinazione lineare di due esponenziali}: il primo termine cattura una componente rapida (raffreddamento superficiale immediato), il secondo una componente lenta (dissipazione termica del nucleo della scatola): + +\begin{equation} + T(t) = T_{\infty} + A_1 \cdot e^{-\frac{t - t_1}{\tau_1}} + A_2 \cdot e^{-\frac{t - t_2}{\tau_2}} +\end{equation} + +con $T_{\infty} = 22.99\,°C$, $t_1 = 115.0\,\mathrm{s}$ e $t_2 = 117.5\,\mathrm{s}$ fissi. La zona $[115.9,\ 117.2\,\mathrm{s}]$ è esclusa con pesi nulli ($\sigma = 10^{10}$). + +\subsubsection*{Parametri stimati} + +\begin{table}[h!] +\centering +\caption{Parametri stimati — doppio esponenziale} +\begin{tabular}{lll} +\toprule +Parametro & Descrizione & Valore \\ +\midrule +$A_1$ & Ampiezza componente rapida & $20.73 \pm 0.93\,°C$ \\ +$\tau_1$ & Costante di tempo rapida & $1.80 \pm 0.17\,\mathrm{s}$ \\ +$A_2$ & Ampiezza componente lenta & $152.44 \pm 0.65\,°C$ \\ +$\tau_2$ & Costante di tempo lenta & $17.60 \pm 0.13\,\mathrm{s}$ \\ +\bottomrule +\end{tabular} +\end{table} + +\textbf{Curva stimata:} +\begin{equation} + T(t) = 22.99 + 20.73 \cdot e^{-\frac{t - 115.0}{1.80}} + 152.44 \cdot e^{-\frac{t - 117.5}{17.60}} \quad [°C] +\end{equation} + +\textbf{Bontà del fit:} $R^2 = 0.9991$ (punti con peso pieno). Rispetto al singolo esponenziale ($R^2 = 0.9938$), il doppio esponenziale migliora significativamente il fit catturando la dinamica iniziale rapida ($\tau_1 \approx 1.8\,\mathrm{s}$). + +\begin{figure}[h!] + \centering + \includegraphics[width=\textwidth]{fit_doppio_esponenziale.png} + \caption{Fit doppio esponenziale: dati raw (blu), componente rapida (rosso punteggiato), componente lenta (verde punteggiato), somma totale (viola tratteggiato), zona esclusa (arancione).} +\end{figure} + \end{document}