# Stability Assessment of Microalgal Photobioreactors for Carbon Dioxide Capture under Dilution Rate Constraints

## Abstract

**:**

## 1. Introduction

- The uniqueness of the equilibrium point of the closed-loop nonlinear system is proved in the presence of dilution rate saturation constraints and a proportional–integral (PI) tracking controller.
- Bounds on the saturation levels of the dilution rate are derived such that a reference microalgal concentration can be achieved. These bounds have never been reported in the literature so far and hence clearly show the novelty of the current research. Moreover, numerical simulations reveal that such bounds are tight.

## 2. Carbon Dioxide Fixation by Microalgae PBRs

#### 2.1. Underlying Chemical Reactions

#### 2.2. Mathematical Model

#### 2.3. Carbon Dioxide Fixation Capability

## 3. PBR Stability under Dilution Rate Constraint

#### 3.1. Control Problem

**Control problem:**Given a fixed CO${}_{2}$ flow rate, a given reference ${X}^{r}>0$ for the microalgal concentration and a saturated interval $[\u03f5,\overline{D}]$ for the PBR dilution rate, find conditions for the achievable tracking and stability of the PBR in the presence of a PI controller for the microalgal concentration.

#### 3.2. Controller Design

#### 3.3. Closed-Loop System Stability Analysis

**Theorem 1.**

**Proof.**

**Remark 1.**

## 4. Numerical Example

#### 4.1. System Responses without Control Actions

#### 4.2. Controlled PBR without Input Constraint

#### 4.3. Controlled PBR without Model Mismatch But with Input Constraint

#### 4.4. Controlled PBR under Model Mismatch and Input Constraint

#### 4.5. Controlled PBR under Model Mismatch, Input Constraint, and Light Disturbance

#### 4.6. Proportionality of Tracking Time to the Reference Value

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 4.**Achieved reference tracking under model mismatch and PI control, without input saturation.

**Figure 15.**TIC concentration under input saturation, model mismatch, and sinusoidal light disturbance.

**Figure 16.**Reference tracking under input saturation, random model mismatch and sinusoidal light disturbance.

**Figure 17.**TIC concentration under input saturation, random model mismatch and sinusoidal light disturbance.

**Figure 18.**Dilution rate under input saturation, random model mismatch and sinusoidal light disturbance.

Notation | Meaning | Unit |
---|---|---|

X | biomass concentration | ${10}^{9}$ cells/L |

S | TIC concentration | mol/L |

D | dilution rate | h${}^{-1}$ |

$\mu $ | specific biomass growth rate | h${}^{-1}$ |

${\mu}_{max}$ | maximum specific growth rate | h${}^{-1}$ |

${Y}_{X/S}$ | conversion yield of carbon into biomass | ${10}^{9}$ cells/mol |

$\gamma $ | volumetric mass transfer coefficient of CO${}_{2}$ | h${}^{-1}$ |

${\left[{\mathrm{CO}}_{2}\right]}^{*}$ | CO${}_{2}$ concentration in equilibrium with the outlet gas phase | mol/L |

$\left[{\mathrm{CO}}_{2}\right]$ | dissolved CO${}_{2}$ concentration in the liquid | mol/L |

I | light intensity accessible per cell | $\mathsf{\mu}$E/(${10}^{9}$ cells)/s |

${\alpha}_{I}$ | light half-saturation constant | $\mathsf{\mu}$E/(${10}^{9}$ cells)/s |

${\alpha}_{S}$ | TIC half-saturation constant | mmol/(${10}^{9}$ cells) |

${I}_{in}$ | incident light intensity | $\mathsf{\mu}$E/(${10}^{9}$ cells)/s |

${I}_{out}$ | outgoing light intensities | $\mathsf{\mu}$E/m${}^{2}$/s |

${A}_{r}$ | PBR illuminated area | m${}^{2}$ |

V | effective PBR volume | L |

${c}_{1}$ | dimensionless constant | |

${c}_{2}$ | dimensionless constant | |

$pH$ | pH of the aqueous environment in the PBR | |

${k}_{1}$ | dissociation constant of CO${}_{2}$ and bicarbonate | mol/L |

${k}_{2}$ | dissociation constant of CO${}_{2}$ and bicarbonate | mol/L |

${P}_{C{O}_{2}}$ | partial CO${}_{2}$ pressure in the gas phase | atm |

H | Henry constant at 25 #xB0;C | atm L/mol |

F | input culture medium flow rate | L/h |

${r}_{C{O}_{2}}$ | CO${}_{2}$ fixation rate | g/L/day |

${\eta}_{C{O}_{2}}$ | CO${}_{2}$ fixation efficiency | % |

P | biomass productivity | g/L/day |

C | carbon content | w/w |

${M}_{C{O}_{2}}$ | molecular weight of CO${}_{2}$ | g/mol |

${M}_{C}$ | molecular weight of carbon | g/mol |

w | CO${}_{2}$ flow rate | g/day |

Parameter | Value | Unit |
---|---|---|

${I}_{in}$ | 90 | $\mathsf{\mu}$E/(${10}^{9}$ cells)/s |

${P}_{C{O}_{2}}$ | 0.05 | atm |

V | 9.6 | L |

${A}_{r}$ | 0.31 | m${}^{2}$ |

${k}_{1}$ | ${10}^{-6.35}$ | mol/L |

${k}_{2}$ | ${10}^{-10.3}$ | mol/L |

H | 29.41 | atm L/mol |

${Y}_{X/S}$ | 1211 | |

$\gamma $ | 1.4 | h${}^{-1}$ |

${\mu}_{max}$ | 1.07 | h${}^{-1}$ |

${c}_{1}$ | 0.493 | |

${c}_{2}$ | −0.925 | |

${\alpha}_{I}$ | 0.08 | $\mathsf{\mu}$E/(${10}^{9}$ cells)/s |

${\alpha}_{S}$ | 3.8 | mmol/(${10}^{9}$ cells) |

F | 0.12 | L/h |

$pH$ | 6.5 |

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**MDPI and ACS Style**

Nguyen, D.H.
Stability Assessment of Microalgal Photobioreactors for Carbon Dioxide Capture under Dilution Rate Constraints. *Sustainability* **2023**, *15*, 15269.
https://doi.org/10.3390/su152115269

**AMA Style**

Nguyen DH.
Stability Assessment of Microalgal Photobioreactors for Carbon Dioxide Capture under Dilution Rate Constraints. *Sustainability*. 2023; 15(21):15269.
https://doi.org/10.3390/su152115269

**Chicago/Turabian Style**

Nguyen, Dinh Hoa.
2023. "Stability Assessment of Microalgal Photobioreactors for Carbon Dioxide Capture under Dilution Rate Constraints" *Sustainability* 15, no. 21: 15269.
https://doi.org/10.3390/su152115269