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Optimizing micro-algae production in a raceway pond with variable depth

We acknowledge the support of Todd Hurst through the Australian Government Research Training Program (RTP) Scholarship

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  • We present a modified model of algae growth in a raceway pond with the additional feature of variable pond depth. This requires an additional state variable to model depth as well as additional control to allow for variable outflow. We apply numerical optimal control methods to this model and show that the lipid yield of the process can be increased by 67% compared to that obtained with a fixed pond depth.

    Mathematics Subject Classification: Primary: 49N90, 37N25; Secondary: 92-08.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 1.  $ f_{in} $

    Figure 2.  $ f_{out} $

    Figure 3.  $ L $

    Figure 4.  $ \bar{I} $

    Figure 5.  $ s $

    Figure 6.  $ x_l $

    Figure 7.  $ \eta_{L} $

    Table 1.  State Variable, Control Variable, System Parameter and Function Definitions

    Variables Definition Units
    $ s $ Nitrogen concentration gN $ \text{m}^{-3} $
    $ q_n $ Nitrogen quota gN $ \text{(gC)}^{-1} $
    $ x $ Carbon biomass concentration gC $ \text{m}^{-3} $
    $ x_l $ Lipid carbon concentration gC $ \text{m}^{-3} $
    $ x_f $ Functional carbon concentration gC $ \text{m}^{-3} $
    $ L $ Pond depth $ \text{m} $
    $ \bar{I} $ Average light intensity μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $
    $ C_{hl} $ Chlorophyll concentration g$ C_{hl} $ $ \text{m}^{-3} $
    $ I_0 $ Incident light intensity μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $
    $ T $ Raceway temperature
    $ \phi_{T} $ Temperature factor affecting growth kinetics
    $ \lambda $ Optical depth
    $ \mu $ Growth rate $ \text{h}^{-1} $
    $ \rho $ Nitrogen uptake rate gN $ \text{(gC h)}^{-1} $
    $ \theta_N $ Chl:N ratio g$ C_{hl} $ $ \text{(gN)}^{-1} $
    $ \xi $ Attenuation factor $ \text{m}^{-1} $
    $ R $ Overall respiration rate $ \text{h}^{-1} $
    $ f_{in} $ Feeding flow rate $ \text{m}^{3} $ $ \text{h}^{-1} $
    $ f_{out} $ Extraction flow rate $ \text{m}^{3} $ $ \text{h}^{-1} $
    $ \eta_{L} $ Efficiency of light absorption
    $ t_f $ final time point h
     | Show Table
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    Table 2.  Parameter Definition and Values

    Parameters Definition Units Value
    $ \alpha $ Protein synthesis coefficient gC$ \text{(gN)}^{-1} $ 3.0
    $ \beta $ Fatty acid synthesis coefficient gC$ \text{(gN)}^{-1} $ 3.80
    $ \epsilon_I $ Dissociation light constant μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ 50
    $ \varphi $ Biosynthesis cost coefficient gC$ \text{(gN)}^{-1} $ 1.30
    $ \gamma $ Fatty acid mobilization coefficient gC$ \text{(gN)}^{-1} $ 2.90
    $ v $ Reduction factor of nitrogen uptake during night 0.19
    $ \tilde{\mu} $ Theoretical maximum specific growth rate $ \text{h}^{-1} $ $ 8.7916\times10^{-2} $
    $ \bar{\rho} $ Maximum uptake rate gC$ \text{(gN h)}^{-1} $ $ 4.16\times10^{-3} $
    $ a $ Light attenuation due to chlorophyll $ \text{m}^{2} $ $ (\text{g } C_{hl})^{-1} $ 2.0
    $ b $ Light attenuation due to background turbidity $ \text{m}^{-1} $ 0.087
    $ g_1 $ Coefficient (7) gN $ (\text{g} C_{hl})^{-1} $ 16.74
    $ g_2 $ Coefficient (7) gN $ (\text{g} C_{hl} \text{ }^\circ \text{C})^{-1} $ 0.39
    $ g_3 $ Coefficient (7) gN (g$ C_{hl} $μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1})^{-1} $ $ 1.4\times10^{-3} $
    $ g_4 $ Coefficient (7) $ \text{ }^\circ \text{C}^{-1} $ 0.0015
    $ K_s $ Nitrogen saturation constant gN $ \text{m}^{-3} $ 0.018
    $ K_{sI} $ Light saturation constant μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ $ 1.5\times 10^{2} $
    $ m $ Hill coefficient 3.0
    $ Q_l $ Saturation cell quota gN$ \text{(gC)}^{-1} $ 0.20
    $ Q_0 $ Minimum nitrogen cell quota gN$ \text{(gC)}^{-1} $ 0.05
    $ r_0 $ Maintenance respiration rate $ \text{h}^{-1} $ $ 4.16\times10^{-4} $
    $ T_{\min} $ Lower temperature for micro-algae growth $ \text{ }^\circ \text{C} $ -0.20
    $ T_{\max} $ Upper temperature for micro-algae growth $ \text{ }^\circ \text{C} $ 33.30
    $ T_{\text{opt}} $ Temperature at which growth rate is maximal $ \text{ }^\circ \text{C} $ 26.70
    $ s_{in} $ Influent nitrogen concentration gN $ \text{m}^{-3} $ 50.0
    $ T_a $ Coefficient (2) $ \text{ }^\circ \text{C} $ -5.75
    $ T_b $ Coefficient (2) $ \text{ }^\circ \text{C} $ 20.75
    $ t_a $ time point h 3.733
    $ t_b $ time point h 4.9
    $ t_c $ time point h 19.1
    $ t_d $ time point h 20.267
    $ I_{0a} $ Coefficient (3) μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ -3.890408560
    $ I_{0b} $ Coefficient (3) μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ -52.13043162
    $ I_{0c} $ Coefficient (3) μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ 141.6278134
    $ I_{0d} $ Coefficient (3) μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ 841.1792340
    $ I_{0e} $ Coefficient (3) μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ 358.8207660
    $ I_{0f} $ Coefficient (3) μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ 3.890411835
    $ I_{0g} $ Coefficient (3) μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ -52.13042142
    $ I_{0h} $ Coefficient (3) μmol photons $ \text{m}^{-2} $ $ \text{s}^{-1} $ -141.6278049
     | Show Table
    DownLoad: CSV
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