TEX BLOCKS

m {\text{yeast}}=\left(\frac{m {\text{sample,dry}}}{v {\text{sample}}}\right)\\,v {\text{final}} y {\text{fermenter}}=\frac{m {\text{yeast}}}{m {\text{molasses}}} \eta=\begin{cases} 1, & \text{if } t \ge 30^\circ\text{c}\\\\ 0, & \text{if } t < 30^\circ\text{c} \end{cases} c=\frac{m}{v}\ \text{\[g/l]} \begin{aligned} m {\text{yeast}} &= \left(\frac{m {\text{sample,dry}}}{v {\text{sample}}}\right) v {\text{final}}\\\\\[4pt] y {\text{fermenter}} &= \frac{m {\text{yeast}}}{m {\text{molasses}}}\\\\\[6pt] \eta(t) &= \begin{cases} 1, & \text{if } 30^\circ\text{c} \le t \le 35^\circ\text{c}\\\\ 0 8, & \text{if } 25^\circ\text{c} \le t < 30^\circ\text{c}\\\\ 0, & \text{if } t < 25^\circ\text{c} \end{cases}\\\\\[6pt] c {\text{yeast}} &= \frac{m {\text{sample,dry}}}{v {\text{sample}}}\ \text{\[g/l]} \end{aligned} \begin{aligned} \mathbf{x} &= \begin{bmatrix} m {\text{sample,dry}} \\\ v {\text{sample}} \\\ v {\text{final}} \\\ m {\text{molasses}} \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{yeast}} &= f 1(\mathbf{x}), \qquad y {\text{fermenter}} = f 2(\mathbf{x}) \end{aligned} \begin{aligned} m {\text{yeast}} &= \left(\frac{m {\text{sample,dry}}}{v {\text{sample}}}\right)v {\text{final}}\\\\\[4pt] \ln(m {\text{yeast}}) &= \ln(m {\text{sample,dry}}) \ln(v {\text{sample}}) + \ln(v {\text{final}})\\\\\[6pt] \left(\frac{\sigma {m {\text{yeast}}}}{m {\text{yeast}}}\right)^2 &= \left(\frac{\sigma {m {\text{sample,dry}}}}{m {\text{sample,dry}}}\right)^2 \+ \left(\frac{\sigma {v {\text{sample}}}}{v {\text{sample}}}\right)^2 \+ \left(\frac{\sigma {v {\text{final}}}}{v {\text{final}}}\right)^2\\\\\[6pt] y {\text{fermenter}} &= \frac{m {\text{yeast}}}{m {\text{molasses}}} \end{aligned} \begin{aligned} &\textbf{given }\quad m {\text{sample,dry}},\ v {\text{sample}},\ v {\text{final}},\ m {\text{molasses}},\ t,\ t,\ t f\\\\\[4pt] &\textbf{concentration }\quad c {\text{yeast}}=\frac{m {\text{sample,dry}}}{v {\text{sample}}}\ \text{\[g/l]}\\\\\[6pt] &\textbf{total yeast }\quad m {\text{yeast}}=c {\text{yeast}}\\,v {\text{final}}\\\\\[6pt] &\textbf{fermenter yield }\quad y {\text{fermenter}}=\frac{m {\text{yeast}}}{m {\text{molasses}}}\\\\\[10pt] &\textbf{temperature factor }\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{a simple growth profile }\quad m {\text{yeast}}(t)= \frac{m {\max}}{1+\exp\\!\left( k\\,\eta(t)\\,(t t 0)\right)}\\\\\[6pt] &\textbf{net production (from 0 to }t f\textbf{) }\quad \delta m {\text{yeast}}=m {\text{yeast}}(t f) m {\text{yeast}}(0)\\\\\[10pt] &\textbf{mass balance check }\quad \epsilon=\frac{\delta m {\text{yeast}}}{m {\text{molasses}}} y {\text{target}}\\\\\[6pt] &\textbf{decision }\quad \text{pass if }|\epsilon|\le 0 02,\ \text{fail otherwise}\\\\\[12pt] &\textbf{uncertainty propagation (relative) }\quad \left(\frac{\sigma {m {\text{yeast}}}}{m {\text{yeast}}}\right)^2= \left(\frac{\sigma {m {\text{sample,dry}}}}{m {\text{sample,dry}}}\right)^2+ \left(\frac{\sigma {v {\text{sample}}}}{v {\text{sample}}}\right)^2+ \left(\frac{\sigma {v {\text{final}}}}{v {\text{final}}}\right)^2\\\\\[8pt] &\textbf{and for yield }\quad \left(\frac{\sigma {y {\text{fermenter}}}}{y {\text{fermenter}}}\right)^2= \left(\frac{\sigma {m {\text{yeast}}}}{m {\text{yeast}}}\right)^2+ \left(\frac{\sigma {m {\text{molasses}}}}{m {\text{molasses}}}\right)^2 \end{aligned} \begin{aligned} \textbf{inputs }\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{unit conversions }\quad & v {\text{sample}}=25\times 10^{ 3}\ \text{l},\qquad m {\text{molasses}}=9 5\times 10^{6}\ \text{g}\\\\\[8pt] \textbf{step 1 (conc ) }\quad & c {\text{yeast}}=\frac{1 25}{25\times 10^{ 3}}=50\ \text{g/l}\\\\\[6pt] \textbf{step 2 (total) }\quad & m {\text{yeast}}=50\times 18{,}000=9 0\times 10^{5}\ \text{g}\\\\\[6pt] \textbf{step 3 (yield) }\quad & y {\text{fermenter}}=\frac{9 0\times 10^{5}}{9 5\times 10^{6}}=0 0947\\\\\[10pt] \textbf{constraint checks }\quad & 0\<y {\text{fermenter}}<0 2,\qquad v {\text{final}}>0,\qquad v {\text{sample}}>0\\\\\[10pt] \textbf{vector form }\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{outputs }\quad & \begin{bmatrix} m {\text{yeast}}\\\ y {\text{fermenter}} \end{bmatrix} \=\mathbf{g}(\mathbf{x}) \end{aligned}