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.