Blocchi TEX

m_{\text{lievito}}=\left(\frac{m_{\text{campione,secco}}}{V_{\text{campione}}}\right)\,V_{\text{finale}}

Y_{\text{fermentatore}}=\frac{m_{\text{lievito}}}{m_{\text{melassa}}}

\eta=\begin{cases}
1, & \text{se } T \ge 30^\circ\text{C}\\
0, & \text{se } T < 30^\circ\text{C}
\end{cases}

c=\frac{m}{V}\ \text{[g/L]}

\begin{aligned}
m_{\text{lievito}} &= \left(\frac{m_{\text{campione,secco}}}{V_{\text{campione}}}\right) V_{\text{finale}}\\[4pt]
Y_{\text{fermentatore}} &= \frac{m_{\text{lievito}}}{m_{\text{melassa}}}\\[6pt]
\eta(T) &= \begin{cases}
1, & \text{se } 30^\circ\text{C} \le T \le 35^\circ\text{C}\\
0.8, & \text{se } 25^\circ\text{C} \le T < 30^\circ\text{C}\\
0, & \text{se } T < 25^\circ\text{C}
\end{cases}\\[6pt]
c_{\text{lievito}} &= \frac{m_{\text{campione,secco}}}{V_{\text{campione}}}\ \text{[g/L]}
\end{aligned}

\begin{aligned}
\mathbf{x} &= \begin{bmatrix} m_{\text{campione,secco}} \\ V_{\text{campione}} \\ V_{\text{finale}} \\ m_{\text{melassa}} \end{bmatrix},
\qquad
\mathbf{f}(\mathbf{x}) =
\begin{bmatrix}
\left(\frac{x_1}{x_2}\right)x_3\\[6pt]
\frac{\left(\frac{x_1}{x_2}\right)x_3}{x_4}
\end{bmatrix}\\[8pt]
m_{\text{lievito}} &= f_1(\mathbf{x}), \qquad
Y_{\text{fermentatore}} = f_2(\mathbf{x})
\end{aligned}

\begin{aligned}
m_{\text{lievito}} &= \left(\frac{m_{\text{campione,secco}}}{V_{\text{campione}}}\right)V_{\text{finale}}\\[4pt]
\ln(m_{\text{lievito}}) &= \ln(m_{\text{campione,secco}}) - \ln(V_{\text{campione}}) + \ln(V_{\text{finale}})\\[6pt]
\left(\frac{\sigma_{m_{\text{lievito}}}}{m_{\text{lievito}}}\right)^2
&=
\left(\frac{\sigma_{m_{\text{campione,secco}}}}{m_{\text{campione,secco}}}\right)^2
+
\left(\frac{\sigma_{V_{\text{campione}}}}{V_{\text{campione}}}\right)^2
+
\left(\frac{\sigma_{V_{\text{finale}}}}{V_{\text{finale}}}\right)^2\\[6pt]
Y_{\text{fermentatore}} &= \frac{m_{\text{lievito}}}{m_{\text{melassa}}}
\end{aligned}

