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

Todd Hurst; Volker Rehbock

doi:10.3934/jimo.2021027

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2022, Volume 18, Issue 2: 1439-1451. Doi: 10.3934/jimo.2021027

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

Todd Hurst and
Volker Rehbock^,

School of Electrical Engineering, Computing, and Mathematical Sciences, Curtin University, Kent Street, Bentley, Perth, Western Australia, 6102

^* Corresponding author: v.rehbock@curtin.edu.au
^* Corresponding author: v.rehbock@curtin.edu.au

Received: May 31, 2020

Revised: October 31, 2020

Early access: February 2021

Published: March 2022

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

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Abstract

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.

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

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Figure 1. $ f_{in} $

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Figure 2. $ f_{out} $

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Figure 3. $ L $

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Figure 4. $ \bar{I} $

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Figure 5. $ s $

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Figure 6. $ x_l $

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Figure 7. $ \eta_{L} $

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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

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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

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