## 4 Results and Discussion
### Model Validation  
The equations shown in the \"Mathematical modelling\" section were implemented in MatLab(r) for the \"Flat-plate geometry model\" and the \"Annular section geometry model\". The kinetic parameters \(\mu_{max}\) and \(l_{k}\) were integrated in both \"Flat-plate geometry model\" and the \"Annular section geometry model\", while the adjustable parameters \(k_{d}\) and \(k_{a}\) were instead obtained by fitting model results with experimental concentration data. These cultures were conducted under a photon flux density of 210 \(\mu\)E s\({}^{-1}\)m\({}^{-2}\) at 25 \({}^{\circ}\)C. The data and parameters employed and obtained from the modelling are shown in Table 2.  
Figure 5: Specific growth rate versus different irradiation intensities of the microcala _Nannochloropsis__galatana_ in the quasi-isoactinic photobioreactor: black circles, experimental data; line, Equation (10).  
The two adjustable parameters were first changed in the flat geometry model, obtaining the best fit for both reactors' thickness with \(k_{d}=0.0046\) h\({}^{-1}\) and \(k_{a}=0.38\) m\({}^{2}\) g\({}^{-1}\). When using these two parameters in the annular geometry model, the best fit is obtained with the same value of \(k_{d}=0.0046\) h\({}^{-1}\), and this is sound as the microalgae strain did not change, while the value of \(k_{a}\) had to be set at 0.20 m\({}^{2}\) g\({}^{-1}\), meaning that, in the case of the annular reactor, the light attenuation factor is a little smaller than in the flat geometry. Considering that the cellular size (cellular diameter) does not significantly vary in the three different cultivations (data not shown), this variation may be due to the cells creating agglomerates differently in the three cultivations. In fact, we observed that the cells in the annular reactor showed bigger agglomerates than in the other two cultivations. A bigger agglomerate reduces the transversal section of light capture (because some cells are not exposed to the light) and therefore reduces the attenuation constant \(k_{a}\). In Figure 6 the growth curves obtained from the model are compared with the relevant experimental data.  
As can be seen, the modelled growth curves show a good fit with the experimental data, meaning that the model can predict the growth in flat-plate reactors of differ  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & & **Value** & & **Unit** \\ \hline  & Flat type 15 & Flat type 30 & Annular & \\ L & 15 & 30 & 35.5 & mm \\ \(\mu_{max}\) & 0.0256 & 0.0256 & 0.0256 & h\({}^{-1}\) \\ \(I_{K}\) & 15.28 & 15.28 & 15.28 & \(\mu\)E s\({}^{-1}\)m\({}^{-2}\) \\ \(k_{d}\) & 0.0046 & 0.0046 & 0.0046 & h\({}^{-1}\) \\ \(k_{a}\) & 0.38 & 0.38 & 0.20 & m\({}^{2}\) g\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Data and parameters employed for the model.  
Figure 6: Comparison between model prediction and experimental data of _Nammochloropsis gaditana_ growth curves in batch photobioreactors: “orange” Flat-plate 15 mm, “green” Flat-plate 30 mm, “blue” Annular photobioreactor. Dotted lines, model predictions; symbols, experimental data.  
ent depths and in an annular geometry, once the attenuation constant parameter \(k_{a}\) has been adjusted.  
The data obtained through this model may be of interest in order to create a link between the metabolic response of _Nannochloroposis gaditana_ and the action of inputs such as the incident light intensity, the reactor thickness and the geometry. Furthermore, this model may be easily implemented on other microalgae, after experimentally measuring the kinetic parameters.