\begin{aligned}
&\textbf{Dati:}\quad
m_{\text{campione,secco}},\ V_{\text{campione}},\ V_{\text{finale}},\ m_{\text{melassa}},\ T,\ t,\ t_f\\[4pt]
&\textbf{Concentrazione:}\quad
c_{\text{lievito}}=\frac{m_{\text{campione,secco}}}{V_{\text{campione}}}\ \text{[g/L]}\\[6pt]
&\textbf{Lievito totale:}\quad
m_{\text{lievito}}=c_{\text{lievito}}\,V_{\text{finale}}\\[6pt]
&\textbf{Resa del fermentatore:}\quad
Y_{\text{fermentatore}}=\frac{m_{\text{lievito}}}{m_{\text{melassa}}}\\[10pt]
&\textbf{Fattore di temperatura:}\quad
\eta(T)=
\begin{cases}
0, & T<25^\circ\text{C}\\
0.8+0.04\,(T-25^\circ\text{C}), & 25^\circ\text{C}\le T<30^\circ\text{C}\\
1, & 30^\circ\text{C}\le T\le 35^\circ\text{C}\\
\exp\!\left(-0.2\,(T-35^\circ\text{C})\right), & T>35^\circ\text{C}
\end{cases}\\[12pt]
&\textbf{Profilo di crescita semplice:}\quad
m_{\text{lievito}}(t)=
\frac{m_{\max}}{1+\exp\!\left(-k\,\eta(T)\,(t-t_0)\right)}\\[6pt]
&\textbf{Produzione netta (da 0 a }t_f\textbf{):}\quad
\Delta m_{\text{lievito}}=m_{\text{lievito}}(t_f)-m_{\text{lievito}}(0)\\[10pt]
&\textbf{Verifica del bilancio di massa:}\quad
\epsilon=\frac{\Delta m_{\text{lievito}}}{m_{\text{melassa}}}-Y_{\text{target}}\\[6pt]
&\textbf{Decisione:}\quad
\text{supera se }|\epsilon|\le 0.02,\ \text{non supera altrimenti}\\[12pt]
&\textbf{Propagazione dell'incertezza (relativa):}\quad
\left(\frac{\sigma_{m_{\text{lievito}}}}{m_{\text{lievito}}}\right)^2=
\left(\frac{\sigma_{m_{\text{campione,secco}}}}{m_{\text{campione,secco}}}\right)^2+
\left(\frac{\sigma_{V_{\text{campione}}}}{V_{\text{campione}}}\right)^2+
\left(\frac{\sigma_{V_{\text{finale}}}}{V_{\text{finale}}}\right)^2\\[8pt]
&\textbf{E per la resa:}\quad
\left(\frac{\sigma_{Y_{\text{fermentatore}}}}{Y_{\text{fermentatore}}}\right)^2=
\left(\frac{\sigma_{m_{\text{lievito}}}}{m_{\text{lievito}}}\right)^2+
\left(\frac{\sigma_{m_{\text{melassa}}}}{m_{\text{melassa}}}\right)^2
\end{aligned}

\begin{aligned}
\textbf{Input:}\quad
& m_{\text{sample,dry}}=1.25\ \text{g},\ 
V_{\text{sample}}=25\ \text{mL},\ 
V_{\text{final}}=18{,}000\ \text{L},\ 
m_{\text{molasses}}=9{,}500\ \text{kg}\\[6pt]
\textbf{Conversioni di unità:}\quad
& V_{\text{sample}}=25\times 10^{-3}\ \text{L},\qquad
m_{\text{molasses}}=9.5\times 10^{6}\ \text{g}\\[8pt]
\textbf{Passo 1 (concentrazione):}\quad
& c_{\text{yeast}}=\frac{1.25}{25\times 10^{-3}}=50\ \text{g/L}\\[6pt]
\textbf{Passo 2 (totale):}\quad
& m_{\text{yeast}}=50\times 18{,}000=9.0\times 10^{5}\ \text{g}\\[6pt]
\textbf{Passo 3 (resa):}\quad
& Y_{\text{fermenter}}=\frac{9.0\times 10^{5}}{9.5\times 10^{6}}=0.0947\\[10pt]
\textbf{Verifica dei vincoli:}\quad
& 0<Y_{\text{fermenter}}<0.2,\qquad
V_{\text{final}}>0,\qquad
V_{\text{sample}}>0\\[10pt]
\textbf{Forma vettoriale:}\quad
& \mathbf{x}=
\begin{bmatrix}
m_{\text{sample,dry}}\\
V_{\text{sample}}\\
V_{\text{final}}\\
m_{\text{molasses}}
\end{bmatrix},
\qquad
\mathbf{g}(\mathbf{x})=
\begin{bmatrix}
\left(\frac{x_1}{x_2}\right)x_3\\[6pt]
\frac{\left(\frac{x_1}{x_2}\right)x_3}{x_4}
\end{bmatrix}\\[10pt]
\textbf{Output:}\quad
& \begin{bmatrix} m_{\text{yeast}}\\ Y_{\text{fermenter}} \end{bmatrix}
=\mathbf{g}(\mathbf{x})
\end{aligned}