Objectives we may look for:

1) Understand the pectin extraction process -> what are the pros and cons of each?
2) What is the Co2/energy produced in each of the extraction processes?
3) How can we optimise the paramaters in each process (eg. temp.) in terms of efficiency/yield/emissions?

What to look for in each process:
1) CO2/energy emissions
2) Time taken for process to occur
3) Complexity/cost
4) Existence of models/parameters involved
5) Yield

Probably 1), 5), 2) are the most important things to look for in descending order

Overview of how pectin is produced (conventionally):

1) Peels/peel powders are put in distilled water, ph adjusted then
2) Heated up using some process (eg. ohmic heating, microwaves, ultrasound, ..)
3) Centrifuged/processed out via filtration, distillation and washing, which uses the resources water, temperature, ethanol, acid used, ..

One way we can optimise is by looking at a single parameter and trying to minimise this. But, we can do multi-objective optimisation:

For example, let is look at the yield and the Co2, by making a graph of eg. energy vs yield, where there will be a scatter plot of different datapoints of different processes. We aim to get a graph where the highest energy and yield together by choosing the datapoints that choose this correlation.

Comparison of the different extraction processes

| Parameters | Microwave | Ultrasound | Ohmic Heating | Pressure-based | Pulsed Electric field | Moderate electric field |
| ---------- | --------- | ---------- | ------------- | -------------- | --------------------- | --- |
| | | | | | | |

\begin{frame}
\frametitle{Description of the data}

We calculated approximations for the $k_1$, $k_2$ and $k_{-1}$ rates based on data collected at pH values of 4, 5 and 6, at temperatures of $50^{o}C$, $55^{o}C$, $60^{o}C$, $65^{o}C$ and $70^{o}C$

Data was based on 10g cellulose suspended in 10ml of water for 4 hours.
At 10 minute intervals, concentration measurements in $g/ml$ were reported for:

\begin{itemize}
\item Substrate
\item Enzyme
\item Enzyme complex
\item Pectin
\end{itemize}

\end{frame}

\begin{frame}
\frametitle{}

We calculated approximations for the $k_1$, $k_2$ and $k_{-1}$ rates based on data collected at pH values of 4, 5 and 6, at temperatures of $50^{o}C$, $55^{o}C$, $60^{o}C$, $65^{o}C$ and $70^{o}C$

Data was based on 10g cellulose suspended in 10ml of water for 4 hours.
At 10 minute intervals, concentration measurements in $g/ml$ were reported for:

\begin{itemize}
\item Substrate
\item Enzyme
\item Enzyme complex
\item Pectin
\end{itemize}

\end{frame}

\begin{equation*}
S+E \xleftrightharpoons[k_{-1}]{k_{1}} C \stackrel{k_2}{\longrightarrow} P+E
\end{equation*}
\begin{aligned}
& \frac{d s}{d t}=-k_1 s e+k_{-1} c \\
& \frac{d e}{d t}=-k_1 s e+\left(k_{-1}+k_2\right) c \\
& \frac{d c}{d t}=k_1 s e-\left(k_{-1}+k_2\right) c \\
& \frac{d p}{d t}=k_2 c
\end{aligned} \\
From conservation of mass for the enzyme we have
\begin{align*}
\frac{d e}{d t}+\frac{d c}{d t} \ \Rightarrow e(t)+c(t)=e_0 \Rightarrow e(t)=e_0-c
\end{align*}
At $t=0 \Rightarrow c(0)=0, e(0)=e_0$
\end{frame}

\begin{frame}{Frame Title}
So we end up with just
\begin{align*}
& \frac{d s}{d t}=-k_1 s (e_0-c)+k_{-1} c \\
& \frac{d c}{d t}=k_1 s (e_0-c)-\left(k_{-1}+k_2\right) c \\
& \frac{d p}{d t}=k_2 c
\end{align*}
Which we numerically solved to obtain
\end{frame}

$$
t=\frac{1}{k_1 e_0} t^*, \quad s=s_0 s^*, \quad c=e_0 c^*, \quad p=e_0 p^*
$$
Substituting gives
\begin{aligned}
k_1 e_0 s_0 \frac{d s^*}{d t^*} & =-k_1 s_0 s^*\left(e_0-e_0 c^*\right)+k_{-1} e_0 c^* \\
\frac{d s^*}{d t^*} & =-s^*\left(1-c^*\right)+\frac{k_{-1}}{k_1 s_0} c^* \\
\frac{d s^*}{d t^*} & =-s^*\left(1-c^*\right)+\lambda c^*,
\end{aligned}
$$
\text{where} $\lambda=k_{-1} /\left(k_1 s_0\right)$.

Similarly
$$
\epsilon \frac{d c^*}{d t^*}=s^*\left(1-c^*\right)-\kappa c^*,
$$
where $\epsilon=e_0 / s_0$ and $\kappa=\left(k_{-1}+k_2\right) /\left(k_1 s_0\right)$.
$$
\frac{d p^*}{d t^*}=\delta c^*,
$$
where $\delta = \frac{k_2}{k_1 e_0}$

Overleaf link: [https://www.overleaf.com/7445446968xcqvtvnzwdpm](https://www.overleaf.com/7445446968xcqvtvnzwdpm)

\begin{frame}{References}
\bibliographystyle{alpha}
\begin{thebibliography}{99}
\bibitem{quantum_cradle} T. Kinoshita, T. Wenger, and D.S. Weiss, Nature {\bf 440}, 900
\end{thebibliography}
\end{frame}

\begin{frame}{References}
\bibliographystyle{alpha}
\begin{thebibliography}{99}
\bibitem{}Maran, J.P., Sivakumar, V., Thirugnanasambandham, K. and Sridhar, R., 2013. Optimization of microwave assisted extraction of pectin from orange peel. Carbohydrate polymers, 97(2), pp.703-709.
\bibitem{}Wang, S., Chen, F., Wu, J., Wang, Z., Liao, X. and Hu, X., 2007. Optimization of pectin extraction assisted by microwave from apple pomace using response surface methodology. Journal of food engineering, 78(2), pp.693-700.

\bibitem{}Phatak, L., Chang, K.C. and Brown, G., 1988. Isolation and characterization of pectin in sugar‐beet pulp. Journal of Food Science, 53(3), pp.830-833.

\bibitem{}Chandel, V., Biswas, D., Roy, S., Vaidya, D., Verma, A. and Gupta, A., 2022. Current advancements in pectin: Extraction, properties and multifunctional applications. Foods, 11(17), p.2683.
\end{thebibliography}
\end{frame}

\begin{frame}{References}
\bibliographystyle{alpha}
\begin{thebibliography}{99}

\bibitem{}Ma, S. and Wang, Z.H., 2013. Pulsed electric field-assisted modification of pectin from sugar beet pulp. Carbohydrate polymers, 92(2), pp.1700-1704.

\bibitem{}Naghshineh, M., Olsen, K. and Georgiou, C.A., 2013. Sustainable production of pectin from lime peel by high hydrostatic pressure treatment. Food chemistry, 136(2), pp.472-478.

\bibitem{}Karbuz, P. and Tugrul, N., 2021. Microwave and ultrasound assisted extraction of pectin from various fruits peel. Journal of food science and technology, 58(2), pp.641-650.
\end{thebibliography}
\end{frame}