2.1 Computational experiment setup
In order to illustrate the proposed model construction framework and to
test the efficiency of the hybrid model in process predictive modelling
and optimisation, computational experiments were used in this work so
that the best process performance can be identified as the benchmark. A
microalgal lutein production process was selected as the specific case
study, as an algal photo-production process is biologically more complex
than a conventional fermentation process. Algal biomass growth and
lutein synthesis are mainly affected by light intensity and nitrate
concentration 21. A complex kinetic model designed in
our previous work was used to generate computational experimental data
for different purposes and is presented in Eq. (1a)-(1d)22. This model can well simulate effects of light
intensity, light attenuation, and nitrate supply on biomass growth and
lutein production. However, given its complex model structure, its
application in process optimisation and bioreactor design is limited,
and identifying its model structure is time consuming.
\begin{equation}
\frac{dc_{X}}{\text{dt}}=\frac{u_{m}}{20}\bullet\left(\frac{I_{0}}{I_{0}+k_{s}+\frac{I_{0}^{2}}{k_{i}}}+\sum_{n=1}^{9}\frac{2\bullet I_{n}}{I_{n}+k_{s}+\frac{I_{n}^{2}}{k_{i}}}+\frac{I_{10}}{I_{10}+k_{s}+\frac{I_{10}^{2}}{k_{i}}}\right)\bullet\frac{c_{N}}{c_{N}+K_{N}}\ \bullet c_{X}-u_{d}\bullet c_{X}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1a)\nonumber \\
\end{equation}\begin{equation}
\frac{dc_{N}}{\text{dt}}=-Y_{N/X}\bullet\frac{u_{m}}{20}\bullet\left(\frac{I_{0}}{I_{0}+k_{s}+\frac{I_{0}^{2}}{k_{i}}}+\sum_{n=1}^{9}\frac{2\bullet I_{n}}{I_{n}+k_{s}+\frac{I_{n}^{2}}{k_{i}}}+\frac{I_{10}}{I_{10}+k_{s}+\frac{I_{10}^{2}}{k_{i}}}\right)\bullet\frac{c_{N}}{c_{N}+K_{N}}\ \bullet c_{X}+F_{\text{in}}\bullet c_{N,in}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(1b\right)\nonumber \\
\end{equation}\begin{equation}
\frac{dc_{L}}{\text{dt}}=\frac{k_{m}}{20}\bullet\left(\frac{I_{0}}{I_{0}+k_{\text{sL}}+\frac{I_{0}^{2}}{k_{\text{iL}}}}+\sum_{n=1}^{9}\frac{2\bullet I_{n}}{I_{n}+k_{\text{sL}}+\frac{I_{n}^{2}}{k_{\text{iL}}}}+\frac{I_{10}}{I_{10}+k_{\text{sL}}+\frac{I_{10}^{2}}{k_{\text{iL}}}}\right)\ \ \bullet\frac{c_{N}}{c_{N}+K_{\text{NL}}}\ \bullet c_{X}-k_{d}\bullet c_{L}\bullet c_{X}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(1c\right)\nonumber \\
\end{equation}\begin{equation}
I(l)=I_{0}\bullet\left(e^{-\left(\tau\bullet c_{X}+K_{a}\right)\bullet l}+e^{-\left(\tau\bullet c_{X}+K_{a}\right)\bullet\left(z-l\right)}\right)\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left(1d\right)\nonumber \\
\end{equation}where \(c_{X}\), \(c_{N}\), and \(c_{L}\) are the concentrations of
biomass, nitrate, and lutein, respectively. \(F_{\text{in}}\) and\(c_{N,in}\) are nitrate inflow rate and concentration, respectively.\(l\) is the distance from light source, \(z\) is width of the reactor
(0.084 m). \(I_{n}\) is the local light intensity at a distance of\(\frac{n\bullet z}{10}\) (\(n=1,\ldots,10)\) m away from the
incident light surface area, while \(I_{0}\) is the incident light
intensity. Parameter values and their physical meanings can be found in22.
This complex model served to act as the “true” experimental system.
Initially, three computational experiments were conducted under a broad
spectrum of operating conditions from nitrate-limiting conditions and
photo-limitation to nitrate-excessive conditions and photo-inhibition.
Data generated in these experiments was then used to construct the
hybrid model. Once constructed, the hybrid model was exploited to
predict and optimise a number of fed-batch processes under different
conditions, and computational experimental verification using this
complex model was executed to verify accuracy of the hybrid model.
Detailed operating conditions of the computational experiments are
listed in Table 1.
Table 1: Operating conditions of all the computational experiments: Exp.
1-3 (batch processes), used for parameter estimation; Offline (fed-batch
processes): initial conditions of the four offline optimisation
processes; Online (fed-batch process): initial conditions of the model
self-calibration experiment.